Introduction to Algebra - Buffalo State Collegestern.buffalostate.edu/newmath/SMSG/45Introduction To Algebra Pa… · Introduction to Algebra Teacher's Commentary, Part I REVISED

School Mathematics Study Group

Introduction to Algebra

Unit 45

Introduction to Algebra Teacher's Commentary, Part I

REVISED EDITION

Prepared under the supervision of

a Panel consisting of:

V. H. Haag

Mildred KeifFer

Oscar Schaaf

M. A. Sobel

Marie Wilcox

A. B. Willcox

Franklin and Marshall College

Cincinnati Board of Education

South Eugene High School,

Eugene, Oregon

Montclair State College,

Upper Montclair, New Jersey

Thomas Carr Howe High School,

Indianapolis, Indiana

Amherst College

New Haven and London, Yale University Press, 1963

Copyright @ 1962 by The Board of Trustees of the Leland Stanford Junior University. Printed in the United States of America.

All rights reserved. This book may not be reproduced in whole or in part, in any form, without written permission from the publishers.

Financial support for the School Mathematics Study Group has been provided by the National Science Foundation.

Chapter

. . . . . . . . . . . . . 1 . SETS AND THE NUMBER LINE

1.1 . Sets . . . . . . . . . . . . . . . . . . . 1.2 . The Number Line . . . . . . . . . . . . .

. . . . . . Answers t o Review Problem Set

Suggested Test Items . . . . . * . . . . . . . . . . Answers to Suggested Test Items

. . . . . . . . 2 NUMERALS. SENTENCES. AND VARIABLES

. . . . Numbers and Their Names

Sentences . . . . . . . . . . . A Property of the Number One . . Some Proper t ies of Addition and

Mult ipl icat ion . . . . . e r n .

. . . The Dis t r ibu t ive Property

Variables . . . . . . . . . . . Answers t o Review Problem Se t

Suggested Test Items . . . . . . Answers t o Suggested Test Ittsms

3 . OPEN SENTENCES AND TRUTH SETS . * . . . 3.1 . Open Sentences . . . a . e . . ,

3.2 . Truth Sets of Open Sentences . . 3.3 . Graphs of Truth Se ts . . . . * . 3 4 . Compound Open Sentences and

Their Graphs . . . . . e m . .

Answers t o Review Problem Set . Suggested Test Items . a . . Answers t o Suggested Test Items

4 . PROPERTIES OF OPERATIONS . . . . . . . . . . . . . . . . . . . . 4.1 Iden t i t y Elements

4.2 . Closure . . . . . . . . . . . . . . . . . 4.3 . Commutative and Associative Proper t ies . . . . . of Addition and Mult ip l icat ion

. . . . . . . . . . . 4.4 Dis t r ibu t ive Property

. . . . . . Answers t o Review Problem Se t . . . . . . . . . . . Suggested Test Items

Answers t o Suggested Test Items . . . . .

Page

1

2 8

15 17 20

23

23

27

29

31 42 46 52

55 57

61 61 65 69

71

79 82 84

87 89

94

97 101

109 113 3-15

CHAPTER PAGE

. . . . . . . . . OPEN SENTENCES AND WORD SENTENCES 119 . . . . . . . 5.1 . Open Phrases t o Word Phrases 120 . . . . . . . 5.2 . Word Phrases t o Open Phrases 123

. . . . . 5.3 . Open Sentences to Word Sentences 127

. . . . . 5.4 . Word Sentences to Open Sentences 130 . . . . . . . . . . . . 5.5 . OtherTranslat ions 134

. . . . . . . Answers t o Review Problem Set 138 . . . . . . . . . . . Suggested T e s t Items 142

. . . . . . Answers t o Suggested Test Items 144

. . . . . . . . . . . . . . . CHALLENGE PROBLEMS (hswers ) 147

. . . . . . . . . . . . . . . . . . 6 THE REAL NUMBERS 159 . . . . . . . . . . . . . . 6.1 The Real Numbers 160

. . . . . . . 6.2 . Order on the Real Number Line 166 . . . . . . . . . . . . . . . . . . 6.3 Opposites 168

. . . . . . . . . . . . . . 6.4 . Absolute Value 175

. . . . . . . Answers t o Review Problem Set 177

. . . . . . . . . . . Suggested Test Items 179

. . . . . . Answers t o Suggested Test Items 181

. . . . . . . . . . 7 . ADDITION OF REAL NUMBERS

7.1 . Using the Real Numbers i n Addition . . . . . 7.2 . Addition and the Number Line

7.3 . Addition Property of Zero; Addition . . . . . . . Property of Opposites

. . . . . . . 7.4 . Properties of Addition

. . . . 7.5 . Addition Property of Equality

. . . . 7.6 . Truth Sets of open Sentences

. . . . . . . . . . 7.7 . Additive Inverse

Answers t o Review Problem Set . . . . . . . . . . . . Suggested Test Items

Answers t o Suggested Test Items . . .

CHAPTER PAGE

. . . . . . . . . . MULTIPLICATION OF REAL NUMBERS

8.1 . Products . . . . . . . . . . . . . . . . . . . . . . . . 8.2 . proper t ies of ~ u l t i p l i c a t i o n

. . . 8.3 . Using the Mult ip l icat ion Proper t ies . . . . . . . . . . . 8.4 ~ u l t i p l i c a t i v e Inverse

8.5 . Mult ipl icat ion Property of Equali ty . . . . . . . . . . 8.6 . ~ o l u t i o n s of open sentences

8.7 . Products and the Number Zero . . . . . . . Answers t o Review Problem Se t . . . . . .

. . . . . . . . . . . Suggested Test Items

Answers t o Suggested Test Items . . . . . PROPERTIES OF ORDER . . . . . . . . . . . . . . .

9.1 . The Order Relat ion f o r Real Numbers . . . 9.2 . Addition Property of Order . . . . . . . .

. . . . . 9.3 . Mult ipl icat ion Property of Order

Answers t o Review Problem Se t . . . . . . . . . . . . . . . . . Suggested Test Items

. . . . . Answers t o Suggested Test Items

. . . . . SUBTRACTION AND DIVISION OF REAL NUMBERS . . . . . . . . . 10.1 The Meaning of Subtraction . . . . . . . . 10.2 . Proper t ies of Subtraction

. . . . . . 10.3 Finding Distance by Subtraction . . . . . . . . . . 10.4 The Meaning of Division

. . . . . . . . . . . . . . . . 10.5 Common Names

10.6 . Fractions . . . . . . . . . . . . . . . . . . . . . . Answers t o Review Problem Se t

Suggested Test Items . . . . . . . . . . . . . . . . Answers t o Suggested Test Items

ANSWERS TO CHALLENGE PROBLEMS . . . . . . . . . . . . . . 332

PREFACE TO TEACHERS

This t e x t has been w r i t t e n f o r t h e n i n t h grade s t u d e n t whose mathematical t a l e n t i s underdeveloped. The s u b j e c t mat ter presented i s e s s e n t i a l l y t h a t which appears i n t h e School Mathematics Study Group t e x t : F i r s t Course - i n Algebra. Th i s is

p a r t of the body of mathematics which members of t h e Study Group bel ieve is important f o r a l l we l l educated c i t i z e n s i n o u r s o c i e t y . I t i s a l s o t h e mathematics which Is important f o r t h e pre-col lege s t u d e n t a s he prepares f o r advanced work i n t h e f i e l d of mathematics and r e l a t e d s u b j e c t s .

It i s the hope of t h e panel t h a t t h i s m a t e r i a l w i l l s e r v e t o awaken the i n t e r e s t of a l a r g e group of s t u d e n t s who have mathematical a b i l i t y which has no t y e t been recognized. It is hoped a l s o t h a t t h i s t e x t w i l l c o n t r i b u t e t o t h e understanding of fundamental concepts f o r those s t u d e n t s whose p rogress i n mathematics has been blocked o r hampered through r o t e l e a r n i n g o r inappropr ia te curr iculum. However t h i s t e x t Is - n o t o f f e r e d a s appropr ia te con ten t f o r t h e slow l e a r n e r s among t h e non college-bound s t u d e n t s .

The mathematics which appears i n t h e t e x t is n o t of t h e II type normally c a l l e d "business" o r vocat ional" mathematics;

nor i s i t intended t h a t t h i s s e r v e a s a te rminal course. Rather , a s t h e t i t l e c l e a r l y s t a t e s , t h i s i s an In t roduc t ion t o Algebra which w i l l provide t h e s t u d e n t wi th many of t h e b a s i c concepts necessary f o r f u r t h e r s tudy .

Some of t h e important f e a t u r e s of t h e t e x t a r e t h e following:

(1) The reading l e v e l i s a p p r o p r i a t e f o r t h e kind of s t u d e n t s f o r whom t h e t e x t i s w r i t t e n .

( 2 ) I n o r d e r t o achieve t h e o b j e c t i v e of in t roduc ing one new concept a t a t i m e s e c t i o n s a r e d iv ided I n t o subsec t ions , each inc lud ing e x e r c i s e s .

( 3 ) New concepts a r e introduced through concre te examples.

( 4 ) Easy d r i l l m a t e r i a l i s inc luded i n t h e e x e r c i s e s .

(5) Chapter summaries and adequate s e t s of review problems a r e provided.

(b ) Terminology i s k e p t t o a minimum.

(7 ) A g l o s s a r y of impor tant terms and d e f i n i t i o n s i s inc luded a t t h e end of each o f t h e f o u r p a r t s .

Some g e n e r a l sugges t ions f o r t h e use of t h e t e x t a r e o f f e r e d below.

Reading. A S is t h e c a s e wi th a l l SMSG t e x t s t h i s t e x t was w r i t t e n

w i t h t h e e x p e c t a t i o n t h a t i t can and w i l l be read by t h e s t u d e n t . S ince many s t u d e n t s a r e n o t accustomed t o reading a book on mathematics, i t w i l l be necessary t o a s s i s t them I n l e a r n i n g t o make t h e b e s t use of t h e book.

Teachers r e p o r t t h a t a t t h e beginning of t h e course they f i n d it b e s t t o read t h e t e x t a loud while s t u d e n t s read s i l e n t l y . When s t u d e n t s e v e n t u a l l y do t h e r ead ing on t h e i r own they need t o be reminded a g a i n and aga in of t h e n e c e s s i t y f o r r e read ing some of t h e sen tences . It i s hoped t h a t by t h e end of t h e y e a r they w i l l have gained a good measure of competence i n reading mathematics.

Check Your Reading. -- The t e x t provides s e t s of q u e s t i o n s t i t l e d Check Your

Reading which a r e concerned w i t h t h e i d e a s i n t h e m a t e r i a l which t h e s t u d e n t has J u s t r e a d .

It would be wise t o s t a r t a c l a s s per iod by reviewing t h e reading q u e s t i o n s from t h e preceding day o r t h e preceding two days. The s t u d e n t who was n o t a b l e t o d i s c u s s a ques t ion when i t was f irst encountered would have t h e oppor tun i ty t o do so i n t h e review.

Problem S e t s . - The t e x t has an ample supply of e x e r c i s e s . They a r e graded

i n each l i s t s o t h a t t h e most d i f f i c u l t a r e a t t h e end of t h e l i s t . I n an e x e r c i s e which has p a r t s t h e t e a c h e r should use a s many of them a s seems best f o r t h e p a r t i c u l a r c l a s s s i t u a t i o n .

Problems have been inc luded which may be omi t ted wi thout any l o s s 3f c o n t i n u i t y . Among them a r e s t a r r e d problems which a r e more d i f f i c u l t t han o t h e r s . Problems o f t h i s type a s w e l l a s t h e challenge problems which appea r a t t h e end of each of t h e f o u r par t s , might wel l be a p p r o p r i a t e f o r t h e " e x t r a c r e d i t " p a r t of the assignments.

This t e x t i s i n f o u r p a r t s . I n t h e d i r e c t i o n s s e n t o u t t o " t ryout" c e n t e r s du r ing t h e p a s t two y e a r s t e a c h e r s were advised to use t h e i r own Judgement a s t o how r a p i d l y they should Introduce t h e m a t e r i a l t o t h e i r s t u d e n t s . The r e p o r t s of t h e teachers i n d i c a t e t h a t i t t a k e s more than one y e a r of s tudy f o r s tudents of average a b i l i t y t o complete t h e f o u r p a r t s success - f u l l y . It i s n o t c l e a r a s y e t how we l l s t u d e n t s of lower than average a b i l i t y can l e a r n a l g e b r a from t h i s t e x t .

A comparison experiment conducted r e c e n t l y by t h e Minnesota National Laboratory showed t h a t c o l l e g e capable s t u d e n t s s tudy ing from t h i s t e x t performed a s w e l l on SMSG u n i t tests a s s t u d e n t s o f l i k e a b i l i t y s tudy ing from t h e t e x t F i r s t Course - i n Algebra.

Chapter 1

SETS AND THE NUMBER LINE

I n t h i s c h a p t e r we use t h e non-negative r a t i o n a l numbers and the bas ic opera t ions upon them a s a f a m i l i a r background f o r the In t roduc t ion o f concepts and procedures which may be new t o the pup i l . We cons ide r b r i e f l y two of t h e i n d i s p e n s l b l e t o o l s f o r our s tudy of t h e s t r u c t u r e of t h e r e a l number system -- s e t s and t h e number l i n e .

One of t h e g r e a t un i fy ing and s impl i fy ing concepts of a l l mathematics, the idea of set , is of importance throughout t h e course i n many ways: i n c l a s s i f y i n g t h e numbers wi th which w e

work, i n examining t h e p r o p e r t i e s of t h e o p e r a t i o n s upon t h e s e numbers. I n so lv ing equa t ions and i n e q u a l i t i e s , i n f a c t o r i n g polynomials, i n t h e s tudy of f u n c t i o n s , e t c .

Since most s t u d e n t s have n o t s t u d i e d about sets be fore e n t e r i n g t h i s course, and s i n c e t h e b a s i c n o t i o n s of se t a r e usual ly grasped q u i t e r e a d i l y , i t seems a good t o p i c , from a motivat ional s t andpo in t , wi th which t o s t a r t t h e course. We

move on quickly from t h i s f irst d i s c u s s i o n of s e t s , however, postponing much work w i t h opera t ions on elements of s e t s and with c losure , s o a s t o g e t quickly t o the p r e s e n t a t i o n of v a r i a b l e ( i n Chapter 2 ) . Th i s is done l a r g e l y because ( 1 ) t e a c h e r s and s tuden t s expect t h e e a r l y i n t r o d u c t i o n of v a r i a b l e and ( 2 ) o u r study of t h e s t r u c t u r e of t h e number system can begin wi th t h e idea of v a r i a b l e .

Next we p lace t h e number l i n e before t h e s t u d e n t . Here again i s a concept t h a t i s o f use throughout t h e course . I t i s the device f o r p i c t u r i n g many of t h e i d e a s about numbers and opera t ions on them. Th i s i s immediately apparent a s t h e graphing of sets i s introduced and is followed i n t h e f i n a l s e c t i o n of the chap te r by a d d i t i o n and m u l t i p l i c a t i o n on t h e number l i n e .

Pup i l s who have s t u d i e d SMSG Mathematics - f o r J u n i o r High School w i l l have had a l i t t l e exper ience w i t h sets and t h e number l i n e . They may be a b l e t o go through p a r t s of t h i s chap te r a l i t t l e more quickly than o t h e r s tuden t s . bu t t h e t rea tment i s s u f f i c i e n t l y d i f f e r e n t t h a t noth ing should be omit ted .

pages 1-3: 1-1

The t e a c h e r i s r e f e r r e d t o Haag, S t u d i e s i n Mathematics, - Volume 111. S t r u c t u r e o f Elementary Algebra, Chapter 2, Sec t ion 1. -

1-1. S e t s . - Though t h e f i r s t sets l i s t e d a t t h e o u t s e t o f t h e chap te r

a r e n o t examples of s e t s of numbers, we move quickly i n t h e t e x t t o c o n s i d e r a t i o n of such s e t s . Though non-numerical s e t s may be

o f i n t e r e s t , a prolonged d i s c u s s i o n o f them would c o n s t i t u t e a d i v e r s i o n f r o m t h e b a s i c purpose of t h e course.

The concept of s e t is in t roduced by making use of t h e s t u d e n t ' s exper ience . You may f i n d i t necessary o r d e s i r a b l e t o g ive s e v e r a l o t h e r examples.

We do n o t in t roduce much of t h e s t andard set n o t a t i o n such a s s e t b u i l d e r n o t a t i o n , c , ) , C , U , Cf , because the t o p i c s t o which t h e s e n o t a t i o n s a r e p a r t i c u l a r l y w e l l adapted a r e probably too widely separa ted I n t h e book f o r r e t e n t i o n . There is , however, no o b j e c t i o n t o t h e t e a c h e r us ing any of these i f he s o d e s i r e s . C e r t a i n l y , i f t h e c l a s s a l r e a d y has a background Inc lud ing set n o t a t i o n , t h e t e a c h e r should make use of it.

Braces a r e in t roduced a s a means of recognizing sets and a s a means of l i s t i n g sets.

Study Guide; page 2: 1. S t r e s s t h e idea t h a t "set" w i l l be used throughout t h e

course .

Answers - t o Problem S e t 1 - l a - pages 2-3: - -' 1. ( a ) ( 9 , 19, 29, 39, 49)

( b ) ( 3 , 13, 23, 33, 43) ( c ) (10, 20, 30, 40)

2. The set i n ( c ) . It has 4 elements .

3. ( a ) ( d , e , f, g, h , i, .I) ( b ) (0 , e~ a ) ( c ) (1, s, p) ( P o i n t o u t t h a t t h e l e t t e r is l i s t e d

only once even though i t occurs more than once. )

pages 3-5: 1-1

Problems preceded by t h e a s t e r i s k * a r e more cha l l eng ing than o t h e r s i n the same s e t of e x e r c i s e s . Such problems a r e included p r imar i ly f o r t h e b r i g h t e r and more cur ious s t u d e n t , and the use of these problems with an e n t i r e c l a s s may consume time needed l a t e r i n t h e y e a r t o complete t h e b a s i c work o f t h e course. The t eacher w i l l have t o decide a s he reaches each such problem whether time and t h e a b i l i t y of h i s s t u d e n t s permit him t o deal wi th t h e problem wi th t h e c l a s s a s a whole.

*5. ( a ) ( C a l i f o r n i a , Connect icut , co lorado] ( b ) (N.Y., N . J . , N.H., N.C., N.D., N . M . , Neb., Nev.1 ( c ) Hawaii

( d ) There a r e no elements i n t h i s s e t .

Pages 3-4. It should be pointed o u t t h a t t h e r e a r e two methods of desc r ib ing s e t s . A set can be l i s t e d wi th t h e elements en- closed i n braces, o r a s e t can be desc r ibed wi th a ve rba l d e s c r i p - t ion . It i s Important t o no te t h a t i n some c a s e s a ve rba l desc r ip t ion and a l i s t i n g are equa l ly adequate i n d e s c r i b i n g a s e t . However, t h e r e a r e s e t s which can be descr ibed only i n one of the two ways. On one hand, f o r example, i s t h e s e t ( 2 , 3, 5, 7, 8}, which i s n o t e a s i l y descr ibed i n words; on t h e o t h e r hand, there is the n u l l s e t , which cannot be l i s t e d and must be described i n some o t h e r manner.

Pages 4-51. We int roduce t h e technique f o r l i s t i n g sets which have many elements and s e t s t h a t a r e i n f i n i t e . We u s e t h e common nota t ion of t h e t h r e e d o t s ". . , I ' which mean "and so f o r t h " o r "continuing i n t h e same pat tern1 ' . Depending upon t h e c l a s s , t h i s no ta t ion may o r may n o t need more exp lana t ion .

The r e p r e s e n t a t i o n o f a s e t by a c a p i t a l l e t t e r is i n t r o - duced. The s tuden t should understand t h a t t h i s is simply a way of naming t h e s e t . We then d e f i n e t h e s e t of count ing numbers, whole numbers, m u l t i p l e s of 3 ( t o c l a r i f y t h e concept o f a mul t ip le ) , even numbers, and t h e set of odd numbers.

pages 6-7: 1-1

Answers - to Problem Set l - l b ; pages 6-7: --

A i s the s e t of a l l even numbers l e s s than 7. B i s t h e s e t of a l l odd numbers. C i s t h e s e t of a l l mul t ip les of 6. D i s t h e s e t of a l l whole numbers. E i s t h e s e t of a l l mul t ip les of 4. F i s t h e s e t of a l l whole ( o r counting) numbers g rea t e r than 11.

G i s t h e s e t of a l l odd numbers g rea t e r than 21

(o r 22).

H is t h e s e t of a l l odd numbers l e s s than 19 ( o r 1 8 ) . I i s t h e set of a l l m u l t i p l e s o f 6 which a r e less than 70 ( o r 6 7 ) . J i s t h e set of a l l even numbers g r e a t e r than 26 and less than 100.

K is t h e s e t o f a l l 30-day months. (other verbal d e s c r i p t i o n s a r e p o s s i b l e ) .

s e t obta ined by dividing each element of t h e s e t o f even numbers by two Is t h e set: (0, 1, 2, . . .),

which i s the set of whole numbers.

pages 8-9: I -1

Many of t h e problem sets i n t h i s c h a p t e r a r e s h o r t , and t h e teacher may wish t o cover more than one problem set I n a day. For most s t u d e n t s , t h e s h o r t problem s e t s should s u f f i c e t o convey the idea of sets which a r e needed i n t h i s course . The t e a c h e r i s

cautioned n o t t o dwell on t h e s e s e c t i o n s a t l e n g t h , n o r t o prolong g rea t ly t h e e x e r c i s e work on s e t s , f o r it i s t h e a l g e b r a i c s t r u c - tu re o f t h e r e a l number system, r a t h e r than t h e s tudy o f s e t s f o r t h e i r own sake, t h a t c o n s t i t u t e s t h e h e a r t of t h e course.

Answers t o Problem S e t 1 - l c ; pages 8-10: - 1. B is a s u b s e t of A C i s n o t a s u b s e t o f A because 30

D is a s u b s e t of A is a n element of C and n o t an E is a s u b s e t of A element of A .

" a subset Of A F is n o t a s u b s e t of A because 0

is an element of F and n o t an element of A .

G i s n o t a s u b s e t of A because 27, 29, . . . a r e elements of G and n o t elements of A .

2. T = [ I , 4, 9 , 1 6 ) R = (1, 41

( a ) No, R does n o t con ta in 2 a s an element. ( b ) Yes ( c ) Yes ( d ) NO. 9, 16 a r e elements of T but n o t of S.

3. K = ( 1 , 2, 3, 4, 9 , 16)

( a ) S, T, R, K a r e a l l s u b s e t s of K .

( b ) R is a s u b s e t o f R. ( c ) K has the most elements .

( d ) R has t h e fewest elements.

4. ( a ) We have de f ined an odd number a s "one more than a n even number." Hence, i f w e add two odd numbers we w i l l have 2 more than a n even n u m b e r ~ w h i c h w i l l be an even number.

pages 9-10: 1-1

( b ) From o u r d e f i n i t i o n of a n odd number, mul t ip ly ing an odd number by an odd number would always r e s u l t i n a n "ex t ra" one, s o t h e product is always odd.

I f T = ( 1 , 2 , 3, 4 ) , t h e s e t of sums of p a i r s of T i s S = (2 , 3 , 4, 5, 6 , 7, 8) . S i s n o t a s u b s e t of T because t h e elements 5, 6, 7, 8 a r e n o t i n T.

I f Q = {o, I ) , then t h e s e t o f products of p a i r s i s P = (0 , 1) . P i s a s u b s e t of Q.

If R = (0, 1, 21, t h e s e t of sums of p a i r s of R is

S = (0, 1, 2, 3 , 4) and t h e s e t P of products of p a i r s of R i s P = (0, 1, 2, 4 ) ; n e i t h e r P n o r S is a subse t of R .

The s u b j e c t of c l o s u r e in t roduced i n t h i s problem w i l l

be dwelt upon thoroughly l a t e r s o it should probably be l e f t a s an i n t e r e s t problem a t t h i s time and should no t

be allowed t o d i s t r a c t t h e c l a s s from more immediate ideas .

( a ) T Is n o t c losed under a d d i t i o n .

( b ) Q, i s c losed under m u l t i p l i c a t i o n but not under a d d i t i o n .

( c ) R is n o t c losed under e i t h e r m u l t i p l i c a t i o n o r a d d i t i o n .

( d ) N i s c losed under both m u l t i p l i c a t i o n and a d d i t i o n .

Comment on problems 2 -- and 3. Experience shows t h a t s t u d e n t s u s u a l l y have d i f f i c u l t y

understanding t h e d i r e c t i o n s given f o r these two problems, r e g a r d l e s s of t h e c a r e with which t h e i n s t r u c t i o n s a r e w r i t t e n . Here we a r e touching f o r t h e f irst time the ideas

I 1 of " i n t e r s e c t i o n " and union" o f two sets. These w i l l be

h i t aga in i n va r ious c o n t e x t s ; thus , i t is n o t necessary f o r t h e t e a c h e r t o make an a l l - o u t production of problems 2 and 3 . The d i f f i c u l t i e s he re can be eased by means of a d ia logue between t e a c h e r s and c l a s s i n which i t is made c l e a r t h a t

pages 10-12: 1-1

1 ) t h e elements i n R i n S c o n s i s t of those elements common t o R , S;

2) the elements i n R - o r i n S c o n s i s t of those elements e i t h e r i n R o r I n S o r both.

A f t e r t h e c l a s s succeeds i n understanding these two opera t ions on s e t s , be s u r e t h a t t h e words - and and - o r remain t h e key words r a t h e r than t h e words "both", "common", ' e i t h e r " , e t c . There i s a good reason f o r t h i s , because very soon i n t h e course (Chapter 3) w e w i l l meet conjunct ions and d i s j u n c t i o n s o f sentences i n which t h e intersections and unions of s e t s w i l l be implied by and or, r e s p e c t i v e l y .

Pages 10-11. The t eacher should be aware of t h r e e common e r r o r s made by s tuden t s i n working w i t h t h e empty s e t . The most common error i s the confusion of (0) and Of, and t h i s i s warned a g a i n s t i n the t e x t , but may need f u r t h e r emphasis by t h e t eacher . A

l e s s s l g n l f l c a n t mistake is t o use t h e words "an - empty set" o r "a - n u l l s e t " Ins tead of " t h e - empty set" o r " t h e - n u l l sett t . There is but one empty s e t though it has many d e s c r i p t i o n s . A t h i r d e r r o r is t h e use of t h e symbol, [Ofl, i n s t e a d of j u s t Of.

The statement t h a t t h e n u l l s e t is a s u b s e t o f every set

may cause some d i f f i c u l t y . The t e a c h e r should p o i n t o u t t h a t t o say t h a t every element of A i s a n element o f B means t h a t there is no element I n A which is n o t I n B. The n u l l set Cf is a subset of t h e set (1 , 2, 3 ) s i n c e Of has no elements which a r e not i n t h e set (1 , 2, 3) .

Answers & Problem S e t 1-Id; pages 11-12:

1. ( a ) A = ( 2 ) ; t h e r e f o r e i t is n o t Of. ( b ) B = Of (4 c = Of ( d ) T h i s Is no t t h e n u l l s e t bu t t h e set (0)

( e l Y( (f) (01, not Of

pages 12-14: 1-2

2. The l is t of s u b s e t s of B is 0' ( 1 ) ( 2 ) ( 3 ) There a r e e i g h t subse t s .

[ I , 21 2 X 2 X 2 = 2 3

There a r e 16 s u b s e t s . 2 x 2 x 2 x 2 = 2 4

4. For a set c o n s i s t i n g of n elements, t h e number of sub- s e t s is 2". Th i s problem is included t o h e l p d iscover t h e s t u d e n t who has t h e a b i l i t y t o genera l i ze . Do no t cons ide r t h i s a s something f o r t h e e n t i r e c l a s s t o master a t t h i s t ime, c e r t a i n l y no t t h e n o t a t i o n 2 .

Pages 12-14. The number l i n e is used a s an i l l u s t r a t i v e and moti- v a t i o n a l device , and o u r d i s c u s s i o n of It is q u i t e i n t u i t i v e and Informal . A s was t h e case wi th t h e preceding s e c t i o n , more ques- t i o n s a r e r a i s e d than can be answered immediately.

P resen t on t h e number l i n e i m p l i c i t l y a r e p o i n t s correspond- i n g t o t h e nega t ive numbers, a s is suggested by t h e presence i n t h e i l l u s t r a t i o n of t h e l e f t s i d e of t h e number l i n e . Since, however, t h e p lan of t h e course is t o move d i r e c t l y t o the c o n s i d e r a t i o n of t h e p r o p e r t i e s o f the opera t ions on t h e non- nega t ive numbers, anything more than casua l r ecogn i t ion o f t h e e x i s t e n c e of t h e nega t ive numbers a t t h i s time would be a d i s t r a c t i o n t o t h e s t u d e n t .

The idea o f successor is important . Suppose you begin with t h e count ing number one. The successor is "one more", o r 1 + 1.

The successor o f 105 is 105 + 1, o r 106; of 100,000,005 i s 100,000,006. This Implies t h a t whenever you th ink of a whole number, however l a r g e , it always has a successor . To the pup i l

pages 13-17: 1-2

should come t h e r e a l i z a t i o n t h a t t h e r e i s no l a s t number. An ---- i n t e r e s t i n g r e f e r e n c e f o r t h e s t u d e n t i s Tobias Dantz ig , Number, the Language Sc ience , pp. 61-64. -

The use of t h e term " i n f i n i t e l y many" on t h e p a r t of t h e s tuden t and t e a c h e r should h e l p t h e s t u d e n t avoid t h e noun " i n f i n i t y , " and w i t h i t t h e t empta t ion t o c a l l " i n f i n i t y " a num-

e r a l f o r a l a r g e number. The emphasis he re is on t h e f a c t t h a t a c o o r d i n a t e is t h e

number which i s a s s o c i a t e d w i t h a p o i n t on t h e l i n e . "coordinate" ,

"assoc ia ted" , and " corresponding to" must e v e n t u a l l y become p a r t of t h e p u p i l l s vocabulary. He must n o t confuse coord ina te wi th p o i n t , no r coord ina te w i t h t h e name o f t h e number.

The d i s t i n c t i o n between number and name of a number comes up he re f o r t h e f i rs t t ime. Do n o t make an i s s u e of i t a t t h i s time, f o r i t i s d e a l t w i th e x p l i c i t l y a t t h e beginning o f Chapter 2.

Answers & Problem S e t 1 -2a; page 15:

1. S = (1, 6, 11, 16, 21, . . ., 46) ( a l i s t d e s c r i p t i o n )

2. ( a ) f i n i t e ( b ) i n f i n i t e ( c ) i n f i n i t e

( d ) f i n i t e ( e ) f i n i t e ( f ) i n f i n i t e

Pages 15-17. Here we p i c t u r e t h e number l i n e , t h e p o i n t s being l abe led wi th r a t i o n a l numbers. You may want t o p o i n t t h i s o u t t o t h e s t u d e n t s a f t e r they have read a t t h e t o p o f page 17. We must be c a r e f u l t o observe t h a t t h e g e n e r a l s t a t e m e n t on page 1 7 concerning r a t i o n a l numbers i s n o t a d e f i n i t i o n , s i n c e i t does no t t ake i n t o account t h e n e g a t i v e numbers. Do n o t make an i s s u e of t h i s wi th t h e s t u d e n t s ; f o r t h e moment we merely want them t o have t h e i d e a t h a t t h e s e numbers a r e among t h e r a t i o n a l s .

It is a l s o p o s s i b l e t o say t h a t a number r e p r e s e n t e d by a f r a c t i o n i n d i c a t i n g t h e d i v i s i o n o f a whole number by a count ing number i s a r a t i o n a l number. T h i s s t a t e m e n t may be o f i n t e r e s t s i n c e I t Is expressed i n terms o f t h e s e r e c e n t l y d e f i n e d sets,

pages 17-19: 1-2

but t h e s ta tement i n t h e t e x t has t h e advantage t h a t t h e

"34" "14" "8" exc lus ion of d i v i s i o n by zero is e x p l i c i t . , - II n"

2 , 7' a r e some p o s s i b l e names f o r these numbers. 2

A r a t i o n a l number may be represented by a f r a c t i o n , but some r a t i o n a l numbers may a l s o be represented by o t h e r numerals, such a s 1.333 ... and 1 .42 . The number l i n e I l l u s t r a t i o n on

I1 n?l "6" '

page 1 6 g ives t h e name "2" a s wel l a s t h e f r a c t i o n s a- , 7 , 11 811

t o name t h e number 2.

The same diagram makes c l e a r t h a t n o t a l l r a t i o n a l numbers a r e whole numbers. The s t u d e n t s may have seen some f r a c t i o n s t h a t do no t r e p r e s e n t r a t i o n a l numbers, such a s , ^ , 7, ^ e t c . They w i l l have t o be reminded t h a t so -ca l l ed "decimal f r a c t i o n s " a r e n o t by t h i s d e f i n i t i o n f r a c t i o n s .

I t i s necessary t o keep the words " r a t i o n a l number" and ' f r a c t i o n " c a r e f u l l y d i s t i n g u i s h e d . L a t e r on i n t h e course, i t w i l l be seen t h a t t h e meaning of t h e term " f r a c t i o n 1 inc ludes any express ion , a l s o invo lv ing v a r i a b l e s , which is i n t h e form of an i n d i c a t e d q u o t i e n t .

Pages 17-19. The idea of l 'densi tyl l of numbers i s being i n i t i a t e d here . By d e n s i t y of numbers we mean t h a t between any two numbers t h e r e i s always ano the r , and hence t h a t between any two numbers t h e r e a r e i n f i n i t e l y many numbers. This sugges ts t h a t on t h e number l i n e , between any two p o i n t s t h e r e Is always ano the r p o i n t , and, i n f a c t , i n f i n i t e l y many p o i n t s . We refer

h e r e t o "points" i n t h e mathematical, r a t h e r than phys ica l , sense-- t h a t i s , p o i n t s of no dimension. Because t h e s tuden t may n o t be th ink ing of p o i n t s i n t h i s way he may n o t i n t u i t i v e l y f e e l t h a t between any two p o i n t s on t h e number l i n e o t h e r p o i n t s

may be l o c a t e d . Therefore , he is shown "betweenness" f o r number1

f i rs t ; then, t ak ing t h e s e numbers a s coord ina tes , he can i n f e r "betweenness" of p o i n t s on t h e number l i n e .

The f a c t t h a t t h e r e a r e p o i n t s on t h e number l i n e which do n o t correspond t o r a t i o n a l numbers should a rouse t h e s tudents1 c u r i o s i t y . Do n o t expand on t h i s a t t h i s t ime, however. I r r a - t i o n a l numbers w i l l be in t roduced a t a l a t e r t ime, a s coordinates of such p o i n t s .

pages 19-20: 1-2

A t t h i s p o i n t i n t h e course, i t is hoped t h a t t h e s t u d e n t w i l l accept t h e f a c t t h a t every p o i n t t o t h e r i g h t of 0 on t h e number l i n e can be ass igned a number. He may n o t accep t t h e f a c t t h a t - not every such p o i n t has a r a t i o n a l number a s i t s coordinate , but t h i s f a c t need not be emphasized u n t i l Chapter 12. H e may a l s o be impat ient t o a s s i g n numbers t o p o i n t s t o t h e l e f t of 0.

For the time being, u n t i l Chapter 6 , we s h a l l concen t ra te on t h e non-negative r e a l numbers. This s e t of numbers, inc lud ing 0 and a l l numbers which a r e coord ina tes of p o i n t s t o t h e r i g h t of 0 ,

we c a l l t h e s e t of numbers of a r i t h m e t i c . A f t e r we e s t a b l i s h t h e p r o p e r t i e s of opera t ions on t h e s e numbers ( i n Chapters 2 and 4) we s h a l l cons ider t h e set of a l l r e a l numbers which inc ludes the negat ive numbers ( i n Chapter 6 ) . Then i n Chapters 7 , 8, 9 , and 10, we s p e l l o u t t h e p r o p e r t i e s of o p e r a t i o n s on a l l r e a l numbers.

Answers t o Problem S e t 1-2b; pages 20-21: -

2. The s t u d e n t should c i r c l e t h e p o i n t s l a b e l e d 0, 1, 2 ,

3 , 4, 5, i n ( a )

0, 1 i n ( b )

pages 20-21: 1-2

3. T h i s problem r e p r e s e n t s a very good r u l e r exe rc i se . If

time i s a f a c t o r you may choose t o omit it .

11 23 47 l a r g e s t , v; l a r g e r T, w, e t c . 2

1 s m a l l e s t , T; 1 1 2 smal le r , m, pr, e t c .

3 5 7 The s t u d e n t may n o t i c e t h e sequence of T, 5, 8,

11 12 13 a n d g i v e t h e answer n, e t c . I f he 12

doesn ' t , p o i n t t h i s ou t .

1 8 is a possible answer.

1 2 - = 2 - 2 4 3 1 ; between 5~ and 5~ is 5~ o r g. 1 2 57' T?-^

O f course o t h e r s a r e p o s s i b l e such a s &, &. I n f i n i t e l y many.

I n f i n i t e l y many. I n f i n i t e l y many.

2' 3 2 1 a r e p o s s i b i l i t i e s ;

p o s s i b i l i t i e s

There is none--no m a t t e r what one is o f f e r e d a s I t next" , ano the r can be found between t h i s number and 2. Th i s should provoke some i n t e r e s t i n g d iscuss ion!

pages 21-22; 1-2 4 8 4, -, , 7 - 3, 3 + 1, 2 x 2 a r e p o s s i b i l i t i e s .

-I

Here w e a r e bu i ld ing t h e idea of o rde r :

The p o i n t wi th coord ina te 3.5 is t o t h e r i g h t of t h e

p o i n t wi th coordinate 2. 3.5 is g r e a t e r than 2.

The p o i n t wi th coord ina te 1.5 is t o t h e l e f t of t h e

p o i n t wi th coord ina te 2. 1.5 is l e s s than 2.

omi t t ing those which name t h e same number. ( b ) i n f i n i t e

( c ) i n f i n i t e

P a ~ e s 21-22. It should be polnted o u t t h a t t h e graph of a s e t

i s simply t h e p o i n t s marked on t h e number l i n e .

Answers - t o Problem S e t 1-2c; pages 22-23; --

2. ( a ) If S = (0, 3, 4, 7 ? a n d T = {O, 2, 4, 6, 8, 101, then K = ( 0 , 4) and M = (0, 2, 3, 4, 6, 7, 8, 101

pages 23-24: 1-2

( c ) The p o i n t s on t h e graph of K a r e p r o j e c t i o n s of t h e

p o i n t s appear ing simultaneously on S and T . The po in t s on t h e graph of M a r e t h e p r o j e c t i o n of every p o i n t on S and every p o i n t on T. The s t u d e n t need n o t , of course , answer i n t h e s e terms.

3. ( a ) A : = I I I I I I a - I - - I I - I I

0 I 2 3 4 5 6 7 8 9 10

( b ) I f C is t h e set of numbers which a r e elements of both A and B (meaning I n A - and i n B ) , then C has no

elements .

( c ) C i s t h e empty s e t ( o r t h e n u l l s e t ) .

Pages 23-24. T h i s l i s t should n o t be used a s a teaching a l d but - as a guide f o r t h e s t u d e n t .

Review Problem s; pages 24-27: The review problems can be used i n a v a r i e t y of ways. They

may be used f o r homework. They may be used f o r t e s t i tems. Problem *12 should n o t be given t o every s t u d e n t . It involves a

pages 24-25

very sub t le idea involving i n f i n i t e subse t s .

Answers - t o Review Problem - Se t ; pages 24-27:

(0, 3, 6 , 9, 12, -, 48 I [o, 3, 6, . . ., 48)

(0, 6, 12, . . ., 48 I

The s e t of mul t ip les of 3 i s not a subset of t h e s e t of mul t ip les of 6 because t h e r e a r e elements i n t h e f irst s e t not appearing i n t h e second. For example, t h e number 9 i s

a mul t ip le of 3 but it i s not a mul t ip le of 6. The s e t of mul t ip les of 6 - i s a subset of t h e s e t of mul t ip les of 3.

The s e t of a l l even numbers g r e a t e r than 8. Other descr ip- t i o n s a r e poss ib le .

h he s e t of a l l odd numbers from 7 t o 59 inc lus ive" 1s one of t he poss ible descr ip t ions .

The empty s e t .

( a ) 18 elements (b) 25 elements ( c ) i n f i n i t e l y many elements ( d ) 34 elements (don' t f o rge t zero)

( e ) 101 elements ( f ) i n f i n i t e l y many elements (g) i n f i n i t e l y many elements

I f S = (5, 7, 9 ) and T = (0, 2, 4, 6, 8, 101

( a ) then K = C( K is a subset of S and of T. A l l t h r e e a r e f i n i t e s e t s .

(b) M = ( 0 , 2, 4, 5, 6, 7, 8, 9, 101 M i s not a subset of S. T i s a subset of M . M i s a f i n i t e s e t .

(4 R = [5, 7 , 91 R i s a subset of M, of S.

(d) A = C(. A has no elements. A i s t h e empty s e t .

( e ) A and K are t h e same.

pages 25-26

( f ) Subsets of f i n i t e s e t s a r e always f i n i t e .

(g) The s e t D of a l l r a t i o n a l numbers from 0 t o 10 i n c l u s i v e i s n o t a f i n i t e s e t . This i l l u s t r a t e s the i n t e r e s t i n g idea t h a t i t i s n o t s u f f i c i e n t t o be a b l e t o name t h e l a s t number t o be a b l e t o count t h e s e t .

S i s a subse t of D.

Every i n f i n i t e s e t does have f i n i t e subse t s . D i s a subse t of D.

I n f i n i t e s e t s can have i n f i n i t e subse t s , fo r example, t h e s e t of count ing numbers i s a subset of t h e s e t of

r a t i o n a l numbers, o r of t h e whole numbers.

10. ( a ) l ~ ~ ~ l - l ~ l ~ l ~ l ~ l l - -

1 I 5

0 T - 2 + 3 3 - " 3 4 F l3 5

= 1 and $Â = 3 a r e whole numbers. 7 1 and 3 a r e count ing numbers. A l l t h e elements of t h e set a r e r a t i o n a l numbers.

22 - i s between t h e whole numbers 3 and 4. 7 22 - i s g r e a t e r t h a n 3.1. 7 The p o i n t wi th coord ina te * l i e s t o the r i g h t of 3.1. T 22 l i e s between 3 . 1 and 3.2. "T

12. The teacher should not f e e l compelled t o use c l a s s time f o r t h i s problem since the ideas may be l o s t on the c l a s s . How- ever, it can lead t o an in te res t ing discussion i f enough of the c lass w i l l benefit from it. The obviously capable indi - vidual i n the c l a s s should have the opportunity t o do I t .

This i s a much more useful de f in i t ion of an I n f i n i t e s e t than has been developed i n the t e x t .

The s e t of multiples of 3 i s a proper subset of the s e t of whole numbers since i t does not include the elements 1, 2, 4, 5, . . . a s a p a r t i a l l is t . One possible one-to-one corre- spondence between the s e t of whole numbers and multiples of

where n represents any whole number.

For the superior student it could be pointed out t h a t mathematicians take t h i s a s a de f in i t ion of an i n f i n i t e s e t : A s e t i s i n f i n i t e if It can be placed i n one-to-one corre- spondence w i t h a proper subset of i t s e l f . .

Suggested Test Items -- ( ~ h e "suggested t e s t itemsi' which follow a re not intended t o

comprise a balanced or complete t e s t , but a re , a s the t i t l e implies, questions which seem sui tab le f o r inclusion i n a t e s t on t h i s chapter. )

1. Are the following s e t s f i n i t e or i n f i n i t e ? I f it i s possible,

l i s t the elements of each.

( a ) The people In t h i s classroom today. (b ) A l l multiples of 3 .

( c ) A l l counting numbers l e s s than 7.

(d) a l l whole numbers which a r e not multiples of 5. 1

( e ) a l l numbers between 0 and T.

2. ( a ) Given s e t S = (0, 1, 2, 3, 4). Find s e t T, the s e t of products of each element of s e t S and 1. I s T a subset of S?

( b ) Given s e t A = (0, 2, 4, 6, 8). Find s e t B, the se t of products of each element of s e t A and 0. Is B a subset of A? Is B the empty se t?

3 . Describe i n words each of the following s e t s

(a) (1, 3, 5, 7, . I (b) (0, 5, 10, 15, - 1 ( 4 [o, 1, 2, 3, 41 (d l f6

4. Given s e t N = (1, 2, 3, 4, 8, 9, 12, 16).

(a) Find the subset R consisting of a l l elements of N which

a r e squares of whole numbers.

(b) Find s e t K of the odd numbers i n s e t N.

( c ) Find s e t A of the squares of the elements of N.

(d) Find s e t B whose elements a r e each 3 more than twice the corresponding element of N.

( e ) Find s e t C, t he s e t of a l l numbers which a r e elements of both N and B.

(f) Find s e t D, the s e t of a l l numbers which a r e elements of e i t h e r N or B or both.

5. Consider each of the following se t s , and f o r those which a re f i n i t e l i s t the elements, i f possible. I f the s e t i s the empty s e t , wr i te the usual symbol, $. (a) A l l counting numbers l e s s than 1.

(b) A l l whole numbers l e s s than 1.

(c ) A l l numbers l e s s than 1.

(d ) All counting numbers such tha t 10 times the number i s grea ter than the number i t s e l f .

( e ) A l l whole numbers such t h a t 10 times the number i s equal t o the number times i t s e l f .

6 . ( a ) Draw a number l i n e and l o c a t e t h e p o i n t s whose

coordinates a re :

( b ) Which of t h e s e coord ina tes are counting numbers? Whole numbers? Ra t iona l numbers?

( c ) On t h e number l i n e how i s t h e po in t wi th coord ina te 5 .4 loca ted wi th r e s p e c t t o t h e p o i n t wi th coord ina te 4? w i t h coordinate 6.2?

3 7. ( a ) Is 7 t o t h e l e f t of on t h e number l i n e ?

( b ) Show t h e graph of t h e s e t K = (0, 3, 7 ) .

( c ) Write 3 o t h e r names t h a t could be used f o r t h e coordi - na te 3.

8. I f A i s t h e s e t of a l l whole numbers l e s s t h a n 20 which a r e not mul t ip les of e i t h e r 2, 3, o r 5 ,

( a ) l i s t t h e elements of s e t A ;

(b) draw t h e graph of s e t A .

1 2 9 . L i s t two numbers between 8 and v. How do you know t h a t t h e y 1 a r e between and

10. S t a t e S , t h e s e t of a l l whole numbers.

( a ) Is it f i n i t e o r i n f i n i t e ? ( b ) Is i t c losed under a d d i t i o n ? Explain why. ( c ) Is I t c losed under m u l t i p l i c a t i o n ? Expla in why.

( d ) Is i t c losed under the opera t ion of f i n d i n g t h e average of two numbers? Show why.

S t a t e T , the s e t of a l l odd numbers, and answer ques t ions ( a ) through ( d ) .

S t a t e R , t h e s e t o f a l l odd numbers less than 8 and answer ques t ions ( a ) through ( d ) .

Answers t o Suggested Test Items - -- ( a ) f i n i t e (Ann, Mary, Peter, . . ., John) Really

depends on the c lass . (b ) i n f i n i t e

( c ) f i n i t e {I, 2, 3, 4, 5, 61 (d) i n f i n i t e ( e ) i n f i n i t e

( a ) T = (0, 1, 2, 3, 4) Yes, T i s a subset of S. ( b ) B = (0) . Yes, B i s a subset of A . Noy B i s not the

empty s e t .

( a ) The s e t of odd numbers.

(b) The s e t of multiples of 5.

(c) The s e t of whole numbers l e s s than 5, or the se t of whole numbers from 0 t o 4, inclusive.

(d) The empty s e t .

(a) 0 (b) to) ( c ) i n f i n i t e s e t (d) i n f i n i t e s e t

(4 (0)

' (b) 2, 3, 4 a r e counting numbers. 0, 2, 3, 4 a r e whole numbers. All a r e r a t iona l numbers.

( c ) This i s t o the r igh t of the point whose coordinate i s 4. It i s t o the l e f t of the point whose coordinate i s 6.2.

3 7. ( a ) Yes, I s t o t h e l e f t of 8 on t h e number l i n e .

2 + 1, 5 - 2, e t c , ( c ) Among s e v e r a l p o s s i b i l i t i e s a r e F, i,

8. ( a ) A = 0, 7, 11, 13, 17, 191

3 4 5 5 6 9 . Among s e v e r a l p o s s i b i l i t i e s a r e E, T^, z, z, e t c .

10. s = ( 0 , 1 , 2 , 3 , . . . I ( a ) I n f i n i t e

( b ) Yes, any element i n s e t S added t o any element i n s e t S produces a n element I n t h e s e t S.

( c ) Yes, same a s above.

( d ) No. + 8 - = and 9 Is n o t a n element of t h e se t 2 of whole numbers.

T = (1, 3, 5, 7, . I ( a ) i n f i n i t e

( b ) No, s i n c e 1 + 3 = 4 and 4 is n o t an element of t h e set T.

( c ) Yes, same a s ( b ) above.

( d ) N O , s i n c e - = 2 and 2 is n o t a n element o f t h e 2 set T.

R = (1, 3 , 5, 71 ( a ) f i n i t e ( b ) No. 1 + 3 = 4 and 4 is n o t an element of R.

( c ) No. 3 x 5 = 15 and 15 Is n o t an element o f R. 1 + 3 ( d ) No. 7 = 2 and 2 is n o t a n element of R.

Chapter 2

NUMERALS, SENTENCES, A N D VARIABLES

For background i n t h e t o p i c s Included i n t h i s c h a p t e r t h e t eacher i s r e f e r r e d t o Haag, S t u d i e s - i n Mathematics, Volume - 111,

S t r u c t u r e - of Elementary Algebra, Chapter 3 , Sec t ions 1 and 2, and Chapter 6, Sect ion 1.

2-1. Numbers and The i r Names.

The aim of t h i s s e c t i o n i s t o b r i n g o u t t h e d i s t i n c t i o n between numbers themselves and t h e names f o r them and a l s o t o in t roduce t h e not ion of a phrase . Along t h e way a number of Important conventions used i n a l g e b r a a r e poin ted o u t .

We do not want t o make a p r e c i s e d e f i n i t i o n of "common - name." The term i s a r e l a t i v e one and should be used q u i t e in fo rmal ly . Note t h a t some numbers do not have what we would wish a t t h i s t i m e

t o c a l l a common name, such a s tf 4, while some may have s e v e r a l common names ( e . g. 0.5 and 2+, 5, e t c . ) . 2'

The i d e a s of i n d i c a t e d - sum and i n d i c a t e d product a r e ve ry handy, p a r t i c u l a r l y i n d i s c u s s i n g t h e d i s t r i b u t i v e p roper ty , and w i l l be used f r e q u e n t l y . They a r e a l s o u s e f u l t o coun te rac t t h e tendency, encouraged I n a r i t h m e t i c , t o regard an express ion such a s " 4 + 2" not a s t h e name of a number bu t r a t h e r a s a command t o add 4 and 2 t o o b t a i n t h e number 6. This p o i n t of view makes I t d i f f i c u l t f o r a p u p i l t o accep t such express ions a s names of anything. I n pass ing , you may wish t o mention t o t h e c l a s s i n d i c a t e d q u o t i e n t s and i n d i c a t e d d i f f e r e n c e s . Some may a l ready be f a m i l i a r wi th " i n d i c a t e d q u o t i e n t a s synonymous wi th

" f r a c t i o n ." You w i l l n o t i c e t h a t t h e word " f a c t o r " i s not in t roduced h e r e

and f o r t h e fo l lowing reason . It i s f e l t t h a t t h e mathematical concept of " f a c t o r " i s such an important one t h a t we should wai t u n t i l t h e s t u d e n t s a r e ready f o r i t s d e f i n i t i o n and a p p l i c a t i o n t o t h e theory of prime f a c t o r i z a t i o n of I n t e g e r s and polynomials i n Chapters 11-15.

If t h e t eacher f e e l s compelled t o use " f a c t o r " a t t h i s p o i n t a s a handy word t o d e s c r i b e t h e numbers involved i n an i n d i c a t e d

product , he should do s o wi th c a u t i o n . Be s u r e t h a t t h e s t u d e n t s

pages 27-28: 2-1

do not th ink of f a c t o r s i n terms of t h e form of a numeral. For example, avoid t h i s k ind of f a u l t y th inking: "2 i s not a f a c t o r of 2 + 4 because 2 + 4 does not involve t h e i n d i c a t e d operat ion of m u l t i p l i c a t i o n . " I n s t e a d , encourage t h i s kind of th inking: "2 i s a f a c t o r of 2 + 4 because t h e r e i s a number, 3 , such t h a t t h e product of 2 and 3 i s 2 + 4." I n genera l , t h e number a i s a f a c t o r of b i f and only i f t h e r e i s a number c such t h a t ac = b . L a t e r we l e a r n why f a c t o r i n g i s mathematically i n t e r e s t i n g only f o r I n t e g e r s o r polynomials.

Note t h e use of quotes t o i n d i c a t e when t h e re fe rence i s t o t h e numeral o r express ion r a t h e r than t o t h e number represen ted , It i s important t o be c a r e f u l about t h i s a t f i r s t . However, s ince good Engl ish does not always demand t h i s kind of d i s t i n c t i o n but r a t h e r a l lows t h e context t o g ive t h e meaning, we tend l a t e r t o become more re laxed about i t and use such forms a s t h e expression 3x - 4y + 7" r a t h e r than " t h e express ion Ã 3 - 4y + 71".

The agreement about t h e p re fe rence f o r m u l t i p l i c a t i o n over a d d i t i o n i s made t o f a c i l i t a t e t h e work wi th express ions and - not a s an end i n i t s e l f . I n c e r t a i n k inds of express ions t h e agreement should a l s o apply t o d i v i s i o n a s wel l a s m u l t i p l i c a t i o n , f o r example when d i v i s i o n i s w r i t t e n i n t h e form 2/3 o r 2 + 3 , r a t h e r than *. We p r e f e r t o avoid t h e s e forms and, i n p a r t i c - 7 u l a r , t o d iscourage t h e use of t h e symbol 'I+''.

The use of parentheses might be compared t o t h e use of punc- t u a t i o n marks i n t h e w r i t i n g of Eng l i sh . Emphasis should be on t h e use of parentheses t o enable us t o read express ions without ambiguity and not on t h e technique of manipulat ing parentheses f o r t h e i r own sake .

Answers t o Oral Exerc ises 2- la ; page 28: -- - Exerc i ses 7, 8, 9, 10 may have more than one answer. For

1 example, .5 and 2 a r e both common names f o r one h a l f . This term II common name" i s in t roduced t o improve on t h e o l d term "simpler name" which i s o f t e n ambiguous. 1. 1 2 4 . 1 7. 4

5 10. 6

pages 28-31: 2-1

13. 4 1 1 4 . 15. 1 3?

These a re possible answers:

16. 8 + 4 17. 1 0 + 5 18. y + T 1 1 19. 1 + 0

Answers - t o Problem Set 2-la; pages 28-29: -- 1. (a ) 5 (d ) 5 1 (4 2

2. Many responses a re possible, such as: 12 - 2

( a ) 9 - 4; \- (e) 5 x 1; 3 + 2

( c ) 1 + 1 + 1 + 1 + 9 ; ^ (g) 2 + 1; ,+

(d) XXX + vI; 30 + 6 (h) 1.7 - .8; .4 + .5

3. Many responses are possible, such as: 30 (a) 5 x 2; 8 + 2; -,-; 11 - 1

(b) 35 x 1; 3$ + $; 35; 100 - 65

Answers t o Oral Exercises 2-lb; page 31: -- - In Exercise 10, since only addition and subtraction a re

involved, the order i s immaterial. The same i s t rue of Exercise 13 because only mult ipl icat ion and divis ion are involved. 1. 17 3 . 4 5. 11 7 - 13

pages 31-35: 2-1

Answers - to Problem -- Set 2-lb; page 31:

1. 14 6 . 3

a

10. 36; here, order would - 4 not make any difference. *

The words "numeral" and "numerical phrase" denote almost th, same thing. A phrase may be a more complicated expression whlchj involves some operat ions; 'tnumeral' includes all these and alad

the common names of numbers. We do not wish to make any fuss 04 this distinction, and are happy if the student learns to use the'

words in this way in the course of the year Just by watching

others use them. We introduce both because people do use both,

and because a term for a numeral which involves some Indicated

operations is sometimes handy.

In the term "numerical phrasett the word "numerical" is ni-

very important and Is used not so much to distinguish it from

word phrase as from an open phrase (one involving one or more

variables) which Is coming. The word "operations" is intended at this point to suggest

the basic operations of arithmetic (multiplication, division, addition and subtraction). In some contexts it may be desirable to admit operations such as finding the square root, forming the absolute value, etc.

Answers to Oral Exercises 2-lc; page 33: -- - Exercises l(f) and l(~) suggest properties that will be

discussed later and should not be overemphasized here except to mention that the order apparently is not important. .

1. (a) Yes (d) Yes (â‚¬ Yes

(b) Yes ( e ) No (h) No ( c ) No (f) Yes (1) No

( J ) Yes

pages 31-33: 2-1 1

9. 2..

10. 4

Answers to Problem Set 2-lb; page 31: - -- 1. 14 6 . 3

1 2. 21F 7. 7

10. 36; here, order would - not make any difference.

The words "numeralt' and "numerical phrase" denote almost the same thing. A phrase may be a more complicated expression which

involves some operations; "numeral" includes all these and also

the common names of numbers. We do not wish to make any fuss over

this distinction, and are happy if the student learns to use the

words in this way in the course of the year just by watching

others use them. We Introduce both because people do use both,

and because a term for a numeral which Involves some indicated operations is sometimes handy.

In the term "numerical phrase" the word "numerical" is not

very important and Is used not so much to distinguish it from a

word phrase as from an open phrase (one involving one or more

variables) which is coming. The word "operations" is intended at this point to suggest

the basic operations of arithmetic (multiplication, division, addition and subtraction). In some contexts it may be desirable

to admit operations such as finding the square root, forming the

absolute value, etc.

Answers to Oral Exercises 2-lc- page 33: -- -J

Exercises l(f) and l(j) suggest properties that will be

discussed later and should not be overemphasized here except to mention that the order apparently is not Important. .

1. (a) yes (d) yes (â‚¬ Yes

(b) Yea ( e l No (h) No (4 No (f) yes (1) No

( J ) Yes

pages 31-33: 2-1 1

9. 27.

10. 4

Answers - t o Problem S e t 2- lb; page 31: --

10. 36; h e r e , o r d e r would - not make any d i f f e r e n c e .

The words "numeral" and "numerical phrase" denote almost t h e same t h i n g . A phrase may be a more complicated expression which Involves some opera t ions ; " n u m e r a l inc ludes a l l t h e s e and a l s o t h e common names of numbers. We do no t wish t o make any f u s s over t h i s d i s t i n c t i o n , and a r e happy i f t h e s t u d e n t l e a r n s t o use t h e words i n t h i s way i n t h e course of t h e yea r j u s t by watching o t h e r s use them. We in t roduce both because people do use both, and because a term f o r a numeral which Involves some i n d i c a t e d o p e r a t i o n s i s sometimes handy.

I n t h e term "numerical phrase" t h e word "numerical" i s not ve ry important and Is used n o t s o much t o d i s t i n g u i s h it from a word phrase a s from an open phrase (one invo lv ing one o r more v a r i a b l e s ) which i s coming.

The word "opera t ions" i s Intended a t t h i s p o i n t t o suggest t h e b a s i c o p e r a t i o n s of a r i t h m e t i c ( m u l t i p l i c a t i o n , d i v i s i o n , a d d i t i o n and s u b t r a c t i o n ) . I n some c o n t e x t s it may be d e s i r a b l e t o admit o p e r a t i o n s such as f i n d i n g t h e square r o o t , forming t h e a b s o l u t e va lue , e t c .

Answers t o Oral Exerc i ses 2 - l c w page 33: -- -3

Exerc i ses l ( f ) and l ( j ) sugges t p r o p e r t i e s t h a t w i l l be d i scussed l a t e r and should no t be overemphasized he re except t o mention t h a t t h e o r d e r apparen t ly i s not impor tant . .

1. (a) yes ( d ) yes (g) Yes (b) Yes ( e l No ( h ) No ( 0 ) No ( f ) yes ( 1 ) No

0) Yes

pages 34-36: 2-1 and 2-2

Answers - t o Problem Se t 2- lc; pages 34-35: , - -

2. ( a ) 2 x ( 3 + 1 )

(b) 2 + ( 4 x 3 )

( c ) (6 x 3 ) - 1

( d ) (12 - 1 ) x 2

3. (a) ( $ x 6 ) + 3

( b ) (2 x 5) + ( 6 x 2 )

( c ) (2 X 3 ) + ( 4 x 3 )

( d ) ( 3 x 8) - 4

( e ) 10 # 10, no

(g) 11 # 15, yes

(h) 15 # 10, y e s

2-2. Sentences.

The words " t rue" and " f a l s e " f o r sentences seem p r e f e r a b l e t o " r igh t t1 and "wrong" o r " c o r r e c t " and " Incor rec t " because t h e l a t t e r a l l imply moral Judgments t o many people . There i s no th ing I l l e g a l , immoral, o r wrong i n t h e usua l sense of t h e word about a f a l s e sentence . The s tuden t should be encouraged t o use on ly " t rue" and " fa l se t1 i n t h i s c o n t e x t .

We have been doing two k inds of t h i n g s wi th our sentences : We t a l k about sentences , and we use sen tences . When we w r i t e -- -

' 3 + 5 = 8" i s a t r u e sentence ,

we a r e t a l k i n 4 about our language; when, i n t h e course of a - s e r i e s of s t e p s , we write

pages 37-39: 2-2

we are usin& the language. Now when we t a l k about the language, we can per fec t ly well t a l k about a f a l s e sentence, i f we find t h i s useful . Thus, it I s qui te a l l r ight t o say

' 3 + 5 = 10" is a f a l se sentence;

b u t it i s f a r from a l l r ight t o use the sentence

i n the course of a proof. When we are ac tua l ly u s i n ~ t h e language, f a l s e sentences have no place; when we a re ta lking about our language, they a re often very useful .

Check Your Readins -- Question 8 should lead t o a discussion of various mnemonic

devices such as "points t o the smaller number In a t rue sentence.

Answers t o Oral Exercises 2-2; page 38: -- - 1. False 6. True 11. True 16. False 2 . True 7 . False 12. False 17. True 3 . True 8. True 13. False 18. False

4 . False 9 . True 14. True 19. False 5. True 10 . False 15. False 20. False

Answers - t o Problem Set 2-2; pages 39-40: -- 1. (a ) False (e) False (1) False

(b ) True ( f ) False ' ( 3 ) True

(4 (g) Tnle (k) False

(d ) True (h) Tlxe

2. (a) 10 - (7 - 3) = 6 ( g ) 3 x (5 + 2) x 4 = 84

(b) 3 x ( 5 + 7) = 36 (h) ( 3 x 5 + 2) x 4 = 68

(c) (3 x 5) + 7 = 22 (i) 3 x (5 - 2) x 4 = 36

(d ) 3 x (5 - 4) = 3 ( J ) ( 3 x 5 ) - ( 2 ~ 4 ) = 7

( e ) (3 x 5 ) - 4 = 11 (k) ( 3 x 5 - 2) x 4 = 52

(f) (3 x 5) + (2 x 4) = 23 1 1 *(I) (12 x y - w) x 9 = 51 In problem 2 both

parentheses and the

( 1 2 X $ ) - ( $ x 9 ) = 3 conventionconcerning 1 1 order of operations

*(n) 12 x ( y - 3-) x 9 = 18 are used.

pages 39-41: 2-2 and 2-3

3. (a) False (b) False ( c ) True ( d ) True ( e ) False

4. ( a ) Four plus eight i s equal t o ten plus f ive. False - (b) Five plus seven i s not equal t o s h plus f ive . True, -- ( c ) Thirteen i s l e s s than eighteen minus 7. False --- (d) One plus two i s g rea te r than zero. True. - -

2-3. A Property -- of the Number 2. This i s the f i r s t time the student encounters the word

"property" used i n a mathematical sense. He w i l l see t h i s word often during the course and our object i s t o play heavily on the word t o indicate a charac ter i s t ic , a pat tern, a behavior which a given element o r operation displays. That is , a property of an object i s something it has which i s a distinguishing character- i s t i c of the object.

The pa r t i cu la r number 1, unlike a l l other numbers, has the peculiar property t h a t the product of 1 and a given number i s the given number. This i s qui te obvious t o a student; thus, we begin our discussion of propert ies with the property which I s

eas ies t t o understand. Later we s h a l l c a l l 1 an iden t i ty f o r multiplication. It i s a l so a valuable property t o have estab- l ished (o r accepted) when we introduce variables l a t e r i n t h i s chapter. Otherwise, we might have d i f f i c u l t y just i fying t h a t

- - and n + 4

are names f o r the same number no matter what number n i s . For the time being w e a r e content t o f ind ce r t a in propert ies

by considering many numerical examples and then s t a t e the general- izat ion in words. In Chapter 4 we s h a l l symbolize these proper- t i e s using variables.

TMre may be a tendency on the part of the student t o r e s i s t the use of the mult ipl icat ion property of one i n the exercises. He may f e e l t h a t he i s being asked t o use a more complicated way of doing things which he already knows how t o do.

pages 41-^4: 2-3

It i s important t o point out t o him t h a t w e a r e not t ry ing t o push a "new" method but r a t h e r t o show the importance of the mu l t i p l i ca t ion proper ty of one. It i s hoped t h a t he w i l l come t o see t h a t t h i s proper ty gives the j u s t i f i c a t i o n f o r the various methods of s impl i fying expressions with which he may be fami l ia r . Once the j u s t i f l c a t l o n Is understood it i s a l l r i g h t , of course, f o r him t o use sho r t - cu t s . Perhaps i t can be s a i d t h a t one has t o e a r n the r i g h t t o use shor t -cu ts . It i s important t o emphasize t h a t i n t h i s sec t ion the mul t ip l ica t ion property of one and the uses of t h i s proper ty a r e more important than the methodology involved i n s impl i fying expressions.

Answers t o Oral Exercises x; page 44: -- 1 5 5 1. ( a ) T X ? = - 20

Answers - t o Problem Set 3; pages 44-45: - 3 12 ( e ) 4 x - = - 3 3

pages 44-46: 2-3 and 2-4

2-4. - Some Properties of Addition and Multiplication. - - The aim of this and the next section is to look at the

fundamental properties of addition and multiplication in terms of specific numbers. We goas far as obtaining a general statement of the properties in English. You should not state the properties at this time using variables. We do not need these formulations at this point and prefer to lead up to variables in a different way In Section 2-6. It is important to emphasize the pattern idea here and you may want to do this by writing something like the following on the board when discussing, for example, the associative property for addition:

(first number + second number) + third number =

first number + (second number + third number). The use of the properties of addition and multiplication as

an aid to computation in certain kinds of arithmetic problems is both interesting and important but is not the main point of these properties. These properties will play much more fundamental a

pages 46-47: 2-4

r o l e I n t h i s course . They c o n s t i t u t e t h e foundation on which t h e e n t i r e s u b j e c t of a lgebra Is b u i l t .

The p r o p e r t i e s w i l l be r e tu rned t o i n Chapter 4 and subsequent c h a p t e r s where t h e genera l s t a t ements us ing v a r i a b l e s w i l l

be g iven . They a r e d iscussed he re not only a s a p a r t of t h e " s p i r a l method" but because t h e d i s t r i b u t i v e p roper ty i s used

i n in t roduc ing t h e concept of v a r i a b l e . From t h e mathematician's p o i n t of view t h e s tatement t h a t

an opera t ion i s a b i n a r y opera t ion on a s e t of elements impl ies t h a t t h e opera t ion can be app l i ed t o every p a i r of elements i n t h e s e t . I n t h i s s e c t i o n we use t h e word b ina ry only t o b r ing ou t t h e f a c t t h a t - t h e opera t ion - i n ques t ion - i s app l i ed t o two -- elements . We do not concern ourse lves he re with t h e ques t ion whether t h e opera t ion can be app l i ed t o every p a i r of elements t h a t can be chosen.

Each of t h e fo l lowing f i v e numerals

i s an i n d i c a t e d sum of two numbers and each names t h e same number. - This l a t t e r f a c t enables us t o w r i t e "4 + 6 + 3 + 8" without any ambiguity. The f a c t remains, however, t h a t a d d i t i o n i s a b i n a r y o p e r a t i o n .

Answers t o Oral Exerc i ses 2-4a; page 47: -- - 1. ( 4 + 2 ) + 7 = 6 + 7 4 + ( 2 + 7 ) = 4 + 9

= 13 = 13

pages 47-48: 2-4

Answers - t o Problem Set 2-4a; pages 47-48: -- 1. ( a ) 4 + ( 2 + 7 ) = ( 4 + 2 ) + 7

1 1 (b) (6 + 1) + 7 - 6 + (1 + 3.)

(c ) 3 + (4 + 11) = ( 3 + 4) + 11 ( d ) ( 5 + 1 ) + 6 = 5 + ( 1 + 6 ) 0 ~ 5 + ( 1 + 6 ) = ( 5 + 1 ) + 6

( e ) (11 + 13) + 121 = 11 + (13 + 121) o r

11 + (13 + 121) = (11 + 13) + 121

2 2. ( a ) ($ + $) + 3 = 1 + I, easier . 1

T + (^ + $1 2 1 4 2

(b) 3 + ( s - + =-) = Ty + 1, easier . 2 1 4 (? + + 5

5 ( c ) ($ + 6) + 2 = 1 + easier . 1 1 5 7 + tp +

(dl & + ($ + +) = & + 1, easier. A + ;I + * 3 1 1 5 3 1

(e ) ( 2 5 + ly) + = 4 + STY easier . 2 + (I8 + 27~)

(f) 2.7 + (13.2 + .8) = 2.7 + 14, easier . (2.7+13.2) + .8

pages 48-50: 2-14

3 1 3 (g) (6 + + g = 8 + (^ + i), neither Is easier. HUB is

hinting at the commutative property of addition which is

coming. Do not emphasize it unless some student wants to pursue It.

3. 179 millimeters Yes.

Although the question Is very easy to answer, the fact

that the answer is "yes" depends upon the property studied in

this section, as can be seen by these or similar calculations.

Answers - to Problem Set 2-4b; pages 50-51: -- Some of these problems might better be given as oral

exercises.

1. True, commutative property of addition 2. True, commutative property of addition

3. False 4. True, commutative property of addition

5. True, but - not because of the commutative property!

6. True, associative property of additlon 7. True, commutative property of addition

8. False 9. True, both properties 10. False

11. True, neither property

12. False

13. True, commutative property of additlon

14. True, commutative property of addition 15. False 16. True, both properties

17. True, commutative property applied twice 18. False

19. True, both properties

page 51: 2-4

20. One purpose of t h i s problem i s t o h e l p t h e s t u d e n t s make a h a b i t of quickly recogniz ing a d d i t i o n combinations which f a c i l - i t a t e computation. Another and more Immediate purpose i s t o help t h e p u p i l s begin t o become aware t h a t t h e s e manipula t ions , which they may have taken f o r granted , are p o s s i b l e because of the a s s o c i a t i v e and commutative p r o p e r t i e s of a d d i t i o n .

I n these problems we do no t ask s p e c i f i c a l l y which p r o p e r t i e s a re used i n going from one s t e p t o t h e n e x t . This Is o f t e n tedious - p a r t i c u l a r l y i n t h e l a t t e r s t e p s o f t h e c a l c u l a t i o n . We do not I n s i s t t h a t t h i s be done a t t h i s t ime f o r w e a r e more concerned with having t h e s t u d e n t recognize t h e use fu lness of t h e p roper t i e s than i n having him pursue a thorough step-by-step

reasoning process from beginning t o end of t h e c a l c u l a t i o n . I n s e v e r a l p a r t s of t h i s problem t h e r e a r e v a r i a t i o n s on

"the e a s i e s t way" t o perform t h e a d d i t i o n s . Comparison of some of these i n c l a s s d i scuss ion should h e l p f u l f i l l t h e purposes of the ques t ion .

(a ) The s tuden t may express h i s answer i n a manner l i k e t h i s : " ~ d d t h e 6 and t h e 4 t o get 10, then 10 and 8 t o make 18." This can be shown s t e p by s t e p i n s e v e r a l

ways, e . g.:

2 1 8 ( b ) $ + - + l + - + 5

page 51: 2-4

Ihis i s a case i n which there i s no "easier" way. ( d ) T + ?J

Neither property i s of help i n t h i s computation, though

several propert ies t o be studied l a t e r , most notably the

d i s t r i b u t i v e property, l i e behind the students' calcula-

t ions.

2 4 ( e ) 2$ + 3y+ 6 + 7r

( f ) Here Is another case i n which nei ther property

10 6 - 68 f a c i l i t a t e s the computation. + - - 15

(g) (1.8 + 2.1) + (1.6 + . 9 ) + 1.2 1.2 + (1.8 + 2.1) + ( .9 + 1.6)

(1.2 + 1.8) + (2.1 + . 9 ) + 1.6 3.0 + 3.0 + 1 . 6

(3.0 + 3.0) + 1 . 6 6.0 + 1 .6

7.6

pages 51-54: 2-^'

(h) (8 + 7 ) + 4 + (3 + 6)

Answers to Oral Exercises 2-4c; pages 53-54: -- - True,

True,

True,

True, Fa1 se

Fa1 se

True,

True,

True,

True,

True,

True,

associative property of multiplication

commutative property of multiplication

commutative property of addition


commutative property of addition commutative property of addition (twice)


commutative property of multiplication and commutative property of addition

associative property of multiplication

associative property of addition

Answers - to Problem Set 2-4c; pages 54-57: -- 1. This problem is intended to serve the same purposes for

the properties of multiplication which Problem 20 of

Problem Set 2-3b served for the addition properties.

(a) 4 x 7 ~ 2 5 4 x 2 5 ~ 7 ( 4 x 25) x 7 100 x 7

700

page 2-4

This problem i s a reminder t h a t add i t ion p rope r t i e s a r e not t o be fo rgo t ten while mu l t i p l i ca t i on proper t i es are a t the cen te r of a t t e n t i o n .

( d ) 2 x 38 x 50 2 x 50 x 38

( 2 x 50) x 38

100 x 38

3800

( f ) ( i x 43) x 6

page 54: 2-11

(j) ( 4 x 8) x (25 x 5 ) 4 x (8 x (25 x 5 ) ) 4 x ((25 x 5 ) X 8) 4 x (25 x (5 x 8 ) ) ( 4 x 25) x (5 x 8 )

100 x 40 4000

(k) ( 3 X 4) X (7 X 25) The student w i l l probably

3 x ( 4 X (7 x 25) ) give an answer such a s 3 X ( ( 7 X 25) X 4) ' ~ u l t i p l y 4 times 25 and 3 X (7 .X (25 X 4 ) ) ge t 100; then multiply 3 3 x (7 x 100) times 7 and ge t 21; then

( 3 x 7 ) x 100 multiply 21 times 100 and

21 x 100 ge t 2100. 'I

21 00

(1) Here i s an exercise i n which there i s no "easiest" way,

that is, regrouping i s not involved. 12 x 14 = 168

This way of doing the calculat ion i s preferable only i n t h a t it involves only one' d i g i t numbers u n t i l the simplest form i s written.

(n) 6 x 8 x 125 6 x (8 x 125) 6 x ~ O O O *

6000

pages 54-55: 2-4

Observe tha t i n t h i s case the or ig ina l form of the problem i s the bes t from which t o work.

The f i r s t forms of each pa r t of the problem a re eas ier t o compute because repe t i t ion of a p a r t i a l product i s involved i n each case. Thus the recurring p a r t i a l products can be copied a f t e r t h e i r f i r s t writing.

These problems are the f i r s t i n which a variable occurs. It I s - not the Intention t o introduce "variable" now, b u t

only t o have the student replace "t" w i t h the correct number. "Variable" w i l l be discussed i n Section 2-6.

( a ) t = 5 (b) t = 8

1 ( c ) t = 15 3

(d ) t = (3 + 1 ) o r lip, the commutative property of mult ipl icat ion i s the important par t .

( e ) t = 3.7 (f) t .= .5

(g) t = 7.2 i- 5 o r 12.2

pages 55-57: 2-4

(h) t = 6. We expect some answers of t = 4, but this is not an example of the commutative property, - since subtraction is not commutative.

- (k) t = 4 (1) t = 6.2 (m) t = 16. Again, t = 4 is wrong. Division is not

commutative.

4. No. Have students give counterexample, such as

(a) 8 + 4 # 4 + 8 (b) 8 - 4 # 4 - 8

( c ) (8 - 4) + 2 # 8 + (4 + 2) (d) (8 - 4) - 2 # 8 - (4 - 2)

Problems 5 through 10 are difficult and are included only for use with the better students.

5. 2 @ 3 = 2 + 2 ( 3 ) = 8

3 @ 2 = 3 + 2(2) = 7 Not commutative since 2@ 3 # 3 Q 2

6. 2 X 5 = ( 2 + 1 ) ~ ( 5 + 1 ) = 1 8 5 X 2 = (5+1) x ( 2 + 1 ) =i8 Yes, it I s commutative. Don't expect the student to prove this, but he should be able to furnish several examples.

*7. (2@3) @4 = (2 + (2 x 3))@4 = 8 + (2 x 4) = 16 2 @ (3 @4) = 2@ (3 -F (2 x 4)) = 2 @ 11 = 2 -F 2(11) = 24

No, it is - not associative.

2 X (5 X 3 ) = 2 X ( ( 5 + 1)(3 + 1)) = 2 X 24 = (2 + 1)(24 + 1) = 75

No, it I s not associative.

*9 and *lo. " ~ e e p the instructions simple" should be the cau-

tion for all except the exceptional student.

pages 57-61: 2-5

2-5. - The Distributive Property.

The properties of addition and multiplication studied in the

previous section appear symmetrical in form and do not really

reveal anything different about the two operations. Here the

student discovers from his number facts that "multiplication is distributive over addition," that is, that there is a definite connection between the operations. Although we mean t h e dis-

tributive property of multiplication over addition," throughout

the course we shall usually shorten this to 'the distributive

property." It is not necessary that the student immediately

grasp the significance of the full statement of the property.

An example is given to show that addition is - not distributive

over multiplication.

Again we use the spiral technique of presentation. One form

of the distributive property, a(b + c) = ab + ac, I s given;

then after some experience with this form it is presented in the

form ab + ac = a(b + c). The emphasis here is on changing back

and forth between indicated sums and indicated products. Later,

in Chapter 4, other forms, (b + c)a = ba + ca, ba + ca =

(b + c)a, are studied and used to simplify certain expressions. Even later, in Chapter 13, the distributive property is applied

to the problem of multiplying polynomials and factoring poly-

nomials. In the meanwhile many examples of the use of the pro-

perty are scattered throughout the exercises.

Answers to Oral Exercises 2-5a; pages 60-61:

1. True, this does illustrate the distributive property.

2. True, this does illustrate the distributive property. 3. True, this does illustrate the distributive property.

4. False 5. False This, in fact, illustrates that addition Is not

distributive over multiplication.

6. True, this does not Illustrate the distributive property. 7. True, this does illustrate the distributive property.

8. Indicated product 11. Indicated product 9 . Indicated sum 12. Indicated sum 10. Indicated sum 13 . Indicated product

pages 61-63: 2-5

Answers t o Problem S e t 2-5a; pages 61-62: - -- 1. 6(8 + 4) = 6(8) + 6(4) 2 . 9(7 + 6 ) = 9(7) + 9(6) 3 . 0(8 + 9 ) = 0(8) + o(9) 4. 9(8 + 11) = 9(8) + 9(11) 5. 5(8 + 4) = 5(8) + 5(4) 6 . 7(2 + 8) = 7(2) + 7(8) 7 . 3(80 + 3 ) = 3(80) + 3 ( 3 ) 8. 4(100 + 7 ) = 4(100) + 4(7)

9. 13(10 + 1 ) = 13(10) + 13(1) 10. 18(20 + 2) = 18(20) + 18(2)

Not t r u e Not t r u e

True

True . D i s t r i b u t i v e p r o p e r t y i s u s e d .

Not t r u e

Yes. D i s t r i b u t i v e p r o p e r t y i s u s e d .

Y e s . D i s t r i b u t i v e p r o p e r t y i s u s e d .

Not t r u e

Answers t o Oral E k e r c i s e s 2-5b; page 63: --

pages 63-65: 2-5

5 . 4(6 + 4) = 4(6) + 4($) 1 6. 6 ( 1 + 1 ) = 6 ( $ ) + 6 ( r ) 7

2 4 + 3 3 + 2

Answers t o Oral Exercises 2-5c; page 65: -- - 1. 2(3 + 5)

1 6 . ^( 5 + 3 )

2 . 18(3.2 + .8) 7. f(8 + 4 )

3 . (3.1)(7 + , 3 ) 8 . i4 ( .6 + . 4 )

4. 6(19.2 + .8) 9. 9(76 + 4)

5. 3(37 + 3 )

Answers t o Problem Set 2-5c; pages 65-66: - -- 1. 110(100) = 11,000 11. 16 + 40 = 56

2. 12($) + 12(+ - 7 1 2 . 0(17 + 83) = 0

3 . 27(1) = 27 13. 88(200) + 88(1) = 17,688 1

4 . $1) - 3 14. g(9) 8 = 8

5. 3 0 ) = 3 15. 9(1) = 9 2

6. 6(-) + 6(J) = 13 16. 7(4) = 28

7. g(20) = 180 17. 8(100) - 800

8. 100(100) = 10,ooo 18. (7($) + 7 ( 9 ) + 7(5) 1 4 2

9 . 5 ( 9 + 5 ( - ) = 77 3

7 0 ) + 7(5)

10. 7(8) + 7($ = 60 7(6) = 42

pages 65-66: 2-5

pages 66-69: 2-5 and 2-6

+ 3 ( 8 + 7 ) *46. ( a ) (p

V a r i a b l e s .

The aim of t h i s s e c t i o n is t o acquaint t h e pup i l with one meaning of t h e word v a r i a b l e . A t t h i s p o i n t we i n s i s t t h a t "n" o r "x" , o r whatever l e t t e r i s used a s t h e v a r i a b l e , must be t h o u m t - of a s t h e name of a d e f i n i t e number al though we may not have ve ry much Information about t h a t number. I n some cases , such as i n t h e example discussed. i n t h e t e x t , t h e number may be unspec i f i ed because what we want t o s a y about it i s t h e same f o r every number i n a given se t . This i s always t h e c a s e when we a r e i n t e r e s t e d i n t h e p a t t e r n o r form of a problem r a t h e r than i n t h e answer. I n o t h e r c a s e s t h e number may be unspeci f ied because we do not know what it i s a t t h e o u t s e t bu t w i l l f i n d it out l a t e r . Var iab les used i n t h i s con tex t a r e u s u a l l y c a l l e d "un- knowns." I n any c a s e t r y t o avoid t h e concept of a v a r i a b l e a s something t h a t v a r i e s over a s e t of numbers.

The d i scuss ion of t h e example would not have been changed i n any e s s e n t i a l way i f we had decided t o denote t h e chosen number by some l e t t e r o t h e r than n .

The set of numbers from which a v a r i a b l e may be s p e c i f i e d i s c a l l e d d o m a i n ' by some, r a n g e by o t h e r s , depending l a r g e l y upon the p o i n t of view from which t h e v a r i a b l e and i t s set a r e be ing seen . There a r e p o i n t s of view, then , t o support t h e choice of e i t h e r term. Since t h e most n a t u r a l connection f o r many t e a c h e r s t o make, when a v a r i a b l e and i t s set a r e mentioned, i s

t o s e e t h a t v a r i a b l e a s t h e i n d e p e n d e n t " v a r i a b l e I n a funct ion r e l a t i o n s h i p , t h e name "domain" f o r i t s s e t comes e a s i l y t o mind. It must be emphasized, however, t h a t t h e v a r i a b l e need not be seen a s t h e i n d e p e n d e n t v a r i a b l e i n a func t ion r e l a t i o n s h i p , but may i n f a c t be considered as t h e "dependent" v a r i a b l e .

page 70: 2-6

Answers t o Oral Exercises 2-6a; page 70: -- -

2n + 4 Five more than some number. Two l e s s than some number. Four times some number. Some number divided by 5. Three more than twice some number. Two l e s s than three times some number. Seven times the r e s u l t of f inding two l e s s than some number. Some number divided by 4 and the r e s u l t increased by 5. The product of f i v e more than some number and two less

than the or ig ina l number.

page 71: 2-6

Answers - t o Problem Set 2-6a; pages 71-72: --

Point out the use of the associat ive and commutative proper- t i e s i n Problem 8.

14. 3n - 7 Encourage students t o use

15. *x+6 d i f fe ren t l e t t e r s ra ther than 11 I t always "nl' or always x .

pages 71-75: 2-6

21. Four more than e igh t times a number. 22. Four l e s s than the quot ient of twice a number divided by

three .

23. The product of e igh t and the d i f fe rence obtained by sub-

t r a c t i n g 5 from 2 times a number. 24. Take a number, sub t r ac t 5, mul t ip ly by 12, add 0,

and divide by three .

The s teps i n t he proposed example a r e

The l a s t phrase i s a numeral f o r n + 2 .

The student may wonder why we i n s i s t on wr i t ing 3n + 12 = 3(n + 4 ) . E i the r method w i l l , of course, l ead t o the same r e s u l t . The completion of t h i s as

and the subsequent s impl i f ica t ion lead t o t he numeral n + 2

with l e s s computation than the f i r s t method. Perhaps i t w i l l s a t i s f y most s tudents who r a i s e the question

i f you point out t h a t the f i r s t method br ings out t he pa t t e rn while the second method tends t o o b l i t e r a t e t h e p a t t e r n .

Some teachers have found it he lp fu l , i n in t roducing the notion of var iab les t o t h e i r s tudents , t o play a number game i n c l a s s i n addi t ion t o t h e mater ia l I n t he t e x t . Another successful method has been t o use such a game a t t he board.

pages 75-76: 2-6

Example: "Choose a number from some s e t S - such a s , f o r i n s t a n c e , t h e whole numbers between 1 and 30 - add 3, m u l t i p l y by 2, and s u b t r a c t twice t h e number chosen."

D i f f e r e n t p u p i l s t r y t h e game a t t h e board wi th d i f f e r e n t numbers, and always o b t a i n 6 . Others may be I n s t r u c t e d t o l e a v e t h e numerals i n I n d i c a t e d form, another may use "number" I n s t e a d of a s p e c i f i c numeral, and y e t o t h e r s may use a v a r i a b l e l i k e n o r ' x from t h e beginning. The board may look l i k e t h i s :

4 5 number n

7 5 + 3 number + 3 n + 3 1 4 2 ( 5 + 3 ) 2(number + 3 ) 2(n + 3 )

or 14 2 - 5 + 6 2(number) + 6 2n + 6 6 2 . 5 + 6 - 2 - 5 2 ( n u m b e r ) + 6 - 2 ( n u m b e r ) 2 n + 6 - 2 n

Or 6 6 6 6

This example uses t h e d i s t r i b u t i v e p roper ty , which t h e s tuden t s have seen, but it a l s o uses a s s o c i a t i v i t y and commutativity with one s u b t r a c t i o n , which t h e y have not seen . The opera t ions with numbers a r e q u i t e s imple, however, and s o t h e "2n - 2n" should r e a l l y not g ive any t r o u b l e . It i s c e r t a i n l y not worth making a f u s s o v e r . If t h e s u b t r a c t i o n is, f o r some reason, l i k e l y t o g ive t r o u b l e , t h e game may always be played wi th an example such a s t h e one i n t h e t e x t which invo lves no s u b t r a c t i o n .

Answers - t o Problem S e t 2-6b; pages 75-77: -- 1. 2 ( t + 3)

5. Both forms are c o r r e c t . The second i s found from t h e f i r s t by uae of t h e a s s o c i a t i v e p r o p e r t y of m u l t i p l i - c a t i o n .

5. Ne i the r form i s c o r r e c t . 2 ( a + b ) and 2a + 2b a r e c o r r e c t forms.

6 . ( a ) 9 + 32 = ? ( l o o ) + 32

pages 76-77:

P(l + rt) = 500(1 + 0.04(3)) = 500(1 + .12) = 500 + 60 = 5bO

- 2 48 a - &&$ = 96 - 4 r - 1

-* is the final number.

n. Yes, we get the original number.

0. Every answer is zero!

True f o r all values of x! True for all values of x! Fa1 se True f o r all values of x! False. Don't be concerned about the negative result, but

caution those who want to think of subtraction as being commutative.

pages 77-79

1 4 . False

15. True

16. True f o r a l l values of x!

17. False

18. False 19. False

20. True f o r a l l values of x!

Answers - t o Review Problem - Set; pages 79-81: 6 5 + 1

1. Many possible answers, f o r example: 4 - 1, 7, 2

2 . A "common name" of a number is a numeral most often

used t o represent the number. For example, "2" 6 i s a common name of 5 - 3, 7, e t c .

3 . We do the multiplication and division f i r s t , then the addition and subtract ion.

5. ( a ) t rue

( b ) f a l s e

( c ) f a l s e

6. "<I'

( d ) f a l s e

( e ) t rue

( f ) f a l s e

8. A binary operation I s an operation tha t i s applied t o

only two numbers a t a time.

9 . ( a ) Yes ( b ) Yes

(4 No

( d ) Yes

( e ) Yes

(f) No

10. ( a ) Associative property of addition

( b ) Neither, i t i a the commutative property of addition

which I s i l l u s t r a t e d .

( c ) Associative property of multiplication

( d ) Neither, the commutative property of multiplication

i s Involved i f we replace "j" by "=" . The sentence, as it stands, i s f a l s e .

page 80

True, commutative property of addition and commutative property of multiplication True, commutative and associative properties of addition. True, multiplication property of one Fa1 se

True, commutative property of multiplication and distributive property Fa1 s e True, multiplication property of one

Fa1 se True, none of the properties are Involved.

19(& + (b) 15(12)

14. A variable is a numeral which represents a definite, but unspecified, number chosen from a given set of numbers.

page 81

Simplified: (4)(2)(n + 1) +

8

n + 8 - The t r i c k i s t o add eight t o each number n.

Suggested -- Test Items

Insert parentheses in each of the following so that the resulting sentence is true:

State the property illustrated by each of the following true sentences:

(a) 7 x 3 = 3 x 7 (d) 7 + (5 + 4) = (7 + 5 ) + 4 (b) 5(6 + 2) = 5(6) + 5(2) (e) 9 + (3 + 4) = (3 + 4) + 9 (c) (8 x 2) x 3 = 8 x (2 x 3)

Which of the numerals listed below are names for 6?

6 + 6 a ) 7 (d 1 1 + (1 + 1) + 15

3

Which of the following sentences are true and which are false?

Show the steps in finding the simplest name for the number indicated:

Show how you would use the associative, commutative, and

distributive properties to perform each of the following computations as simply as possible:

(a) 61 + (17 + 3 9 (b) (12.8)(7) + (12*8)(3) (c) (5 x 13) x 20

A certain number n is multiplied by 5, then increased by 3, and this result is multiplied by 2. Which of the following

open phrases describes this statement?

Given that the domain of x is the set (0,1,2), find the

value of the phrase

for each value of x.

Use the numbers 3 , 7 , and 5 to illustrate

(a) the associative property of multiplication, (b ) the distributive property.

Show how the distributive property can be used to find each of the following products :

a ) 4 x 5^ (b) 15 x 1006

Answers - to Suggested Test Items -- (a) 5 x (4 + 3) = 35 (d) 7 x (2 + (2 x 3)) = 56 (b) (5x 4) + 3 = 23 (e) (7 x 2) + (2 x 3) = 20

(4 7 (a) the

(b) the ( c ) the ( d ) the (e) the

( 2 + 2) x 3 = 84 (f) ((7 x 2) + 2) x 3 = 48

commutative property of multiplication distributive property associative property of multiplication associative property of addition commutative property of addition

The numerals in (b), (d), and (e) are names for 6. 6 + 6 12

(a) 7 =

(c) 3 x 1 + 1 = 3 + 1 = 4

(a) true (b) true ( c ) false

(d) true ( e ) true (f) false

3 (a) 6y + (17 + 3:) = $ + (3y + 17) commutative property of

addit ion

= (6y + %)+ 17 associative property of addition

(12.8) (7) + (12.8) (3) = 12.8(7 + 3) distributive property

(5 x 13) x 20 = (13 x 5) x 20 commutative property of multiplication

= 13 x (5 x 20) associative property of multiplication

= 13 x 100 = 1300

The phrase "2(5n + up in the following

) distributive property 8

3)" is the correct one. It is built

sequence :

3, 2(5n + 3). 0 + 4 x = 0, we get 0 + Ã‘5 = 2.

1 + 4 5 x = 1 , weget 1 + ~ = 1 + ~ .

2 + L 2 + 3 x = 2, we get 2 +,T = 5.

(b) 15 x 1006 = 15 x (1000 + 6) = 15(1OOO) + 15(6) = 15000 + 80 = 15080

1 ( c ) 6 x (6 + 4) = ( 6 x $) + (6 x

= 3 4 - 2

= 5

Chapter 3

OPEN SENTENCES AND TRUTH SETS

The p r o p e r t i e s of opera t ions which were* verba l i zed i n Chapter 2 w i l l be formalized i n Chapter 4 i n symbolic form. I n prepara t ion f o r t h i s fo rmal iza t ion we first e n r i c h o u r vocabu- l a r y . The concept of a sentence , from Chapter 2, i s enlarged i n three ways: ( 1 ) We i n c r e a s e t h e v a r i e t y of r e l a t i o n s which our sentences can express , s o t h a t i n e q u a l i t i e s a r e included along with equat ions . ( 2 ) We w r i t e open sen tences which involve v a r i a b l e s , and f o r which t h e no t ion o f a t r u t h set becomes important . It I s e s s e n t i a l t h a t t h e s t u d e n t cons ide r both equat ions and i n e q u a l i t i e s a s sen tences , as o b j e c t s of a lgebra with equal r i g h t t o o u r a t t e n t i o n , and a s equa l ly i n t e r e s t i n g and u s e f u l types of sentences . ( 3 ) We cons ide r compound sentences a s w e l l a s s imple sentences . While n o t a l l of these concepts a r e immediately necessary f o r s t a t i n g t h e p r o p e r t i e s of t h e opera t ions on t h e numbers of a r i t h m e t i c , i t is worthwhile t o in t roduce them toge the r , and they w i l l be used many times throughout t h e course .

Although t h i s chap te r i s devoted e n t i r e l y t o sen tences , i t must be emphasized t h a t we do n o t s tudy sentences f o r t h e i r own sakes. A s always, our main goal is t h e understanding o f t h e p r o p e r t i e s of t h e opera t ions , and sentences happen t o be u s e f u l language devices f o r recording these p r o p e r t i e s . S tuden t s quickly become enamoured of t h e process of s o l v i n g sentences . This i s good, but be s u r e t h a t t h i s enthusiasm i s d i r e c t e d beyond t h e mere fun of manipulat ing sentences . A f t e r a l l , sentences a r e only p a r t of t h e language, b u t no t t h e subs tance , of a lgebra .

The t eacher may want t o read, a s a genera l r e fe rence f o r the work of t h i s chapter , Haag, S t u d i e s - i n Mathematics, Volume I11 , S t r u c t u r e k f 7 m, Chapter 2, S e c t i o n 2. -

3-1. Open Sentences. The experimentat ion wi th t h e example "2x + 3 = 18" is

supposed t o sugges t a sys temat ic way of guess ing va lues of t h e

pages 83-85: 3-1

v a r i a b l e which w i l l make t h e sentence t r u e . The method might a l s o sugges t how one might decide whether o r n o t a l l such values have been found. For example, a value of x g r e a t e r than

1 w i l l g i v e a number g r e a t e r than 2(%) + 3 and a value smal ler 1 than QÃ w i l l g i v e a number l e s s than 2(%) + 3. The p r o p e r t i e s

of o r d e r which a r e suggested he re w i l l be taken up l a t e r i n Chapter 9.

Answers t o Oral Exerc i ses - 3-la; page 85:

True True True True True

2

3 - 2

1

3 - 2

3

2

1

4

3 7 - 2

True F a l s e F a l s e F a l s e

3 8 7 2 3"

9 2 0

1

11 T 7 - 2 10 3

Answers & Problem S e t 3 - l a ; page 85:

1. (a) 5 i s n o t a t r u t h number of t h e sentence .

( b ) 16 i s a t r u t h number.

( c ) 3 is a t r u t h number.

( d ) 1 is a t r u t h number.

( e ) 5 is a t r u t h number. h he a l e r t s t u d e n t may observe t h a t 4 i s a l s o a t r u t h number f o r t h i s sentence . )

pages 85-88: 3-1

( f ) 5 is n o t a t r u t h number.

(g) 1 2 is a t r u t h number.

2. For f i n d i n g a t r u t h number f o r each of t h e s e sen tences , emphasize reasonable guess ing procedures t h a t " c e n t e r in t t on the t a r g e t . Systematic s o l u t i o n of equat ions w i l l be d e a l t with i n l a t e r chap te r s .

Answers -- t o Oral Exerc i ses 3 - lb ; page 88: - 1. ( a ) True (1) F a l s e

( b ) F a l s e ( J ) True ( c ) F a l s e ( k ) F a l s e ( d ) F a l s e ( 1 ) True ( e ) True (m) T r u e ( f ) T r u e ( n ) F a l s e ( g ) F a l s e ( 0 ) F a l s e ( h ) True ( p ) F a l s e

pages 89-90: 3-1

Answers

[ 5) A l l

9f A 11

A l l

( 3 ) The

( 0

The The A l l

The 2 I

[ G I 9s The A l l

A l l

The

( 0 1

Answers

1.

t o Problem S e t 3 - lb ; page 89: - -- numbers g r e a t e r than 4

number g r e a t e r than 5 numbers l e s s than 8

s e t of a l l numbers

s e t of a l l numbers set of a l l numbers numbers g r e a t e r than 2

s e t o f a l l numbers

set of a l l numbers numbers l e s s than 27 numbers l e s s than 5 set of a l l numbers except ze ro

(f) 144 (g) 64 (h) 196 0) 1

( J ) 0

pages 90-93: 3-1 and 3-2

Answers & Problem S e t 3 - l c ; page 91:

1. 3 6. o

3-2. Truth S e t s of Open Sentences.

An open sentence involving one v a r i a b l e has a " t r u t h set" defined as the set of numbers f o r which it I s true. We have no need a t t h i s time t o in t roduce a name f o r t h e s e t which makes a sentence f a l s e . The p h ~ a s e " s o l u t i o n s e t " is a l s o used f o r " t r u t h set," p a r t i c u l a r l y f o r sentences which a r e i n t h e form of equat ions. We s h a l l use " s o l u t i o n s e t " l a t e r , bu t we want t h e s tuden t t o use " t r u t h set" long enough t o g e t i t s f u l l s l g n i f i - cance.

U n t i l the i n t r o d u c t i o n of t h e r e a l numbers i n Chapter 6, when a sentence is w r i t t e n and no domain is s p e c i f i e d , t h e domain may be i n f e r r e d t o be t h e set of numbers of a r i t h m e t i c f o r which the given sentence has meaning. Note, however, t h a t when t h e s tuden t begins t o t r a n s l a t e "word problems" i n t o open sen tences , he w i l l sometimes f i n d inheren t i n t h e problem, bu t n o t s p e l l e d ou t f o r him, some f u r t h e r l i m i t a t i o n upon the domain. Thus t h e agreement s p e c i f i e d i n t h e t e x t regarding t h e domain refers t o sentences only, and should n o t be extended t o inc lude "word problems. "

The t 'eacher may want t o t ake a moment o f c l a s s time t o be c e r t a i n t h a t t h e s t u d e n t s remember c l e a r l y t h e s e t of numbers of

pages 93-94: 3-2

a r i t h m e t i c . T h i s understanding can be re in fo rced soon ( i n the n e x t s e c t i o n ) by t h e graphing of t h i s set.

Answers -- t o Oral Exerc i ses 3-2a; pages 93-94:

1. (0) 8. (01 15. (21

2. (1,2) 9. 9: 16. (21

3. (11 10. (0) 17. (21

4. (0,1) 11. 9: 18. (1,2)

5. (2) 12. (1,2) 19. (2)

6 . (1) 13. (1,2] 20. fif

7. (2) 14. Of

Answers &g Problem S e t 3-2a; pages 94-96:

3. (a) T = (2,3,4,5,6) *(d ) T is t h e s e t of numbers

F = (7,8,91 g r e a t e r than 4 and less than 7. F is t h e s e t of

( b ) T = (0) numbers g r e a t e r than o r = (10,20,30, tO,50) equal t o 7.

* ( e ) T is t h e set of numbers ( c ) T = (3,5) less than 7. F is the

F' = (7,9,11) s e t of numbers g r e a t e r than o r equal t o 7, but l e s s than 10.

*(f) T is the empty s e t . F is t he s e t o f numbers g r e a t e r than 8.

4. ( a ) T = ( 0 , 1 , 2 , 3 , 4 ) * ( e ) T = ( 1 , 2 , 3 , 4 , 5 , 6 ) o r T

(b) T = ( 6 ) i s t he se t of counting

(4 T = or numbers l e s s than 7. * (d ) T = Of * ( f ) T = (0,1,2,3,4,5,6)

6. I n t h i s exe r c i s e encourage s t uden t s t o g ive a v a r i e t y of examples.

( a ) Examples a r e x + 5 = x + 4; x + 2 < x + 1

( b ) Examples a r e x = 5; x + 7 = 10

( c ) Examples a r e 2y + 4 = 2(y + 2) ; x + 3 = 3 + x

( d ) Examples a r e x > 5; 3x + 2 > 1 4 This exe r c i s e might be a good one f o r c l a s s d i scuss ion .

7. ( a ) T = (2) ( c ) T = [3,4,5) ( b ) T = t h e empty set ( d ) T = ( 0 , l )

8. (a ) T = ( 2 ) ( c ) T = s e t o f a l l numbers 5 g r e a t e r than and less

than 5. ( b ) T = ($1 ( d ) T = s e t o f a l l numbers

g r e a t e r than 0 and less

than 2.

The s e t s i n ( a ) and ( b ) a r e f i n i t e .

9. I n connection wi th t he se exe r c i s e s t he t e ache r should bear i n mind t h a t formal methods f o r s o l u t i o n of equations and i n e q u a l i t i e s have n o t been developed a s y e t , s i nce they depend upon p r o p e r t i e s of t h e r e a l numbers t o be presented i n l a t e r chap te r s . Somewhat

sys temat ic guesswork i s the s t u d e n t ' s method and the s t r e s s should r e s t more upon t h e f a c t t h a t t h e value i n q u e s t i o n i s indeed i n t h e t r u t h set, than upon t h e dev ice used t o d i scover it.

I n e x e r c i s e s ( b ) , ( c ) , ( 5 ) and ( k ) above, the s tuden t may need t o be reminded t h a t d i v i s i o n by zero has been excluded i n t h e formation o f r a t i o n a l numbers. L a t e r i t w i l l be s t r e s s e d t h a t s i n c e zero has no r e c i p r o c a l , a n express ion wi th denominator 0 does n o t r ep resen t any number.

Answers - t o Problem -- S e t 3-2b; page 97:

1. ( 1 ) 2. The set of a l l numbers g r e a t e r than 1

3. The s e t of numbers from 0 t o 1 I n c l u s i v e 4. The set of a l l numbers g r e a t e r than o r equal t o 1

5. The s e t of numbers l e s s than 1

b- Of 7. The set of a l l numbers 8. The s e t of a l l numbers 9. The set o f a l l numbers

10. The s e t of a l l numbers g r e a t e r than o r equal t o 1

11. The s e t of a l l numbers less than o r equal t o 2

pages 97-98: 3-2 and 3-3

*12. The s e t of numbers - lo o r g r e a t e r 7 1 *13. The s e t of numbers l e s s than o r equal t o 5-

*14 ( F 1

15. 0'

3-3. Graphs of Tru th S e t s . --- We s h a l l soon s t a r t saying "graph of t h e open sentence"

Ins tead of t h e more clumsy but more n e a r l y p r e c i s e ' g raph o f t h e t r u t h s e t of t h e open sentence . I1

I n graphing sentences whose t r u t h s e t Is $8 do n o t f u s s over the ' p l o t t i n g of t h e empty set . E i t h e r no graph a t a l l o r a number l i n e wi th no p o i n t s marked is a l l r i g h t .

For convenience i n doing problems involving t h e number l i n e , you might f i n d i t h e l p f u l t o d u p l i c a t e s h e e t s of number l i n e s f o r the p u p i l s * use .

Answers - t o Problem S e t 3-3; pages 98-99:

We have no t included oral work i n t h i s s e c t i o n because w e f e e l t h a t t h i s can r e a d i l y be centered around t h e examples i n the t e x t , which t h e t eacher should review c a r e f u l l y wi th t h e c l a s s .

I f t h e s t u d e n t shows on h i s graph an arrow t o t h e l e f t f o r

p a r t s ( e l , (i), ( J ) , and ( o ) , simply p o i n t o u t t h a t t h e domain of t h e v a r i a b l e f o r these sentences is the numbers of a r i t h m e t i c . L a t e r when t h e r e a l numbers, t h e opera t ions upon them, and some of t h e p r o p e r t i e s o f t h e s e opera t ions a r e known, t h e s t u d e n t w i l l be a b l e t o work with confidence wi th sentences i n t h i s extended domain.

2. ( a ) Yes

( b ) No

( c ) Yes--assuming t h a t t h e d o t on t h e number l i n e has t h e 7 coord ina te 5 .

( d ) NO, t h e graph is of whole numbers only--no such r e s t r i c - t i o n has been p laced upon x. This is a good time t o re-emphasize t h a t t h e domain o f a v a r i a b l e when unspeci- f i e d is t h e s e t of a l l numbers o f a r i t h m e t i c f o r which t h e sen tence has meaning.

( e ) Yes

pages 99-102: 3-3 and 3-4

3. Accept and encourage a va r i e ty of responses I n these exercises . Possible answers a r e : ( a ) x = 2: 2x + 4 = 8 ( c ) x 2 4; 3x + 5 2 17 (b ) x # 2 ; x < 2 o r x > 2 (d ) x 2 1 ; 5 x + 1 2 b

3-4. Compound Open Sentences and Their Graphs.

The s tudent has d e a l t with simple sentences, f inding t h e i r t r u t h s e t s and graphing these . With compound sentences, a s with simple sentences, the emphasis should be on what c o n s t i t u t e s a

t r u t h value r a the r than on any technique of f inding the t r u t h values. Frequent use of the compound sentence is made throughout

the course, so t h a t f u r t h e r p rac t i ce i n t h i s a rea awai ts the

student . The word "clause" is used t o denote a sentence which is p a r t

of a compound sentence, j u s t a s In the corresponding s i t u a t i o n In English. The word is convenient but not very important.

Answers t o O r a l Exercises 3 - h ; page 102: -- 1. Yes

Yes Yes The l e f t c lause "8 - 1 = 7" and the r i g h t c lause "5 + 4 = 9" a r e both t rue . Therefore, the compound sentence is t rue .

2. Yes No No The c lause "11 + 12 = 25" i s f a l s e . Therefore, the compound sentence "13 - 7 = b and 11 + 12 = 25" is f a l s e .

Both c lauses a r e f a l s e . Therefore, t h e compound I1 1 sentence % + 3$ = 9 and 9 + 18 = 37" is f a l s e .

Pages 102 -103: 3-4

4. ( a ) f a l s e The clause "8 + 19 = 17" is fa lse .

( b ) f a l s e The clause "16 # 8 + 8" i s f a l s e .

( c ) f a l s e The clause " 9 - 6 = 2" i s f a l s e .

(d ) f a l s e Both clauses a re f a l s e .

( e ) f a l s e Both clauses a r e f a l se .

Answers - t o Problem Set 3-4a; pages 102-103: -- both clauses a re t rue . both clauses a r e f a l s e . both clauses a re t rue. second clause is fa l se . both clauses a re t rue . f i r s t clause i s fa l se . second clause is fa l se . second clause is fa l se . second clause i s f a l s e . second clause is f a l s e . both clauses a r e t rue. both clauses a r e t rue .

The t ex t defines the t r u t h s e t of a compound sentence with

the connecting word "or" a s consisting of a l l those numbers t h a t a r e i n -- a t l e a s t one of the t r u t h s e t s of the clauses which make up the compound sentence. It is par t icu lar ly Important,

pages 103-105: 3-4

not only i n the i n t e r e s t of c l a r i t y he re , but f o r t h e sake o f h i s l a t e r work i n mathematics, t h a t t h e s t u d e n t be g iven a c a r e f u l i n t r o d u c t i o n t o t h e phrase " a t l e a s t . " To have him

explore such synonomous phrases a s "no t less than" may h e l p p i n down the i d e a .

Answers -- t o Ora l E x e r c i s e s 3-4b; page 105: - Yes Yes

Yes Both c l a u s e s a r e t r u e .

Yes No Yes The first c l a u s e is t r u e .

No No No Both c l a u s e s a r e f a l s e .

t r u e The first c l a u s e i s t r u e . true The second c l a u s e is t r u e . t r u e The second c l a u s e is t r u e . f a l s e Both c l a u s e s a r e f a l s e . t r u e Both c l a u s e s a r e t r u e ; t h e s t u d e n t should no te , however, t h a t s i n c e t h e f i r s t c l a u s e is true, t h e

t r u t h of t h e sen tence is e s t a b l i s h e d wi thout c o n s i d e r a t i o n of t h e second c l a u s e .

pages 105-106: 3-4

Answers - t o Problem -- Set 3-4b; pages 105-106:

1. Each sentence which is t rue is t rue because a t l e a s t ,

one clause i s t rue.

( a ) T ( f ) F, both clauses a re f a l s e ( b ) T (I31 T ( c ) Fj both clauses ( h ) T

are f a l s e

( d l T (1) T

(4 T ( j ) F, both clauses a re f a l se

2. ( a ) The s e t consisting of 5 and a l l numbers greater than 6

(b) The s e t consisting of a l l numbers l e s s than o r equal to 3

( c ) The s e t of a l l numbers

(4 Of 1 ( e ) The s e t of a l l numbers l e s s than 7 and the

number 1

3. ( a ) T r u e

( b ) False because "5(8) < 5" is f a l s e ( c ) True

( d ) False because both clauses a r e f a l s e ( e ) True

page 108: 3-4

Answers & Problem - Set 3-4c; page 108:

(The teacher may want to note that exercise 8 is the first graphing exercise-including examples in the text-in which the truth sets of the clauses of a compound sentence have a common value. If the student recalls the meaning of the phrase "at least," he will not find this troublesome. )

pages 108-110: 3-4

Further exercises i n graphing t r u t h s e t s of compound sentences with connective "or*' can be obtained from the preceding section, Oral Exercises 3-4b and Problem Set 3-4b.

Answers t o Oral Exercises 3-4d; page 110: --

Answers - t o Problem - Set 3-4d; page n o :

The empty se t .

The empty se t .

pages 110-111: 3-4

l , The empty se t . , . , 0 I 2 3 4 5 6

Answers - t o Problem - Set 3-4ej page 111:

The s e t of numbers equal , , - k-

t o o r grea ter than 3 0 I 2 3 4

The set which includes 1 A - a Ã . 1

and all numbers grea ter 0 I 2 3 4

than 3

pages 111-112: 3-4

The s e t of a l l numbers l e s s than 5 o r g rea te r than 7

The s e t of a l l numbers equal t o o r less than 3

The s e t of a l l numbers except 3 and 4

The s e t of all

nwnbers grea ter than 2

14. The s e t of a l l numbers . b

0 I 2 3 4

15. The s e t of a l l numbers l e s s than

two, a l l numbers grea ter than 4, and 3

17. The s e t of numbers between 2 and 3 I , /Â¥ > n ./ I I .

0 I 2 3 4 5 6

Summary.

The summary i s intended t o help the student make a quick r e c a l l of the concepts tha t have been studied i n the chapter.

pages 112-113

Answers - t o Review Problem Set ; pages 112-115: - It i s expected tha t the Review Problem Set may help the stu-

dent t o Improve h i s overal l understanding of mathematical sentences by giving him opportunity t o work w i t h a mixture of sentences less sorted in to "types" than the problem s e t s throughout the chapter have been.

1. ( a ) Yes (f) No ( b ) yes (g) Yes (c) No. 3 ( 4 ) - 2 < 7 is f a l s e (h) No (d) Yes (1) Yes

(e) Yes (The student should note tha t t h i s exercise i s

( J ) Y e s

an instance where the commutative property of addition enables him t o answer without ari thmetic calculations. )

(b) The s e t of a l l numbers (s) ( c ) The s e t of a l l numbers (h) The s e t of numbers equal

t o o r grea ter than 3

(d l (0,1) (i) The empty s e t

( 4 W ( j ) The s e t of a l l numbers grea ter than 2 and l e s s than 7

(j I : ; ;

0 1 2 3 4 5 6 7 8 0 1 2 3 The empty s e t .

page 113

(b) The set of a l l numbers l e s s than 13 a : : : : : : : ; : : : : - / :

0 2 4 6 8 10 12 13 14

( d ) The empty s e t I ,

0 I 2 3

( e ) The s e t of a l l numbers l e s s than 3 Â¥Ã‘Ã‘Ã‘iÃ‘Ã‘Ã

0 I 2 3

( h ) The s e t of a l l numbers grea ter than 6 I I , \

w

0 I 2 3 4 5 6 7

(1) The s e t of all numbers

(k) The s e t of al l numbers l e s s than 3 n

.d

0 I 2 3 4

(1) [I) I - I

0 I 2 3

(m) The set of all numbers , b

0 I 2 3 4

pages 113-115

( n ) The empty s e t I . 0 I 2 3

5. ( a ) True

( b ) True ( c ) False ( d ) False ( e ) False

6. ( a ) 11

( f ) True (g) False ( h ) True ( i) True (j) False

7. The sentences in ( a ) , ( c ) , ( d ) , and ( e ) are true for every value i n the domain.

8. ( a ) False

( b ) True

(c ) True

2 3 6 9. (a ) 7 x T =

3 ( b ) r ; ~ $ = %

11. ( a ) 3(10 + 3) = 3(10) + 3(3)

(b) 5(8) + 3 ( 8 ) = (5 + 3)8 1 .

( c ) ( 4 + -)6 = 4(6) + $6)

( d ) True ( e ) True ( f ) True

( d ) 3a + 5a = (3 + 5)a

( e ) 5a + 5b = 5(a + b)

(f) a (6 + b) = a(6) + ab

Suggested Test Items -- 1. Which of the following sentences a r e t rue and which a re

f a l s e ?

( a ) 5 s 6 + l ( c ) 8 + 2 > 3

2. Whlch of the following sentences a re t rue and which are f a l s e ?

6 - 3 3 ( a ) 3 > and 6 i 2 - 1

( c ) 6 + 1 = 4 + 3 o r 6 2 7

3 5 ( d ) + + $ > ~ + 7 and 4 2 5 . 3

3. Which of the following sentences a r e t rue and which a re f a l s e ?

( a ) (18 - 10) - 4 = 18 - (10 - 4)

(b) (18 - 10) - 4 # 18 - (10 + 4) ( c ) 3 + 4 < 8 o r 6 + 5 > 5 + 6 ( d ) 7 + 0 = 7 and 7(0) = 7 (e) 4 > 6 o r 5 + 2 = 1 0

( f ) 7 > 3 o r 17.813 + .529 = 8.777 + 18.442

4. Determine whether each sentence i s t rue f o r the given value of the var iable .

( a ) 3 t + 4 = 15; 2 ( c ) 20 - 2x 2 10; 5

(b) 4x - 3 < 7; 7 ~ 1 6 x 3 ~ (4 F + T # F + ~

5. If the variables have the values assigned below, determine whether the sentence i s t rue . (a) 3x = 4 + y, x I s 2 and y i s 2

(b) 5 x < 2 + y , x i s 3 and y i s 8

6. L i s t the t r u t h s e t of each of the following open sentences. The domain Is the s e t of numbers indicated. ( a ) x + 3 = 3x - 5; (2,4,6) (b) $ + 2 - 3~ = 0; (0,1,2,3)

7. Determine the truth sets of the following open sentences:

(a) 3x + 4 = 25 (c) 2x + 3 = 2 x + 5

(b) 2 x + 1 < 3 (d) 4 + x < 2 x + 1

8. Draw the graphs of the truth sets of the open sentences:

9. Draw the graphs of the truth sets of the compound open sentences: (a) x > 3 and x < 4 (c) x = 5 o r x < 4 (b) x 2 5 and x > 4 (d) x < 3 and x > 4

10. Which of the open sentences A, B, C, D, and E below has the same truth set as the open sentence "p 2 7"?

11. Write open sentences whose truth sets are the sets graphed below:

12. For each of the sentences In column I, select' the appropriate graph of its truth set in column 11.

I I1 (a) 6x = 18 I : : : = : : : Ã

0 1 2 3 4 5 6 7

(b) Y < 3 l : = : ; ~ ; ; 0 1 2 3 4 5 6 7

(4 b # 2 v : ~ ; : : : ; ^ 0 1 2 3 4 5 6 7

(dl t > 4 t : : ; ! ; ; ;

0 1 2 3 4 5 6 7

(e) d 2 2 and d < 5 b : : ~ : ; ; ;

0 1 2 , 3 4 5 6 7

(f) x > 4 1 ; : : c : : : b

0 1 2 3 4 5 6 7

(g) w < 2 and w > 4 1 : : : : : ; ;

0 1 2 3 4 5 6 7

13. If the domain of the variable is the set U = [2,4,6,8,10,12), find truth sets for the following open sentences: (a) 3x + 1 = 13 (c) 2 x < 2 0 and x + 4 = 4 + x (b) 2x = 10 (d) 2 x + 1 = 7 or 2 x - 1 = 3

True False

False Fa1 s e

False False

True

No No

Yes

(41

(7 1

Answers - to Suggested Test Items -- (4 True

( d l True

(4 TJxle (d) False

(d) False (e) False (f) True

(b) the numbers of (d) the numbers greater than or arithmetic less than 1 equal to 3

I a the empty set 0 I 2 3 4 5 6

10. A, C, E

11. Possible answers a r e :

(a) x < 5 (c) x = l or x = 4 (b) x > 2 and x < 4 (d ) X > 3

12. (a ) , ! : : : (d) t : : ; , . : : : Ã ˆ

0 1 2 3 4 5 0 1 2 3 4 5 6 7

c 1 *-+ (f ) I ! : : ; : : : Ã

0 1 2 3 4 5 0 1 2 3 4 5 6 7 I : : : ! : ! :

(g) 0 1 2 3 4 5 6 7 the empty set

Chapter 4

PROPERTIES OF OPERATIONS

I n in t roduc ing t h i s chap te r , it is perhaps adv i sab le f o r us , a s t eachers , t o cons ide r a b a s i c d i f f e r e n c e between t h i s course and the a r i t h m e t i c wi th which t h e s t u d e n t has p rev ious ly worked. The p r i n c i p a l concern of t h i s course i s a sys temat ic s tudy of numbers and t h e i r p r o p e r t i e s , and a r i t h m e t i c would seem t o have had much t h e same purpose. Ari thmetic o f t e n c o n s i s t s of a r a t h e r mechanical a p p l i c a t i o n of a l a r g e number o f r u l e s f o r computing c o r r e c t l y wi th g e t t i n g an answer" a s t h e o b j e c t i v e . On t h e o t h e r hand, w e a r e i n t e r e s t e d i n unders tanding r a t h e r thoroughly why numbers and o p e r a t i o n s on numbers behave a s they do. A r a t h e r well def ined search is made he re f o r important genera l p r o p e r t i e s of t h e numbers and t h e a r i t h m e t i c opera t ions with which t h e s t u d e n t is a l ready f a m i l i a r . I n s h o r t , we a r e i n t e r e s t e d i n what is sometimes r e f e r r e d t o a s t h e " s t r u c t u r e " of t h e "system" of numbers. Other words which convey some of t h e same meaning a s " s t r u c t u r e " a r e " p a t t e r n " , "form", and II organizat iont1 .

It i s i n e v i t a b l e t h a t many o f t h e genera l p r o p e r t i e s of

numbers and of t h e opera t ions we apply t o them a r e a l r e a d y q u i t e f a m i l i a r t o t h e s t u d e n t , even t h e s lower one, from t h e s tudy of a r i t h m e t i c . The p r o p e r t i e s a r e f a m i l i a r , however, only from s p e c i f i c i n s t a n c e s and no t a s e x p l i c i t p r i n c i p l e s .

I n Chapter 2, t h e aim was t o have t h e s t u d e n t d i scover some of these p r o p e r t i e s by means of ques t ions and examples. I n t h e p resen t chap te r , t h e p r o p e r t i e s a r e s t u d i e d f u r t h e r and a r e formalized. The p r o p e r t i e s which w e have sought t o e l i c i t from s t u d e n t s i n t h i s way a r e :

(1) Commutative and a s s o c i a t i v e p r o p e r t i e s f o r both a d d i t i o n and mu1 t i p l i c a t i o n

(2) D i s t r i b u t i v e proper ty o f m u l t i p l i c a t i o n over a d d i t i o n

( 3 ) Addition proper ty of 0 ( 4 ) M u l t i p l i c a t i o n p roper ty of 1

(5) M u l t i p l i c a t i o n proper ty of 0

P r o p e r t i e s ( 3 ) and ( 4 ) above s t a t e , i n terms we would never use wi th t h e s t u d e n t before he i s ready f o r them, t h a t 0 and 1

a r e , r e s p e c t i v e l y , t h e a d d i t i v e and m u l t i p l i c a t i v e i d e n t i t i e s . Property (5) above is included i n t h e l i s t even though it can be deduced from t h e o t h e r p r o p e r t i e s .

I t is worth n o t i n g t h a t i n t h i s c h a p t e r we a r e cons ider ing t h e p r o p e r t i e s only i n r e l a t i o n t o t h e non-negative r e a l numbers, wi th which t h e s t u d e n t is a l ready f a m i l i a r . We c a l l these t h e numbers of a r i t h m e t i c . La te r , it w i l l be seen t h a t t h e same p r o p e r t i e s hold f o r r e a l numbers.

The s t u d e n t , condi t ioned a s he is t o a r i t h m e t i c , may wel l a s k , "why bother?" when confronted wi th t h e fo rmal iza t ion of t h e s e p r o p e r t i e s . This ques t ion may be f o r e s t a l l e d somewhat by e x e r c i s e s which a r e i n t e r e s t i n g i n t h e i r own r i g h t and by t h e t e a c h e r ' s own e s t a b l i s h e d devices . O f course , t h e r e a l answer t o t h e q u e s t i o n "why bother?" c o n s i s t s , t o a l a r g e e x t e n t , of what has been s a i d i n t h e paragraphs preceding t h i s one regarding o u r concern wi th s t r u c t u r e .

nnother major goal of t h i s chap te r is t h e development of a good dea l of technique i n the s i m p l i f i c a t i o n of a l g e b r a i c express ions , a conspicuous f e a t u r e of any beginning a lgebra course . Here, however, w e a r e in t roduc ing t h e s e techniques i n conjunct ion wi th t h e p r o p e r t i e s of numbers and opera t ions . Algebraic s i m p l i f i c a t i o n is p r a c t i c e d a t t h e time t h e p r i n c i p l e s upon which such s i m p l i f i c a t i o n r e s t s a r e f irst developed, and many t imes t h e r e a f t e r . These p r i n c i p l e s a r e p r e c i s e l y the p r o p e r t i e s of numbers which t h e s t u d e n t is t o d i scover i n t h i s

chap te r . The t e a c h e r may want t o read , a s a genera l reference f o r

t h e work of t h i s chap te r , Haag, S t u d i e s - i n Mathematics, Volume - 111, S t r u c t u r e of_ Elementary Algebra, Chapter 3 , S e c t i o n 2.

4-1. I d e n t i t y Elements.

I d e n t i t y Element Addition.

It may w e l l be a d v i s a b l e t o spend more time wi th s lower s tuden t s c i t i n g s p e c i f i c numerical i n s t a n c e s of t h e a d d i t i o n property of ze ro , such a s :

These may help t h e s t u d e n t a p p r e c i a t e t h e s i g n i f i c a n c e of t h e I1 s ta tement , For every number a , a + 0 = a" .

Note t h a t t h e open sentence "a + 0 = a" is t r u e f o r a l l values of t h e v a r i a b l e . Such a sen tence conveys " s t r u c t u r e " o r "pat tern1 ' information about t h e number sys tent.

The a s s o c i a t i o n between t h e " r e s u l t being i d e n t i c a l wi th t h e number t o which ze ro is added" and t h e name " i d e n t i t y e l e m e n t may be worth emphasizing. Slower s t u d e n t s f r e q u e n t l y need t h e a i d of such a s s o c i a t i o n s i n l e a r n i n g new words, and they a r e seldom success fu l i n making t h e a s s o c i a t i o n s themselves.

Answers -- t o Oral Exerc i ses 4-la; page 118:

1. (01 6. (11 2. (01 7. t h e s e t of a l l numbers

3 . (0) 8. the s e t of a l l numbers

4. ( 0 ) 9. t h e s e t of a l l numbers

5 . (7) 10. Of

Mul t ip l i ca t ion Proper ty -- of One.

The m u l t i p l i c a t i v e i d e n t i t y element has been in t roduced a f t e r t h e a d d i t i v e I d e n t i t y element, r a t h e r than simultaneously, i n o r d e r t o g ive t h e s t u d e n t ample time t o a s s i m i l a t e t h e ideas .

The d i f f e r e n t numerals f o r t h e number one a r e mentioned i n t h i s s e c t i o n t o h e l p t h e s t u d e n t a p p r e c i a t e t h e f a c t t h a t t h e

P r o p e r t i e s ( 3 ) and ( 4 ) above s t a t e , i n terms we would never use with t h e s t u d e n t before he is ready f o r them, t h a t 0 and 1

a r e , r e s p e c t i v e l y , t h e a d d i t i v e and mu1 t i p l i c a t i v e i d e n t i t i e s . Property ( 5 ) above is included i n t h e l i s t even though i t can

be deduced from t h e o t h e r p r o p e r t i e s . I t i s worth no t ing t h a t i n t h i s chap te r we a r e cons ider ing

t h e p r o p e r t i e s only i n r e l a t i o n t o t h e non-negative r e a l numbers wi th which t h e s t u d e n t is a l ready f a m i l i a r . We c a l l these t h e numbers of a r i t h m e t i c . L a t e r , it w i l l be seen t h a t the same p r o p e r t i e s hold f o r - a l l r e a l numbers.

The s t u d e n t , condi t ioned a s he is t o a r i t h m e t i c , may wel l a s k , "Why bother?" when confronted wi th t h e fo rmal iza t ion of t h e s e p r o p e r t i e s . This ques t ion may be f o r e s t a l l e d somewhat by

e x e r c i s e s which a r e i n t e r e s t i n g i n t h e i r own r i g h t and by the t e a c h e r ' s own e s t a b l i s h e d dev ices . O f course , t h e r e a l answer t o t h e q u e s t i o n "why bother?" c o n s i s t s , t o a l a r g e e x t e n t , of what has been s a i d i n t h e paragraphs preceding t h i s one regarding o u r concern w i t h s t r u c t u r e .

Another major goal of t h i s chap te r is t h e development of a good dea l of technique i n t h e s i m p l i f i c a t i o n of a l g e b r a i c express ions , a conspicuous f e a t u r e of any beginning a lgebra course . Here, however, w e a r e in t roduc ing these techniques i n conjunct ion wi th t h e p r o p e r t i e s of numbers and opera t ions . Algebraic s i m p l i f i c a t i o n i s p r a c t i c e d a t t h e time t h e p r i n c i p l e s upon which such s i m p l i f i c a t i o n r e s t s a r e f i r s t developed, and many t imes t h e r e a f t e r . These p r i n c i p l e s a r e p r e c i s e l y the p r o p e r t i e s of numbers which t h e s t u d e n t is t o d i scover i n t h i s chap te r .

The t e a c h e r may want t o read , a s a genera l reference f o r t h e work of t h i s chap te r , Haag, S t u d i e s - i n Mathematics, Volume - 111, S t r u c t u r e of Elementary Algebra, Chapter 3, S e c t i o n 2.

Pages 117-118: 4-1

4-1. I d e n t i t y Elements.

I d e n t i t y Element Addit ion.

It may wel l be a d v i s a b l e t o spend more t i m e w i t h s lower s tuden t s c i t i n g s p e c i f i c numerical i n s t a n c e s of t h e a d d i t i o n proper ty of ze ro , such a s :

These may h e l p t h e s t u d e n t a p p r e c i a t e t h e s i g n i f i c a n c e of t h e I1 s ta tement , F o r e v e r y number a , a + 0 = a I 1 .

Note t h a t t h e open sentence a + 0 = a" i s t r u e f o r a l l values of the v a r i a b l e . Such a sentence conveys " s t r u c t u r e 1 o r "pa ttern1I informat ion about t h e number system.

The a s s o c i a t i o n between t h e " r e s u l t being i d e n t i c a l wi th the number t o which zero is a d d e d and t h e name " i d e n t i t y element1 may be worth emphasizing. Slower s t u d e n t s f r e q u e n t l y need t h e a i d of such a s s o c i a t i o n s i n l e a r n i n g new words, and they a r e seldom s u c c e s s f u l i n making t h e a s s o c i a t i o n s themselves.

Answers -- t o Oral Exerc i ses 4-la; page 118:

1. ( 0 ) 6 . ( 1 ) 2. (01 7. t h e s e t of a l l numbers

3 . (0 ) 8. t h e s e t of a l l numbers

4. (0 ) 9. t h e set of a l l numbers

5. (71 l o . Cf

M u l t i p l i c a t i o n P r o p e r t x of One.

The m u l t i p l i c a t i v e i d e n t i t y element has been in t roduced a f t e r t h e a d d i t i v e i d e n t i t y element, r a t h e r than simultaneously, i n o r d e r t o g ive t h e s t u d e n t ample time t o a s s i m i l a t e t h e i d e a s .

The d i f f e r e n t numerals f o r t h e number one a r e mentioned i n t h i s s e c t i o n t o h e l p t h e s t u d e n t a p p r e c i a t e t h e f a c t t h a t t h e

Pages 118-119: 4-1

m u l t i p l i c a t i o n p roper ty of one i s a proper ty of thenumber and has noth ing t o do w i t h t h e numeral chosen t o r epresen t t h e number one. Thus,

11 x ( 1 ) = xl' and " x ( ~ ) 4 = x"

both express t h e proper ty .

Answers -- t o Ora l Exerc i ses - 4-lb; page 119:

1. [ I ) 6. t h e s e t of a l l numbers

2. ( 1 ) 7. t h e set of a l l numbers

3 . Of 8. (31 4. ( 1 ) 9 - ^ 5. 10. (11

11. ( 0 )

M u l t i p l i c a t l o n Property of Z e ~ o .

The m u l t i p l i c a t i o n proper ty of zero i s no t , l i k e t h e o t h e r s , a fundamental proper ty of t h e r e a l number system; it would not , f o r example, appear among t h e axioms f o r an ordered f i e l d . I t

can be der ived from t h e d i s t r i b u t i v e proper ty , t h e a d d i t i o n proper ty of zero , and t h e e x i s t e n c e of a n oppos i t e (which comes i n Chapter 6 ) . A s a m a t t e r of i n t e r e s t , a d e r i v a t i o n of t h i s proper ty is given below:

For any number a , cons ide r t h e express ion

Then a(l + 0 ) = a(1) + a ( 0 ) by t h e d i s t r i b u t i v e proper ty ,

but a ( l + 0 ) = a ( 1 ) by t h e a d d i t i o n proper ty of 0

then a ( 1 ) + a ( 0 ) = a ( 1 ) .

To conclude t h a t a ( 0 ) i s 0, we must add - a ( l ) ( t h e oppos i t e , o r a d d i t i v e inverse of a ( 1 ) ) t o both a i d e s of t h e equat ion above, thus ob ta in ing

Pages 119-122: 4-1

t h e (-a( l) + a ( l ) ) + a ( 0 ) = - a ( l ) + a ( 1 ) by t h e a s s o c i a t i v e proper ty of a d d i t i o n ,

and 0 + a ( 0 ) = 0

a ( 0 ) = 0

by t h e a d d i t i o n p roper ty o f o p p o s i t e s ,

by t h e a d d i t i o n proper ty of 0.

Answers t o Oral Exerc i ses 4 - l c ; page 120: -- - 1. a d d i t i o n proper ty of ze ro

2. m u l t i p l i c a t i o n proper ty of one

3. m u l t i p l i c a t i o n proper ty of ze ro

I n Exerc ise 3 of Oral - Exerc i ses 4-ld and i n Exerc i ses 3 , - 4, 5 and 6 of Problem S e t 4-ld, and i n f u t u r e work i n which t h e -- m u l t i p l i c a t i o n proper ty of 1 is used i n adding f r a c t i o n s o r

r a t i o n a l express ions , t h e s t u d e n t should be encouraged t o mention

o r "wr i t e out" t h e numeral he has used f o r 1. It is important

t h a t the computations which depend upon t h i s p roper ty be c l e a r l y

t i e d t o i t . Here, a s a t many p o i n t s i n t h e course , only a thorough a p p r e c i a t i o n o f t h e connection between concept and

manipulation e n t i t l e s t h e s t u d e n t t o t ake " s h o r t c u t s . " Before

ass igning Exerc ise 5 of Problem -- S e t 4 - lb you may want t o remind

m the s t u d e n t s t h a t o r - ft; i s a numeral f o r 1 f o r a l l

values of the v a r i a b l e s except 0 .

Answers -- t o Oral Exerc i ses 4-ld; pages 122-123: - 1. ( a ) 6, a d d i t i o n p roper ty of zero

( b ) 125, m u l t i p l i c a t i o n proper ty of one

( c ) 0 , m u l t i p l i c a t i o n proper ty of ze ro

a d d i t i o n p roper ty of ze ro ( d l 3, ( e ) 2.81, m u l t i p l i c a t i o n p roper ty of one

( f ) 0 , m u l t i p l i c a t i o n p roper ty of ze ro

(g ) 5.2, a d d i t i o n proper ty of ze ro and m u l t i p l i c a t i o n

proper ty of one

Pages 122-123: 4-1

2. ( a ) True ( c ) False (e) True (g) False

(b) False (d) True (f) True (h) False

Answers to Problem -- Set ^-Id; pages 123-124:

1. (a) 0. Multiplication property of zero (b) b. Multiplication property of one (c) n. Addition property of zero (d) n + 1. No property

2. (a) False, Numerals represent the same number because of the multiplication property of one,

(b) True. Multiplication property of one

( c ) False. Numerals represent the same number because of the multiplication property of one.

( d ) False. The sentence is not true when a is zero. (e) False. The addition property of zero. i.e., for every

number m, m + 0 = m. (f) False for every value of rn

(g) False for every value of m 18 18 18 13 234 -x 1 = - x - = 3. (a) ^= 5 13

m m (c) - = - X l = ~ X ^ = a a x x x 5 5x

x x (d) ~ = ~ X l = $jx7=% 3 3~

a 2a (g) 2 = 2 X 1 = 2 X m = - a a

Pages 123-124: 4-1

4 ^ (f) Ã‘= 1 x 1 - -

3 3

Pages I@-126: 4-1 and 4-2

4-2. Closure.

Th i s s e c t i o n i s concerned wi th two r e l a t e d ideas . The f i rs t is important and i s one t h a t should be impressed s t rong ly on the s t u d e n t . It i s t h i s : i f - a is number of a r i t h m e t i c and - b i s number of a r i t h m e t i c , we can add a and b and we can mul t ip ly a and b. Th i s means t h a t we can f r e e l y write numerals such a s "3a1', "2b1', 113a + 2b1', "abl' , e t c . Each of t h e s e has a meaning; t h e r e is a number which i t r e p r e s e n t s . Moreover, t h e s t u d e n t must be reminded over and over again t h a t a n express ion such a s "a + b" is a numeral r a t h e r than a command t o do a r i t h m e t i c . I n c o n t r a s t , we r e c a l l some cases i n which s u b t r a c t i o n can n o t be performed, and we remember t h a t d i v i s i o n by ze ro i s Impossible. Thus "a - b" has meaning only i f a Is

" c" g r e a t e r than o r equal t o b, and has meaning only i f d is n o t 0 .

The second idea i s t o in t roduce by examples the not ion of a s e t of numbers being c losed under a n opera t ion ; a concept which t h e s t u d e n t has met informal ly i n previous e x e r c i s e s . The t e x t does n o t g ive a formal d e f i n i t i o n of t h i s s i n c e it might be too t e c h n i c a l . What we have i n mind i s t h i s : a p a r t i c u l a r s u b s e t A of t h e numbers of a r i t h m e t i c i s c losed under a p a r t i c u l a r o p e r a t i o n ( e . g . , a d d i t i o n , s u b t r a c t i o n , e t c . ) i f t h e fo l lowing s ta tement is t r u e : i f - a Is any element i n A and & is any element i n A , the opera t ion can be app l i ed t o and - b and, moreover, t h e number which is produced is an element of A . For example, t h e set ( 0 ) Is c losed under a d d i t i o n s i n c e 0 + 0 = 0,

i t is c losed under s u b t r a c t i o n s i n c e 0 - 0 = 0, and it i s c losed under m u l t i p l i c a t i o n s i n c e 0 0 = 0. It is not closed

Pages 126-129: 4-2

under d i v i s i o n s i n c e d i v i s i o n by ze ro cannot be done. The s e t ( 1 ) is c losed under m u l t i p l i c a t i o n and under d i v i s i o n . It is no t closed under a d d i t i o n s i n c e 1 + 1 = 2, and it i s n o t c losed under s u b t r a c t i o n s i n c e 1 - 1 = 0 . Here t h e o p e r a t i o n s can be performed but t h e numbers produced a r e n o t i n t h e set [ I ) . The s e t of even numbers (0 ,2 ,4 , ...I i s c losed under a d d i t i o n and under multiplication. It is n o t c losed under s u b t r a c t i o n and it is not closed under d i v i s i o n . These o p e r a t i o n s cannot always be performed f o r t h i s s e t ; moreover, i n some c a s e s where they can be performed they do n o t y i e l d an even number.

Answers t o Oral Exerc i ses %; pages 128-129:

1. yes 3 . yes 5 . ( d l Yes Yes

2. no 4. yes

Yes Yes

Answers & Problem S e t 4-2; pages 129-131:

1. Addition M u l t i p l i c a t i o n Div i s ion Sub t rac t ion

( a ) c losed c losed n o t c losed n o t closed ( b ) c losed c losed n o t c losed n o t c losed ( c ) c losed c losed c losed n o t c losed

(Can't y e t s u b t r a c t a l a r g e r from a s m a l l e r . )

( d ) c losed c losed n o t c losed n o t c losed ( e ) n o t c losed c losed n o t c losed n o t c losed ( f ) c losed c losed no t c losed n o t c losed (g) closed n o t c losed n o t c losed n o t c losed ( h ) n o t c losed n o t c losed n o t c losed n o t c losed

Sub t rac t ion is n o t c losed u n t i l we extend t h e number system t o include nega t ive numbers, but t h i s need n o t be mentioned a t t h i s time t o t h e s t u d e n t . Div i s ion is n o t c losed f o r any s e t conta in ing 0 . Th i s a g a i n can be played down o r ba re ly mentioned a t t h i s time. J u s t say "we don' t d i v i d e by 0" and l e t I t go a t t h a t . L a t e r it w i l l be shown t h a t 0 has no r e c i p r o c a l .

2. A is n o t c losed under a d d i t i o n s i n c e 2 is n o t i n s e t A .

A i s c losed under multiplication s i n c e every element i n the t a b l e Is an element of A

Const ruct t a b l e s .

We s e e t h a t t h e r e a r e elements i n each t a b l e t h a t a r e not elements of A . Thus, A is c losed under e i t h e r opera t ion .

4. The s e t s i n p a r t s ( d ) and ( e )

5. A l l 4 sets

*6. C i s c losed under a d d i t i o n , b u t n o t under m u l t i p l i c a t i o n . Notice t h a t t h e s e are opera t ions of t 'mul t ip l i ca t ion ' t and "addi t ion" t h a t have been de f ined by t h e table, and a r e not r e l a t e d t o t h e opera t ions w e w i l l be d e a l i n g wi th involving numbers.

*7. The opera t ion "+I' i s n o t commutative s i n c e f o r example

b + a = b and a + b = c. Therefore, a + b 4 b + a. The opera t ion "x" commutative as can be shown by trying

a l l cases , b u t i s more r e a d i l y seen by observing t h e symmetry of t h e t a b l e about t h e d iagonal row of elements.

Class time should n o t o r d i n a r i l y be used f o r d i scuss ion of t h e s t a r r e d problems s i n c e t h e o r d e r l y p rogress of t h i s course i s n o t dependent upon a s u c c e s s f u l s o l u t i o n of such problems.

Pages 131-132: 4-3

4-3 , Commutative A s S ~ ~ i a t i V e P r o p e r t i e s a Addit ion - and

Mu1 t i p l i c a t i o n .

The commutative and a s s o c i a t i v e p r o p e r t i e s of a d d i t i o n and m u l t i p l i c a t i o n were d i scussed i n Chapter 2 by means o f numerical examples. I n t h i s s e c t i o n , t h e p r o p e r t i e s a r e s t a t e d i n open sentences . Actual ly t h e p r o p e r t i e s a r e t r a n s l a t e d from word s ta tements i n t o t h e language o f a lgebra . The t r a n s l a t i o n process of which t h i s i s a n example is considered more s y s t e m a t i c a l l y i n Chapter 5. Some s t u d e n t s may f i n d it easy t o by-pass t h e word statement and go d i r e c t l y from t h e numerical examples t o t h e a lgebra ic s ta tement of t h e p r o p e r t i e s . I n f a c t , t h i s could have been done i n Chapter 2 except t h a t we d i d n o t then have v a r i a b l e s and so had t o f a l l back on word s t a t ements . The comparison of the word s ta tements wi th t h e a l g e b r a s t a t ements shows t h e g r e a t advantage of t h e l a t t e r i n both c l a r i t y and s i m p l i c i t y .

Notice the form of t h e s t a t ements of t h e p r o p e r t i e s . I f we had s t a t e d the commutative proper ty of a d d i t i o n f o r example, without q u a n t i f i c a t i o n of t h e v a r i a b l e s a s

we would have had no i n d i c a t i o n whether t h i s open sentence is : t r u e f o r some, none, o r a l l t h e va lues of a and b. Thus we

quan t i fy the v a r i a b l e s and s t a t e : or every number a and every number b y a + b = b + a . " I n t h i s way we say t h a t t h e open sentence i s t r u e f o r every a and every b.

Examples l i k e " ( 2 + 3 a ) + 2b = 2b + ( 2 + 3a)" and '2m + 3n = 3n + 2m" a r e included (pages 131-132) s i n c e s t u d e n t s o f t e n have t r o u b l e a p p r e c i a t i n g t h e g e n e r a l i t y of t h e s ta tement a t b = b + a . " These examples a r e inc luded t o h e l p head o f f t h i s s o r t of d i f f i c u l t y .

The purpose of t h e l a s t p o r t i o n of t h e s e c t i o n is t o emphasize t h a t there a r e opera t ions which a r e n e i t h e r commutative nor a s s o c i a t i v e .

We sugges t t h a t t h e problem sets i n 4-3 be done as o r a l exe rc i ses wi th ample d i scuss ion . Perhaps t h i s i s t h e b e s t way t o avoid tedium and t o g e t t o t h e r o o t of misunderstandings.

Pages 132-134: 4-3

There seems to be no extra value to be gained by individual work on these exercises.

Answers to Oral Exercises 4-3a; page 132: - 1. 3. 5. 6.

Answers Problem 4-3a; pages 132-133:

1. True, commutative property of addition 2. True, commutative property of addition 3. True, commutative property of addition 4. True, commutative property of addition 5. False, since (3a + 2b) may be a number different

from (3b + 2a)

Answers to Oral Fxercises -9 4-3b- pages 134-135:

True, commutative property of addition

True, associative property of addition True, associative property of multiplication True, commutative property of multiplication True, none of the properties False

False True, commutative property of multiplication True, commutative property of addition and

commutative property of multiplication False False False False True, commutative property of addition and commutative property of multiplication

Page 135: 4-3

The s tuden t may no t conscious ly go through a l l t h e s t e p s I n

t h e e x e r c i s e s above, but i f he i s u n c e r t a i n of an answer t h e a b i l i t y t o s p e l l o u t t h e steps should r e a s s u r e him.

Answers - t o Problem -- S e t 4-3c; pages 135-136:

3 1. Divis ion i s n o t a s s o c i a t i v e . (18 + 6 ) + 2 =

18 + (6 + 2) = 6

Page 136: 4-3

Subtraction is not commutative. "a - b = b - a. It Consider x - 3 and 3 - x.. Whatever number of arithmetic is chosen for x, one of these expressions is not a number

I1 of arithmetic while the other is. Hence x - 3" and "3 - xt' cannot name the same number. .

True. Associative property of addition True. Commutative property of multiplication True. Associative and commutative properties of multiplication True. Commutative property of multiplication True. Commutative property of addition and commutative property of multiplication True. Commutative property of multiplication and associative property of addition True. Commutative and associative property of multiplication and associative property of addition False. The left side may be written (b(2) + c) + 2a, which nay be a number different from ( b(2) + c) 2a. True. Commutative property of addition and commutative property of multiplication True. Associative and commutative property of addition and commutative property of multiplication

True. Commutative and associative property of multiplication

Yes NO for example, (8 * 12) * 16 = 13

8 % (12 * 16) = 1.1

No for example, 5 t 7 = 5 7 % 5 = 7

Yes This can be illustrated as follows:

Pages 137-138: 4-4

D i s t r i b u t i v e Property.

The d i s t r i b u t i v e proper ty , l i k e t h e o t h e r s be fo re it, i s s t a t e d he re a s an open sentence, again b u i l d i n g upon numerical experiences i n Chapter 2. The p roper ty i s s t a t e d i n f o u r d i f f e r e n t forms t o l a y t h e foundation f o r some of i t s f u t u r e a p p l i c a t i o n s . However, t h e s tuden t should understand t h a t t h e r e i s only one d i s t r i b u t i v e p r o p e r t y under cons ide ra t ion .

The examples should be c a r e f u l l y d i scussed , wi th emphasis on t h e f a c t t h a t t h e s e a r e a p p l i c a t i o n s of t h e d i s t r i b u t i v e property. I n example 4, you w i l l n o t e t h e phrase "s impler form". We would l i k e t o use t h i s phrase t o d e s c r i b e t h e end r e s u l t . Although i n most i n s t a n c e s it i s q u i t e obvious t h a t one form i s simpler than another , it appears t o be v i r t u a l l y imposs ib le t o g ive a good d e f i n i t i o n of "simple". Therefore, w e w i l l be s a t i s f i e d t o use t h e express ion i n concre te s i t u a t i o n s where t h e r e i s no p o s s i b i l i t y of confusion and w i l l n o t a t tempt t o g i v e a genera l d e f i n i t i o n . The important Idea h e r e i s t h a t , when we use t h e b a s i c p r o p e r t i e s t o w r i t e a phrase i n ano the r ( s imple r , more compact, more u s e f u l , e a s i e r t o write, e a s i e r t o read , e t c . ) form, t h e r e s u l t i s a phrase which names t h e same number a s t h e given phrase.

A g r e a t d e a l of p r a c t i c e i s given wi th t h e d i s t r i b u t i v e proper ty i n t h e problem sets of Sec t ion 4-4. However, t h e r e i s

no need t o d e s p a i r i f t h e s t u d e n t s seem t o have something less than a mastery of t h e p r i n c i p l e . Following t h e s p i r a l method of development, t h e p roper ty i s used i n t h e same and d i f f e r e n t con tex t s throughout f u t u r e chapters ; a g r e a t e r degree of mastery might we l l await those later chapters .

Answers -- t o Oral Exerc i ses 4-ha; page 138 : - 1. ( a ) i n d i c a t e d product ( g ) i n d i c a t e d sum

( b ) i n d i c a t e d product ( h ) i n d i c a t e d sum

( c ) i n d i c a t e d sum ( i ) i n d i c a t e d sum

( d ) I n d i c a t e d product ( j) i n d i c a t e d sum

( e ) i n d i c a t e d sum ( k ) i n d i c a t e d product

( f ) i n d i c a t e d sum (1) i n d i c a t e d sum

Pages 138-142: 4-4

Answers -- t o Ora l Exerc i ses 4-'4bi page 139:

Answers - t o Problem S e t 4-4b; pages 141-142: -- 1. ( a ) T r u e

( b ) True ( c ) True

The aim of Exerc i se 1 i n t h i s problem s e t is t o have the s t u d e n t recognize t h e t r u t h of each sentence n o t because both s i d e s can be reduced t o t h e same common name, but because t h e sentence is an example o f a true p a t t e r n . You may have t o remind your s t u d e n t s t o do t h i s .

2. ( a ) F a l s e

(b) True ( c ) F a l s e

( d ) F a l s e ( e ) True (f) F a l s e

Page 142: 4-4

(g) True. ( h ) True. ( 1 ) True .

5 . The student may want to work some o f these problems more

quickly by " c o l l e c t i n g terms". He may want t o write "7b" immediately f o r part ( a ) . Make sure that he earns the r ight t o use these short-cuts.

(a) 5b + 2b = ( 5 + 2)b = 7b ( b ) 4a + a(7) = 4a + 7a = ( 4 + 7)a = l l a ( c ) c (2 ) + c ( 3 ) = c ( 2 + 3) = 4 5 ) = 5c

1 1 3 4 (d) 7 + + = (=Â¥+-)rn=

(e ) .4n + .6n = ( * 4 + . 6 )n = In = n ( f ) 8.9b + 3.2b = ( 8 . 9 + 3.2)b = 12.1b (g) 3y + y = 3y + l y = ( 3 + l ) y = 4y (h) m + 2 m = 1 m + 2m = ( 1 + 2)m = 3rn ( 1 ) 2a + 3b ( J ) 3.7n + n ( . 4 ) = 3.711 + .4n = (3 .7 + . 4 ) n = 4.1n

Pages 142-143: 4-4

Answers -- to Oral Exercises 4-4c; pages 142-143:

1. True

2. False

3. False 4. False 5. True

6. True

7. False 8. True

9. False 10. False

2. (a) 2a + 2b = 2(a) + 2(b) = 2(a + b)

(b) 2mn + 5n = (2m)n + 5(n) = (2m + 5)n

(c) 2mn + 2n = 2 m + 2n(l)

= (2n)m + (2n)l = 2n(m + 1)

(d) 6bc + 6c = 6cb + 6c = 6c(b) + 6~(1) = 6c(b + 1)

(f) cx + 4cy = c(x) + ( 4 4 ~ = c(x) + ('(C)(~))Y = c(x) + c(4y) = c(x + 4y)

(h) 3ab + 9a2 = 3ab + (3a) (3a) = 3a(b) + (3a)(3a) = 3a(b + 3a)

(1) 3x + 3x2 = 3x + 3 ( x ) ( x )

= 3 x ( l ) + (~x)(x) = 3 x ( l + X )

( d ) 3 c ( 3 b + 2 )

( e ) 4b(3a + 2c)

( f ) 6 a ( i + a )

5 . ( a ) A r e c t a n g l e has two equal s i d e s and two equa l ends and

s o i ts pe r ime te r i s found by adding t h e number of

inches i n t h e l e n g t h and t h e number of inches i n t h e

width and m u l t i p l y i n g t h e r e s u l t by 2.

2 ( 7 + 3) = 2(10)

= 20 The p e r i m e t e r i s 20 inches .

The amount o f money c o l l e c t e d is $750. We could have

found t h e amount c o l l e c t e d a t each window and then

added t h e two amounts. T h i s would c e r t a i n l y be more

complicated.

Answe~s t o Opal Exe~cises - 4-4d; page 145:

( a + c ) ( a + 4) = ( a + c )a + ( a + c)4 (x + a ) ( x + 3) = (x + a)x + (x + a ) 3 (x + l ) ( a + b ) = (x + l ) a + (x + l ) b (3a + 4) ( a + 5) = (3a + 4)a + (3a + 4)5 (7 + x ) ( x + 7 ) = (7 + x)x + (7 + x)7 (3a + b ) ( c + d) = (3a + b)c + (3a + b)d (mn + x ) ( a + b ) = (mn + x)a + (mn + x)b (ab + c) (b + c ) = (ab + c)b + (ab + c)c (8 - x) (8 + x) = (8 - x)8 + (8 - x)x

Answers - t o Problem -- Set 4-4d; pages 145-146:

8. (x + 2) + (x + 5) = fix + 2) + x) + 5 = ( x + ( x + 2 ) ) + 2 + 5 1 = ((lx + IX) + l ) x + 2 ) + 5 = (2x + 2) + 5 = 2x + ( 2 + 5) 2 x + 7

Pages 145-146: 4-4

(mn + b)(rnn + a) = (mn + b)mn + (mn + b)a = (mn) + bmn + rnna + ba

(xy + a)(y + b) = (xy + a)y + (xy + a)b = xy2 + ay + xyb + ab

If a is 5 and x is 2, then

(a +2)(a +x) = (5 + 2)(5 + 2) = 49

If a is 5 and x is 2, then

a2 + 2a + ax + 2x = (5)2 + 2 ( 5 ) + (5)(2) + 2(2) = 49

Therefore, 2 (a + 2)(a + x) = a + 2a +ax + 2x when a is 5 and x is 2.

If x is 3 and a is 0,

(2x + 3)(x + a) = (6 + 3)(3 + 0) = 27

If x is 3 and a I s 0,

2x2 + 3x + 2ax + 3a = ~ ( 3 ) ~ + 3(3) + 2(0)(3) + 3(0) = 27 2 Therefore, (2x + 3)(x + a) = 2x + 3x + Sax + 3a

when x is 3 and a is 0.

page 146: 4: 4

Summary.

This summary o f p r o p e r t i e s i s very impor tant . We want the s t u d e n t t o begin t h i n k i n g of t h e number system more and more o f t e n i n terms of t h e b a s i c p r o p e r t i e s s o t h a t even tua l ly almost a l l o p e r a t i o n s he does wi th numbers w i l l be performed wi th t h e s e p r o p e r t i e s i n mind. Th i s i s a development which w i l l no t t ake p l a c e f o r most s t u d e n t s very qu ick ly ; however, by the end of t h e y e a r I t i s hoped t h a t t h e major i ty w i l l have progressed t o w i t h i n s i g h t o f t h e g o a l .

Pages 146-147

The l is t of p r o p e r t i e s obta ined a t t h i s p o i n t is no t complete. We s t i l l must in t roduce t h e nega t ive numbers and o b t a i n t h e p r o p e r t i e s of o rde r . The l i s t w i l l be completed f o r our purposes by t h e end of Chapter 12.

Answers - t o Review Problem s; pages 147-150 :

1. - Zero i s t h e I d e n t i t y element of a d d i t i o n . For every number a , a + 0 = a Addit ion p roper ty of ze ro

For every number a, a ( 0 ) = 0 Mu1 t i p l i c a t i o n p roper ty o f ze ro

2. - One is t h e i d e n t i t y element of m u l t i p l i c a t i o n . For every number a , a ( l ) = a M u l t i p l i c a t i o n p roper ty

o f one

$ , {m , (m n o t 0 ) a r e n u m e ~ e r a l s f o r one.

3 4 3 4 (c) ~ + 7 = ~ x l + ~ x l = g x ~ 3 + 3 x f f = & + - 3 = 3 4 8

1 1 48 48 4. ( a ) - = 2 Fxw=g5 4 4 1 5 = (b) T = %J x 60 T5 - 2 2 m 2 m ( c ) r = r x - . -

m 5m

( d ) 2 = 5 x 5 5 m n n 5 5 5 5 2m - 1 0 m ( e ) - = - x n n 5 n T 2 m n

Page 149

6. Set of whole numbers ending In 0 Is closed under addition,

is closed under multiplication, and is not closed under subtraction.

7. For every number a, every number b, and every number c

a + b = b + a ab = ba Commutative properties

(a+b) + c = a + (b+c) (ab)c = a(bc) Associative properties

a(b + c) = ab + ac Distributive property

False. If x is 1, m is 2, y is 3, and n is4,

then (1 + 3)(2 + 4) = (1 + 2)(3 + 4) is a false sentence.

True. Commutative property of addition

True. Commutative property of addition True. Commutative property of multiplication

False. If x is 2, y is 3, and m is 1, then 2 + 3(1) = 2(3) + 1 is a false sentence. False. If x is 2, y is 3, and m is 1, then

2(3 + 1) = 3(2 + 1) is a false sentence. True. Commutative property of multiplication True. Distributive property and commutative property

of multiplication False. If x is 2, y is 3, and m is 1,

2 + 3(1) = (2 + 3)(2 + 1) is a false sentence. True. Associative property of multiplication

10. ( a ) o 2

( b ) 5 How hard did the s tudents work on these?

(4 0 Are they us ing the propert ies of 0 and

( d l 0 1 to make t h e i r work easy?

axy + 8xy

Page 150

5 ( c ) 4 x 1 8 = 4 x ( 1 +

= ( 4 x 1 ) + ( 4 x 2)

t r u t h s e t : (11 t r u t h s e t : ( 1 , 2 )

The t r u t h s e t o f t h e f i r s t s en tence is a s u b s e t of t h e

t r u t h s e t of t h e second sen tence .

t r u t h s e t : A l l numbers t r u t h s e t : All numbers

l e s s than 2

The f i rs t t r u t h s e t is a s u b s e t o f t h e second t r u t h s e t .

7 t r u t h s e t : (F) t r u t h s e t : A l l numbers l e s s

than 2

Ne i the r Is a s u b s e t o f t h e o t h e r .

Suggested Tes t Items -- 1. Show how t o use t h e m u l t i p l i c a t i o n p r o p e r t y of 1 t o f i n d

common names f o r 1 2 + 7 1 2 3 l + T

( a ) ?J+ q ( b ) â€ ( 4 1 1 - 7

2. Which of t h e fo l lowing sentences a r e t r u e f o r every va lue

of t h e v a r i a b l e s ? ( ~ i v e reasons f o r your answers.) ( a ) x ( 2 -I- 3) = ( 2 + 3 ) x ( e ) (3a + c ) -I- d = ( c + d ) + 3a ( b ) b ( a + 2) = a(b + 2) ( f ) ( 2 x 1 ~ = ~ ( x Y )

2 2 ( c ) a 2 b 2 = b a (g) a ( b - b) = a

( d ) (4x + y)3 = 4x(y + 3) ( h ) a + (b - b) = a

3. Each sentence below i s t r u e f o r every va lue of t h e v a r i a b l e s . I n each case decide which p r o p e r t i e s enable u s t o v e r i f y t h i s f a c t . ( a ) x ( y + z ) = xy + xz

(b) XY + (ay + c ) = (xy + a y ) + c ( c ) abcd = ab(cd)

(d) X ~ + X Z = YX -I- zx

(4 ( a b ) ( c d ) = ( d c ) ( b a ) ( f ) x + o = x

( g ) o ( x ) = 0 (h ) l ( x ) = x

4. Show how it i s p o s s i b l e t o use t h e d i s t r i b u t i v e p r o p e r t y t o find common names f o r t h e following i n an easy Way.

( a ) (212) (101) (dl (13)(29)

5* For each s t e p except t h e last i n t h e following, s t a t e which property of t h e operations is used t o der ive it from the precedtng step.

3(2x + y) + 5x = (6x + 3y) + 5x = (3y + 6x1 + 5x = 3y + (6x + 5x1 = 3y + (6 + 5 ) x = 3y + l l x

6* Use the p rope r t i e s of t h e operations t o w r i t e t he following open phrases in simpler form. (a) 6x + 3x

(b) 41a + 37b + 82a + 14b

( c ) .3x + l m 4 y + 7.- + 1.12 + 2 . 3 ~ 2 16 4

(dl i $ + 7 + T ~ + $ x + F ~

7 * Find t h e t r u t h s e t s of t h e following sentences* ( a ) 4x = o (d) 4(a + 4) = 12

4 4 ( e ) V + ~ = T V + T

( f ) ( 4 - 4 ) w = ( w - w)4

8* Change ind ica ted products t o indicated sums, and indicated sums t o ind ica ted products, using the d i s t ~ i b u t T v e property*

(a) 5% + 5~ (d) 2(a + 2) + x ( a + 2) (b) (U + 2 4 4 ( 4 (a + a@ + 1) (c) 3a(2 + 4b) (f) (x + Y ~ ( X + 1)

Answers to Suggested Test Items - -- 1. ( a ) $+y3 2 x l + T x l 3 = T x v + v x T 2 4 3 3 - - - + - = - 8 9 17 12 12 12

True. Commutative property of multiplication

False. ba + b(2) and ab + a(2) are different

numbers If a # b. True. Commutative property of multiplicatlon

False. 12x + 3y and 4xy + 12x are different numbers 3 unless 3y Is 4xy, i .e. , unless y Is 0 or x is T.

True. Associative and commutative properties of

add1 tlon

True. ~ssoclatlve property of multlpllcatlon

False. By the multlpllcatlon of zero, for any number

a, a ( 0 ) = 0 .

True. Addition property of zero

Distributive property

Associative property of addition

Assoc1atlve property of multiplicatlon

Commutative property of multlpllcation

(e) Commutative property of multiplication

(f) Addition property of zero

(g) Multiplication property of zero

(h) Multiplication property of one

4. (a) (212) (101) = 212(100 + 1) = 21,200 + 212 = 21,412

3 (c) 60(2 + f ) = (60)(2) + (60)() = 50 + 36 = 86

5. 3(2x + y) + 5x = (6x + 3y) + 5x distributive property

= (3y + 6x) + 5x commutative property of addition

= 3y + (6x + 5x) associative property of addition

= 3y + (6 + 5 ) ~ distributive property = 3y + llx

6. (a) 6x + 3x = (6 + 3)x = gx

(b) 4la + 37b + 82a + l4b = ( h a + 82a) + (37b + l4b) = (41 + 82)a + (37 + l4)b = 123a + 51b

(c) .3x + 1.4~ + 7.1~ + 1-12 + 2.3~ =

(.3x + 7.1~) + (1.4~ + 2.3~) + 1.12 = (-3 + 7.1)~ + (1.4 + 2.3)~ + l.lz = 7.4~ + 3.7~ + 1-12

(dl 0 (4 6 (f) the set of all numbers

Chapter 5

OPEN SENTENCES AND WORD SENTENCES

The purpose of t h i s c h a p t e r i s t o h e l p develop some a b i l i t y

i n w r i t i n g open sentences f o r word problems. We work f irst J u s t

with phrases . We do some inven t ing of Engl ish phrases t o f i t

open phrases a t t h e s t a r t t o t r y t o h e l p g ive a more complete

p i c t u r e of what t h i s t r a n s l a t i o n back and f o r t h i s l i k e . Then

we t r a n s l a t e back and f o r t h sentences invo lv ing both s ta tements of e q u a l i t y and i n e q u a l i t y .

I n o rde r t o concen t ra te on t h e t r a n s l a t i o n p rocess , we p r e f e r

a t present not t o become involved i n f i n d i n g t r u t h s e t s of t h e

open sentences . We never the less have asked ques t ions i n t h e

word problems. This seems necessary i n o r d e r t o p o i n t c l e a r l y t o

a v a r i a b l e , t o make t h e experience more c l o s e l y r e l a t e d t o t h e

problem so lv ing t o which t h i s t r a n s l a t i o n process w i l l be a p p l i e d , and t o b r i n g f o r t h t h e sentence o r sentences we a r e r e a l l y wanting

the s tudent t o t h i n k o f . Thus i n t h e f irst example i n t h e exposi-

t i o n of Sect ion 5-4, i f i n s t e a d of saying, "How long should each

p iece be?" we s a i d , "Write an open sentence about t h e l e n g t h s of

the p ieces , " t h e s tuden t might wel l answer, " I f one p i e c e i s n

inches long, then n < 44," o r he might even answer, "n > 0. ' '

These a r e t r u e enough sentences , but t h e y miss t h e exper ience w e

want t h e s tuden t s t o have. Some s t u d e n t s may f e e l t h e urge t o go on and f i n d t h e

answer ' . I n t h a t case you should l e t them try, but don' t l e t

' t f inding t h e answer" become a d i s t r a c t i o n a t t h i s p o i n t . T e l l t h e s tudents t h a t we w i l l be developing more e f f i c i e n t methods of

f inding t r u t h s e t s of sentences l a t e r on, bu t f o r t h e p resen t we

s h a l l go no f u r t h e r than w r i t i n g t h e open sen tence .

I n a few of t h e problems i n t h i s course t h e r e i s superf luous information which Is not necessary f o r doing t h e problem. This

i s I n t e n t i o n a l . We hope t h a t occas ional exper ience wi th such

I r r e l e v a n t ma te r i a l w i l l h e l p make t h e s tuden t more aware of information which - i s r e l e v a n t .

An at tempt i s made throughout t h e c h a p t e r t o b r i n g out t h e

point t h a t , i n t r y i n g t o so lve a problem about phys ica l e n t i t i e s ,

one must f i r s t s e t up a mathematical model. Having made t h e

pages 153-154: 5-1

mathematical a b s t r a c t i o n s corresponding t o t h e physica l measures and t h e i r r e l a t i o n s h i p s , one can then w r i t e one o r more open sen tences , and d i r e c t h i s a t t e n t i o n t o f i n d i n g a s o l u t i o n t o t h i s mathematical problem. Once such a s o l u t i o n i s obta ined, i t can be r e l a t e d once aga in t o t h e o r i g i n a l phys ica l problem.

5-1. Open Phrases t o Word Phrases . -- I n t r a n s l a t i n g from open phrases t o word phrases -- you may

p r e f e r t o say "English" phrases -- many word phrases a r e poss ib le . Encourage t h e s t u d e n t s t o use t h e i r imaginat ions and b r ing i n a s g r e a t a v a r i e t y of t r a n s l a t i o n s a s p o s s i b l e . It i s c l e a r t h a t t h e broader t h e i r exper ience i n t h i s type of t r a n s l a t i o n , t h e broader w i l l be t h e base from which they s t a r t t h e reverse process , t r a n s - l a t i o n from word phrases t o open phrases i n t h e next s e c t i o n . Thus, i f supervised s t u d y time i s a v a i l a b l e , it may be advisable f o r t h e t e a c h e r t o work wi th t h e s tuden t a s he begins Problem Se t 5-1, s o t h e s t u d e n t may be helped t o t h i n k of a v a r i e t y of word t r a n s l a t i o n s f o r t h e given open phrases . I f t h e s tudent says t h a t he cannot t h i n k of any d i f f e r e n t t r a n s l a t i o n s , t h e t eacher can ask him ( a s was done i n t h e t e x t ) t o respond t o t h e ques t ion , Number of what? ' and almost any answer t o t h i s ques t ion i s a s u b s t a n t i a l beginning f o r a t r a n s l a t i o n .

I n t h e l a s t example of t h e t e x t i n Sect ion 5-1, it may be necessary f o r t h e t e a c h e r t o work with p a r t i c u l a r c a r e with the c l a s s on t h e phrase "number of p o i n t s B i l l made i f he made 3 more than twice a s many a s Jim." It seems impossible t o s impl i fy t h i s language f u r t h e r , and y e t t h i s i s t y p i c a l of a kind of wording t h a t o f t e n bewilders a s lower s t u d e n t . The t eacher should s t r e s s t h a t 2x + 3 i s a number.

P o s s i b l e t r a n s l a t i o n s of If-I' i nc lude " l e s s than , " " t h e d i f f e r e n c e o f , " " s h o r t e r than , " e t c . You may have t o warn your s t u d e n t s t h a t , s i n c e s u b t r a c t i o n i s not commutative, they must watch which number comes f irst i n us ing " l e s s than ."

You w i l l sooner o r l a t e r f i n d a s tuden t who i s confused about t h e d i f f e r e n c e between g r e a t e r t h a n o r m o r e t h a n which c a l l s f o r "+I1, and " i s g r e a t e r than" o r " i s more than" which c a l l s f o r " > I 1 . Be prepared t o make t h i s d i s t i n c t i o n c l e a r . Thus, his t u r k e y weighs f i v e pounds more than t h a t one" could c a l l f o r t h e

page 155: 5-1

phrase "t + 5", while "This t u r k e y weighs more than twenty pounds" could c a l l f o r t h e sentence "p > 20".

Answers -- t o Oral Exerc ises 5-1; page 155: - This e x e r c i s e i s intended t o provide exper ience .in t r a n s -

l a t i n g open phrases when t h e t r a n s l a t i o n of t h e v a r i a b l e i s g iven . I t 'I

Pay p a r t i c u l a r a t t e n t i o n t o t h e t r a n s l a t i o n s of s i n c e t h i s i s a new not ion not d iscussed I n t h e read ing . S ince t h e m u l t i p l i - c a t i v e inverse and t h e d e f i n i t i o n = a - l i e wel l ahead i n t h e t e x t , it w i l l probably be necessary simply t o make p l a u s i b l e

t 1 t o t h e s tudent t h a t - = - t , r e l y i n g on some examples t o do t h i s . 2 2

One more than t h e number of q u a r t s of b e r r i e s t h a t can be picked i n one hour Two l e s s than t h e number of q u a r t s of b e r r i e s t h a t can be picked i n one hour Twice t h e number of q u a r t s of b e r r i e s t h a t can be picked i n one hour Three more than twice t h e number of q u a r t s of b e r r i e s t h a t can be picked i n one hour One h a l f t h e number of q u a r t s of b e r r i e s t h a t can be picked i n one hour One more than t h e number of r ecords you can buy f o r $3 Two l e s s than t h e number of records you can buy f o r $3 Twice t h e number of r ecords you can buy f o r $3 Three more than twice t h e number of r ecords you can buy

f o r $3 Half as many records a s you can buy f o r $3 One more than t h e number of f e e t i n t h e d iameter of a

given c i r c l e Two l e s s than t h e number of f e e t i n t h e d iameter of a given c i r c l e

Twice t h e number of feet I n t h e d iameter of a given c i r c l e Three more than twice t h e number of f e e t i n the d iameter of a given c i r c l e

One half t h e number of f e e t I n t h e d iameter of a given c i r c l e

page 155: 5-1

Answers - t o Problem S e t 5-1; page 155: -- The t r a n s l a t i o n s given below a r e , of course, sugges t ions

on ly . Encourage s t u d e n t s t o use d i f f e r e n t t r a n s l a t i o n s . Perhaps you w i l l want t o handle t h e s e problems a s o r a l e x e r c i s e s . It i s adv i sab le t h a t t h e s t u d e n t should w r i t e ou t t h e t r a n s l a t i o n s f o r some of t h e problems but not t o t h e po in t where it becomes t e d i o u s . I n Problem 1 2 t h e phrase should be t r a n s l a t e d a s it s t a n d s . 8x i s a d i f f e r e n t phrase from x + 7x.

Be s u r e t h a t t h e s t u d e n t ' s response, o r a l o r w r i t t e n , shows t h a t he i s aware t h a t t h e v a r i a b l e r e p r e s e n t s a number. I n t h i s s o r t of problem, f o r example, t h e v a r i a b l e - w i s not "width" but ' t h e number of f e e t i n t h e width," - x i s not "books" but " t h e number of books Mary has , " - b i s not " t h e boy" but " t h e number of y e a r s i n the boy's aget'. Notice a l s o t h a t a c l e a r , c o r r e c t , and smoothly f lowing way t o s a y t h e l a s t phrase i s " the boy Is

yea r s o ld" . If n i s t h e number of books George read i n J u l y , then t h e phrase i s "7 more than t h e number of books George read i n ~ u l y " . I f n i s t h e number of pennies Mary had when she went t o t h e s t o r e , then t h e phrase i s ' t h e number of pennies Mary has l e f t a f t e r she spends 7 of them f o r candy1'.

If x i s t h e number of inches i n Tom's he igh t on h i s e igh th b i r thday , then t h e phrase i s ' t h e number of inches i n Tom's h e i g h t on h i s n i n t h b i r t h d a y i f he grows 2 inches dur ing t h e y e a r " .

I f x i s t h e number of people t h a t a c e r t a i n bus can hold, then t h e phrase i s "the number of people I n t h e bus i f t h e r e a r e two empty p l a c e s " .

I f n i s t h e number of couples a t t e n d i n g a dance, then t h e phrase i s " t h e number of people a t t h e dance".

If n i s t h e number of mi les from A t o B, then t h e phrase i s "one more than t h e number of mi les from A t o B and back".

If n is t h e number of h a t s Linda has , then t h e phrase i s

' t h e number of h a t s Joyce has i f she has one l e s s than twice t h e number Linda h a s " .

pages 155-156: 5-1 and 5-2

I f n i s t h e number of people i n a c e r t a i n c i t y , then t h e

phrase i s " t h e number of people owning c a r s i f one t h i r d of t h e number of people i n t h a t c i t y own c a r s " . The t e a c h e r

might mention t h e r e s t r i c t i o n on t h e domain of n .

I f n i s t h e number of oranges on t h e t a b l e , then t h e phrase i s t h e number of oranges Tom p u t s i n h i s basket i f h is

mother f i r s t pu t s another orange on t h e t a b l e and Tom then

t akes one-third of t h e oranges and p u t s them i n t o h i s basket1 ' ,

If r i s a c e r t a i n number which Harry chooses, then t h e phrase Is ' t h e number Harry g e t s i f he doubles t h e number he

chose and then adds 5 t o t h e r e s u l t " .

If r Is the number of points made b y Frank I n h i s first

game, then 2 r - 5 i s t h e number made by Joe i f he scores

f i v e l e s s than twice a s many a s Frank.

If x i s a c e r t a i n number, then x + 7x i s t h e sum of t h a t number and one seven t imes a s g r e a t .

If t i s t h e number of s t u d e n t s i n Mr. White's c l a s s , then t ~ + 3 i s t h e number i n Miss Brown's c l a s s when h e r c l a s s has

t h r e e s t u d e n t s more than h a l f a s many a s M r . White's. Again

t h e r e i s a r e s t r i c t i o n on t h e domain of t .

I f r i s t h e number of d o l l a r s Mary has i n h e r purse , then

3r + 1 i s t h e number of d o l l a r s B i l l has when he has one

d o l l a r more than t h r e e t imes a s much a s Mary.

If t i s t h e number of miles covered by t h e Jones fami ly on

t h e first of t h e i r summer t r i p s , 2t - i s t h e number of 3

miles covered I n one t h i r d of t h e second t r i p i f i t i s t o be

one mile l e s s than twice t h e l e n g t h of t h e f i r s t t r i p .

Word - Phrases Phrases . Great c a r e I s taken throughout t h e c h a p t e r t o p o i n t ou t t h a t

a v a r i a b l e r e p r e s e n t s a number. We have seen t h a t no m a t t e r what - - physical problem we may be concerned wi th , when w e make a mathematical t r a n s l a t i o n we are t a l k i n g about numbers.

On t h i s po in t it may seem t h a t , i n t h e example invo lv ing l i n e

segments given I n Sect ion 5-2 of t h e text , we v i o l a t e our

pages 154-155: 5-2

i n s i s t e n c e upon t h e f a c t t h a t t i s a number. Care should be taken t o emphasize i n t h i s example t h a t t i s indeed a number. The phras ing I n t h e problem, however, i s of a k ind t h a t t h e s t u d e n t s are going t o s e e . They might as wel l get used t o It and understand t h a t even though we t a l k t h i s way we a r e us ing a v a r i a b l e t o r e p r e s e n t a number, not a s a l i n e segment.

Some of t h e problems i n t h i s c h a p t e r may involve more than one v a r i a b l e o r may suggest t h e use of more than one v a r i a b l e . This should be allowed t o happen c a s u a l l y . I n t h e c a s e of open sen tences you may have oppor tun i ty t o show t h e p o s s i b i l i t y of a compound sen tence . It i s t o o e a r l y t o be a b l e t o show t h e n e c e s s i t y of a compound sentence f o r a unique s o l u t i o n . Since we a r e not a t p r e s e n t look ing f o r answers it w i l l no t be necessary t o worry y e t about how we w i l l f i n d t h e t r u t h s e t . Nevertheless, t h e s t u d e n t should be encouraged t o use one v a r i a b l e only when- e v e r he Is a b l e t o , s o t h a t , f o r example, consecut ive whole numbers would be represen ted by x, x + 1, and x + 2, r a t h e r than by x, y and z. I f t h e examples i n t h e t e x t have been a t a l l e f f e c t i v e and i f t h e t r a n s l a t i o n s of t h e previous s e c t i o n s were done s a t i s f a c t o r i l y , then i t seems l i k e l y t h a t t h e use of more than one v a r i a b l e w i l l be, f o r most s t u d e n t s , a s o r t of l a s t r e s o r t measure. I n many of t h e s e c a s e s t h e t e a c h e r can a i d t h e s t u d e n t i n t h i n k i n g through t h e problem again s o a s t o permit t h e s t u d e n t i n e f f e c t t o r e d e f i n e one v a r i a b l e I n terms of another .

Answers -- t o Oral Exerc i ses E; pages 157-158:

Help t h e s t u d e n t t o n o t i c e t h a t when t h e v a r i a b l e Is given I n t h e problem, i t i s no t necessary f o r him t o t e l l about It, but i f t h e problem does not g i v e t h e v a r i a b l e , it i s t h e s t u d e n t ' s r e s p o n s i b i l i t y t o choose a l e t t e r and t e l l what it r e p r e s e n t s . Exerc i ses 8 through 13 r e q u i r e t h e s tuden t t o choose t h e

v a r i a b l e . Encourage t he use of d i f f e r e n t l e t t e r s of t h e a lphabet . By t h i s means i t i s hoped t h a t s t u d e n t s w i l l r e a l i z e t h a t t h e meaning o r d e f i n i t i o n of t h e symbol Is t h e important considera- t i o n r a t h e r than t h e cho ice of t h e symbol t o be used.

For many of t h e fo l lowing problems t h e r e a r e implied r e s t r i c - t i o n s on t h e domain of t h e v a r i a b l e . While w e o r d i n a r i l y l e t such r e s t r i c t i o n s remain implied because they seem q u i t e obvious,

pages 157-158: 5-2

t h e r e would be some value i n o c c a s i o n a l l y d i s c u s s i n g wi th t h e s tuden t s what r e s t r i c t i o n s a c t u a l l y e x i s t . For i n s t a n c e , i n Exerc ises 8 and 9 , t h e domain of t h e v a r i a b l e i s t h e s e t of

1 mul t ip les of - QQ ; t h a t i s , when t h e v a r i a b l e r e p r e s e n t s a z / number of d o l l a r s , t h e domain cannot Inc lude numbers l i k e 7 .

In Exerc ises 17 and 18, t h e domain i s t h e s e t of whole numbers; . i n Exerc ise 19, t h e s e t of m u l t i p l e s of 5 ; i n Exerc i se 20, t h e

1 A s e t of mul t ip les of - 3'

Of course, a t t h i s p o i n t i n t h e course we a r e r e s t r i c t i n g ourse lves t o t h e numbers of a r i t h m e t i c , but most of t h e problems of t h i s chap te r by t h e i r n a t u r e g ive on ly non-negative numbers i n t h e domain anyway.

k + 7 25t ; lOOn

n + 5 n - 5 5n n + 5 1 4x I f q i s t h e number of d o l l a r s i n t h e bank, then t h e phrase I s q + 7. If s i s t h e number of d o l l a r s i n t h e bank, then t h e phrase i s s - 7. If Sam i s b yea r s o l d , then t h e phrase i s b + 4 . If Sam's age I s m years , then t h e phrase i s m - 3 . If Sam i s q yea r s o l d , then t h e phrase i s 2q.

1 If Sam i s c y e a r s o l d , then t h e phrase i s p. 12x

Answers - t o Problem S e t 5-2; pages 158- 160; -- The teacher should be prepared t o t each o r r e t e a c h t h e i d e a s

of per imeter and a r e a which a r e used i n t h e s e problems.

pages 158-1519: 5-2

If n I s t h e number of d o l l a r s Fred has , ( o r , " i f Fred has n d o l l a r s " )

3n + 7

If n i s t h e number of d o l l a r s Ann has ,

3 7 - 7

If w i s t h e number of Inches i n t h e width of t h e rec tang le , ( o r , " i f t h e r e c t a n g l e i s w inches wide") 2w

If n i s t h e number, n + 2n ( ~ e a v e i t i n t h i s form, 3n i s - not a t r a n s l a t i o n of

t h e phrase .)

If c i s t h e count ing number, c + ( c + 1) + ( c + 2 ) ( l e a v e i t t h a t way)

I f q i s t h e even number,

q + ( q + 2 )

If n i s "some" number, ( n + 3 ) 2 o r 2(n + 3 )

I f n i s "some" number, 2n + 3

If t h e r e c t a n g l e i s . n inches wide, n ( n + 1 0 ) (YOU may have t o remind them how t o f i n d a r e a . Be c e r t a i n t h a t n i s

t h e number of inches i n t h e width, not j u s t n is t h e wid th . )

If w i s t h e number of inches i n t h e width of t h e rec tang le ,

w + ( w + 1 0 ) + w + ( w + 1 0 ) (Here t h e y may t h i n k 2 t imes t h e number of inches i n t h e width and 2 t imes t h e number of inches i n t h e l eng th , s o accep t 2w + 2(w + l o ) . ) I f s i s t h e number of u n i t s i n t h e s i d e of t h e square , s + s + s + s O r 4s i f they a r r i v e a t t h i s by t h i n k i n g of 4 t imes t h e number of u n i t s i n t h e s i d e

l O O t + 25(t + 2 ) i s t h e number of c e n t s .

l O d + 25(d + 5 ) i s t h e number of c e n t s . 1 60n + 70(n + ^Â¥ i s t h e c o s t i n c e n t s .

pages 158-1519: 5-2

If n I s t h e number of d o l l a r s Fred has , ( o r , " i f Fred has n d o l l a r s " )

3n + 7

If n i s t h e number of d o l l a r s Ann has ,

3 7 - 7

If w i s t h e number of Inches i n t h e width of t h e rec tang le , ( o r , " i f t h e r e c t a n g l e i s w inches wide") 2w

If n i s t h e number, n + 2n ( ~ e a v e i t i n t h i s form, 3n i s - not a t r a n s l a t i o n of

t h e phrase .)

If c i s t h e count ing number, c + ( c + 1) + ( c + 2 ) ( l e a v e i t t h a t way)

I f q i s t h e even number,

q + ( q + 2 )

If n i s "some" number, ( n + 3 ) 2 o r 2(n + 3 )

I f n i s "some" number, 2n + 3

If t h e r e c t a n g l e i s . n inches wide, n ( n + 1 0 ) (YOU may have t o remind them how t o f i n d a r e a . Be c e r t a i n t h a t n i s

t h e number of inches i n t h e width, not j u s t n is t h e wid th . )

If w i s t h e number of inches i n t h e width of t h e rec tang le ,

w + ( w + 1 0 ) + w + ( w + 1 0 ) (Here t h e y may t h i n k 2 t imes t h e number of inches i n t h e width and 2 t imes t h e number of inches i n t h e l eng th , s o accep t 2w + 2(w + l o ) . ) I f s i s t h e number of u n i t s i n t h e s i d e of t h e square , s + s + s + s O r 4s i f they a r r i v e a t t h i s by t h i n k i n g of 4 t imes t h e number of u n i t s i n t h e s i d e

l O O t + 25(t + 2 ) i s t h e number of c e n t s .

l O d + 25(d + 5 ) i s t h e number of c e n t s . 1 60n + 70(n + ^Â¥ i s t h e c o s t i n c e n t s .

page 162: 5-3

128

If n Is t h e number of f e e t i n t h e length of a p iece of

board, t h e t r a n s l a t i o n i s :

A second p i e c e , which i s 62 f e e t long, i s e i g h t

f e e t more than twice t h e l e n g t h of t h e f i r s t .

I f x i s t h e number of vo tes Joe rece ives , the

t r a n s l a t i o n is: The number of vo tes Joe r e c e i v e s , decreased by

f i v e , equa l s 12 , t h e number of vo tes received

by John.

I f n i s t h e number of u n i t s i n t h e l eng th of one p iece

of paper , t h e t r a n s l a t i o n i s : The l e n g t h of paper needed t o make two p o s t e r s i s

30 inches , i f one p o s t e r is one inch more than

twice t h e l eng th of t h e o t h e r .

If r i s t h e number of u n i t s i n t h e l eng th of a

r e c t a n g l e , t h e t r a n s l a t i o n is: The a r e a of a r e c t a n g l e i s 18 square u n i t s ,

i f t h e width i s t h r e e u n i t s l e s s than t h e l e n g t h .

If r i s t h e number of u n i t s i n t h e length of a shee t

of c o n s t r u c t i o n paper , t h e t r a n s l a t i o n is :

A s h e e t of c o n s t r u c t i o n paper does not have an

area of 18 square inches , i f i t s width is

3 Inches s h o r t e r than its l e n g t h .

I f t i s t h e number of yards gained i n t h e first play

of a f o o t b a l l game, t h e t r a n s l a t i o n i s : The team gained twenty yards i n two p l a y s . I n

t h e second p l a y t h e team gained one yard l e s s

than 3 t imes t h e number of yards gained i n

t h e f i r s t p l a y .

I f t Is t h e number of d o l l a r s Mike has , t h e t r a n s l a t i o n

18:

The number of d o l l a r s Robert has i s one d o l l a r

l e s s than t h r e e t imes t h e number Mike h a s . John,

who has 20 d o l l a r s , does no t have t h e same num-

b e r of d o l l a r s a s Robert and Mike have t o g e t h e r .

page 162: 5-3

Answers - t o Problem - Set m; page 162:

We give suggested t r a n s l a t i o n s f o r t h e open s e n t e n c e s ; a s s ign ing them t o t h e p u p i l s w i l l produce a g r e a t v a r i e t y of

t r a n s l a t i o n s . One way of t e s t i n g t h e c o r r e c t n e s s of t h e s tuden t

t r a n s l a t i o n s might be t o d i s t r i b u t e them about t h e c l a s s and

have t h e p u p i l s try t r a n s l a t i n g them back i n t o open sen tences .

This would a l s o se rve t o g ive t h e pup i l s a s t a r ^ on the work

of t h e fo l lowing s e c t i o n by having them f i r s t t r a n s l a t e p u p i l -

made problems i n t o open sen tences .

Let n be t h e number of books i n B i l l ' s desk . Five

t imes t h e number of books i n B i l l ' s desk i s 25. Let y be t h e number of y e a r s i n Harryts age now.

Five yea r s from now Harry w i l l be twenty y e a r s o l d .

Let t be t h e number of inches i n t h e l e n g t h of t h e

board. Af te r f i v e inches i s sawed o f f a board t h e

remaining p iece i s 20 inches long .

Let t be t h e number of d o l l a r s i n t h e t o t a l amount.

Each of f i v e persons rece ived 20 d o l l a r s when t h e

money was d iv ided .

Let n be t h e number of d o l l a r s Frank h a s . John has

t h r e e d o l l a r s . Two t imes t h e number of d o l l a r s Frank

has p l u s what John has i s 47 d o l l a r s .

Let n be t h e number of f i r e c r a c k e r s Frank bought.

John bought twice a s many f i r e c r a c k e r s a s Frank d i d .

Af te r he used 3 he had 47 l e f t . Let x be t h e number of inches i n one s i d e of t h e square .

The per imeter of a square i s 90 i n c h e s .

Let n be t h e number of d r e s s e s Jean h a s . Mary had

4 t imes as many d r e s s e s a s J e a n . Alice had 7 t imes

a s many as Jean . Together Alice and M a r y had 44.

Let k be t h e number of hours Harry and B i l l rode .

Harry and B i l l rode t h e i r b ikes f o r t h e same l e n g t h of

t ime. Harry t r a v e l e d 5 mi les p e r hour, B i l l t r a v e l e d 1 2 miles p e r hour. They t r a v e l e d i n oppos i t e d i r e c t i o n s and were 51 miles a p a r t a t t h e end of t h i s pe r iod of

t ime.

Let n be t h e number of f e e t i n t h e width of t h e

r e c t a n g l e . The l e n g t h of a r e c t a n g l e i s twice t h e width . I t s a r e a i s 300 square f e e t .

pages 162-166: 5-5 and 5-4

11. Let n be the number of f ee t i n the width of the

rectangle . The length of a rectangle i s two fee t more than the width. I t s area i s 300 square f e e t .

12. Let w be the number of f e e t i n the longer side of the rectangle . One s ide of a rectangle i s four f ee t l e s s than the o ther . I t s area i s 16 square f e e t .

*13. Let x be the number of do l l a r s John has. J i m has one d o l l a r more than three times the number John has. Together they have 46 do l l a r s .

*lh . Let y be the number of blocks B i l l walked. John walked 3 blocks a f t e r walking twice as f a r as B i l l .

Tom walked 3 blocks a f t e r walking the same distance as B i l l . John and Tom walked a t o t a l of 50 blocks.

5-4, Word Sentences t o Open Sentences. - - In t h i s lesson, we turn our a t ten t ion t o verbal problems.

You w i l l not ice tha t a question i s asked i n each of the problems. Ear l i e r i n the commentary i t was pointed out tha t the question serves t o help the student f e r r e t out the number he i s Interested i n -and t o make the most f r u i t f u l t r ans la t ion .

The "guessing" method employed in the examples of t h i s lesson i s usually an ef fec t ive one f o r students who are troubled by the abstract ion of switching from a word problem t o an open sentence. You may want t o make even grea ter use of t h i s approach than indicated i n the t e x t . For many students, t h i s guessing technique may remain the best way t o make t rans la t ions indepen-

dent ly. The short exposition on page 165 concerning " i s l e s s than"

and " i s 5 l e s s than" r e s u l t s from past experience i n which many students tend t o see these phrases a s saying essent ia l ly the same th ing . Thus, such meaningless t rans la t ions of "5 i s 4 l e s s than 9" as "5 = 4 < 9" have a r i sen . Hence the warning t o the student a t the end of t h i s sect ion.

Answers -- t o Oral Exercises - 5-4; pages 166-167:

The emphasis.in Exercises 3-19 i s on the t rans la t ion t o sentences. Exercises 8, 9, and 11 through 22 involve variables.

pages 166-167: 5-4

It i s not essent ia l t h a t the t r u t h s e t be found, b u t I f the s tudents want t o do so, permit them t o have t h i s fun.

1. ( a ) n (any variable may be chosen) ( b ) n - 8

Â

(d n(n - 8 ) ( d ) n ( n - 8 ) = 180

2 . ( a ) r represents the number of Inches of length of the short piece.

( b ) r + 3 (4 r + (r + 3 ) ( d l r + (r + 3 ) = 39

30 = 17 + 13 1 4 17 - 3 14 < 10 1 4 = 10 - 7 This i s a f a l s e sentence, b u t i t Is -

a sentence. 42 = 32 + 10

2 m = m + 3 s = 2 s - 5 s < 2 1 + 5 15 Â¥ 3x - 2~ 5 = 4 ~ + Y I f n I s the number, n = 2n - 3 . I f r i s the number, r < 5r + 3 . I f q Is the number, 3q > 2q + 5.

Answers - t o Problem - Set e; pages 167 - 170:

Most of these problems do not have in teg ra l solut ions. This i s t o prevent the student from guessing the r igh t answer before he has written the open sentence. The emphasis I s on the open sentence, not the t r u t h s e t .

page 167: 5-4

Severa l important p o i n t s might be mentioned again a t t h i s

t ime I n an e f f o r t t o a n t i c i p a t e and f o r e s t a l l t r a n s l a t i o n e r r o r s

by t h e s t u d e n t s :

1. The ques t ion asked i n t h e problem i s t h e most e f f e c t i v e

guide t o t h e s tuden t i n t h e d e f i n i t i o n of t h e v a r i a b l e . (Note

t h a t t h e v a r i a b l e need not always be t h e number which i s the

answer t o t h e problem, though t h i s w i l l o f t e n be t h e c a s e . )

2 . Any o t h e r numbers needed i n t h e problems should be

s t a t e d i n terms of t h e one named by t h e v a r i a b l e . Thus we say,

" I f t h e s h o r t e r p iece i s x inches long, t h e longer p iece i s

(x + 3 ) Inches long." O f course some s i t u a t i o n s may n a t u r a l l y l end themselves t o t h e use of two v a r i a b l e s . A s we have s a i d before , t h e r e i s no ob jec t ion i n t h i s c h a p t e r t o inc luding an

occas iona l example of t h i s s o r t .

3 . There should be a d i r e c t t r a n s l a t i o n i n t o an open

sen tence . Thus i n Problem 1 of t h i s Problem S e t , while we could

change t h e sen tence t o 3x = 80, such a sentence i s not a d i r e c t

t r a n s l a t i o n of t h e problem. It does not r e a l l y t e l l t h e s t o r y .

A good t e s t of a d i r e c t t r a n s l a t i o n i s t o s e e whether, with t h e

d e s c r i p t i o n of t h e v a r i a b l e , t h e sentence can be t r a n s l a t e d

r e a d i l y back i n t o t h e o r i g i n a l problem.

The form i n which t h e s t u d e n t i s t o w r i t e t h e s e problems i s suggested i n t h e examples i n t h e t e x t . Some freedom of form i s d e s i r a b l e , of course , bu t c e r t a i n l y a c l e a r d e f i n i t i o n of t h e

v a r i a b l e should appear a long wi th t h e sentence . Frequently t h e

s t u d e n t w i l l f i n d i t h e l p f u l t o w r i t e o u t phrases , e s p e c i a l l y the

more complicated ones, i n terms of t h e v a r i a b l e , before w r i t i n g

t h e sen tence . Thus a t y p i c a l example might have t h i s appearance:

1. If n i s t h e number,

then 2n i s twice t h e number,

and n + 2n = 80.

Other problems w i l l o c c a s i o n a l l y be w r i t t e n out i n t h i s manner i n

t h e answers below; i n most c a s e s , however, only t h e sentence i s

w r i t t e n , s i n c e t h e form i s s i m i l a r i n a l l problems.

pages 168-170: 5-4

Let n be t h e number of n i c k e l s , and d t h e number of dimes. Then d + 2 Is t h e number of q u a r t e r s . We o b t a i n t h e open sentence

The f a c t t h a t n and d can be only p o s i t i v e I n t e g e r s makes it p o s s i b l e t o determine seven s o l u t i o n s . The p o s s i b l e va lues f o r d a r e 1, 2, 3, 4, 5, 6, and 7, and t h e corresponding values f o r n are 48, 41, 34, 27, 20, 13 and 6. (x + 3 ) + (2x + 3 ) - 30 x + 3 - 3x - 3 , i f t h e t a b l e i s x f e e t wide 5(x + 20) = 5x + 100, i f t h e speed of t h e f r e i g h t i s x mi les p e r hour . Notice t h a t a l l va lues of x are t r u t h numbers of t h e open sentence . Let x be the number of q u a r t e r s ; then 3x i s t h e

pages 170-171: 5-4 and 5-5

number of d imes and 2x t h e number of n i c k e l s . Let y be the number of c e n t s John h a s . Then

The s o l u t i o n s e t would c o n s i s t of an i n f i n i t e s e t of p a i r s of p o s i t i v e i n t e g e r s of t h e form (x , 65x).

5-5. Other T r a n s l a t i o n s .

Here we extend t h e s t u d e n t ' s exper ience wi th t r a n s l a t i n g t o

sentences i n v o l v i n g i n e q u a l i t i e s . The term " i n e q u a l i t y " a s well a s t h e word "equat ion" i s not in t roduced i n t h e t e x t u n t i l Chapter 9 . The e x p o s i t i o n I n t h i s s e c t i o n of t h e t e x t p a r a l l e l s t h e e a r l i e r p r e s e n t a t i o n of sentence t r a n s l a t i o n i n Sec t ions 5-3 and 5 - 4 .

While w e a r e not a t t h e moment concerned wi th f i n d i n g t h e t r u t h sets of sentences , it i s l i k e l y t h a t t h e s t u d e n t w i l l be a t l e a s t o c c a s i o n a l l y I n t e r e s t e d I n d i scover ing t h e "answers" t o t h e problems f o r which he has w r i t t e n sen tences . Thus it i s almost c e r t a i n t h a t i t w i l l be no t i ced and pointed ou t t h a t i n e q u a l i t i e s f r e q u e n t l y have many numbers i n t h e i r t r u t h s e t s , i n s t e a d of J u s t one, a s was o f t e n t h e case wi th equa t ions . Here t h e i d e a of t h e open sentence a s a " s o r t e r " of t h e domain of t h e v a r i a b l e can be re-emphasized. A l l e lements of t h e domain which make t h e sentence t r u e a r e p o s s i b l e "answers" t o our word problem, and those e l e - ments of t h e domain which make t h e sentence f a l s e cannot be ' answers" t o t h e problem. Though we l a c k a d e f i n i t i v e s i n g l e 11 answer," we have a c l e a r l y de f ined s e t of "answers," i . e . , t h e t r u t h s e t of t h e sen tence .

Answers t o Oral Ekerc i ses 5-5a; page 171: -- Â

( ~ h e s e a r e p o s s i b l e t r a n s l a t i o n s . ) The t e a c h e r may want t o omit some of t h e l a t t e r e x e r c i s e s of t h i s s e t , p a r t i c u l a r l y if it seems t h a t prolonged background d i s c u s s i o n i s needed regarding t h e geometry upon which t h e t r a n s l a t i o n s would be based.

1. If a Is t h e number o f boys i n c l a s s , t h e number of boys i n c l a s s i s less than 3 .

2 . If a i s t h e number of d o l l a r s i n Joe ' s pocket , t h e number of d o l l a r s i s g r e a t e r than 5 .

page 171: 5-3

3 . I f n i s the number of books needed, the number of books increased by one i s grea ter than 17.

4. I f n i s the number of points made by Harry, one point more than the number i s l e s s than 17.

5. I f t I s the number of penci ls , four more than three times the number i s l e s s than 12.

6 . I f John i s x years old, John's age i s grea ter than 10 and l e s s than 15.

7 . I f m i s the number of hours required t o do a job, the time required is a t l e a s t 3 hours and no more than 1 2

hours. 8. If n i s the number of yards gained on the f i r s t play,

and the second play gained f ive yards more than the f i r s t , the sum of the yardage gained on the two plays i s greater than 35 yards.

9 . I f a i s the score earned by Mary, b is the score earned by Jane, and c i s the score earned by Mike, Mary's score i s grea ter than Jane's and the sum of Mary's and Jane's scores i s grea ter than Mike's.

10. If a bag of change contains n nickels, two more dimes than nickels, and one l e s s quarter than nickels, the sum of money i n the bag is grea ter than 4 d o l l a r s .

11. I f a i s the number of uni t s i n the base of a t r i ang le and i f the height is two uni t s more, the area of the t r iangle is grea ter than 20 square u n i t s .

1 2 . I f 1 is the number of u n i t s i n the width of a rectangle and i f the length i s one u n i t more, the area i s not more than 37 square u n i t s .

13. I f the radius of a c i r c l e i s increased by one, the area of the new c i r c l e I s a t l e a s t 40 square u n i t s .

1 4 . If the height of a cylinder I s two un i t s grea ter than the radius, a , of the base, the volume of the cylinder i s l e s s than 17 cubic un i t s .

15. A cylinder wi th height 4 has a radius shorter than the height of a box. The box has a base with area 6 . The

volume of the cylinder is a t l e a s t as great as the volume of the box.

pages 171-172: 5-5

16. I f x pounds of s a l t a re used t o make a 1@ solution,

the amount of s a l t i n the solution i s 9 pounds.

17. I f x pounds of candy s e l l i n g f o r $.30 a pound i s mixed

with some weighing two pounds more and se l l ing a t $ .40

a pound, the mixture is worth $3.40. 18. I f the height of a t r i ang le i s one u n i t more than the

base, b , the area i s no more than 15 square uni t s .

19. I f the width of a rectangle i s two inches l e s s than the

length, the perimeter Is l e s s than 19.

20. I f one side of a t r i ang le i s twice the f i r s t s ide , a ,

and the t h i r d side i s one l e s s than three times the 1 f i r s t , the perimeter i s greater than 12Ã‘

Answers - t o Problem Set 5-5a; page 172: - The following are suggested t r ans la t ions . Encourage a variety

of t r ans la t ions .

1. I f t i s the number of boys i n the club, the number of

boys i n the club I s l e s s than 6 . 2 . I f t i s the p r i ce of a sweater i n do l l a r s , the price of

the sweater i s grea ter than 6 do l l a r s .

3 . I f y i s the number of students i n the c l a s s , the c lass

w i l l have l e s s than 60 students when 15 more join.

4 . I f y i s Jimmy's score on a t e s t , he w i l l have a score

of more than 60 i f he gets 15 points bonus.

5 . I f y i s the number of cars on the parking l o t , a l o t

10 times as b ig could hold more than 80 ca r s .

6. I f r i s the pr ice of a stamp i n pennies, 25 stamps

would cost l e s s than 2 do l l a r s .

7 . I f x i s the length of a section of fence i n f e e t , two

sect ions of fence plus a gate t h a t i s 5 f e e t long w i l l

cover more than 50 f e e t .

In the ea r ly par t of our work w i t h t rans la t ion we

have been t ry ing t o emphasize the idea t h a t the variable

represents a number by being reasonably precise i n the

language. Thus we have been saying, "the number of

do l l a r s i n the pr ice of the sweater" or the "number of

inches i n the length of a r e c t a n g l e . As we go on, we

w i l l allow ourselves t o become more relaxed i n order t o

pages 172-176: 5-5

"peak more f luen t ly . Thus, we may say, "x i s the l e n g t h

of a section of fence i n f ee t " when there i s no doubt tha t we mean "x i s the number of f e e t i n the length of

a section of fence".

8. I f x i s the weight i n pounds of a sack of f l o u r , two sacks of f lour plus 5 pounds of sugar weigh l e s s than

50 pounds.

9 . I f a i s the number of years i n Mary's age, i f Jane i s

twice as old as Mary, and i f Sal ly i s three times as old

a s Mary, the sum of t h e i r ages I s more than 48.

* lo . Same as above.. .sum of t h e i r ages i s grea ter than or

equal t o 4 8 .

Answers t o Oral Exercises 5-5b: pages 174-175: -- 1. I f x i s the number of do l l a r s John has, x > 50.

2. I f y i s the number of students l i v i n g i n the c i t y ,

y < 150. 3 . I f r i s the height of the plane i n f e e t , r - < 30,000.

4 . If s I s the height of the plane i n f e e t , s - < 50,000

and s > 5280. 5 . If q i s John's weight in l b s . , q + ( q + 10) > 220. 6. If b I s the number of brothers Jane has and c i s the

number of brothers Mary has, c > b .

Answers t o Problem Set 3-5b; pages 175-176: - - 1. I f x i s the number of do l l a r s Tom has, x > 200.

2. I f y i s the number of people t h a t went t o the park,

y < 100.

3 . If t i s the number, 4t + 9 t > 100.

4. I f r I s thenumber, 7 r > 4 5 .

5. I f n i s the number, 8n - 3n < 10. 1 x > 26. 6 . I f x I s the number, 3 x + -

7. If h i s the a l t i t u d e i n f e e t a t Denver, h > 5000.

8. I f m i s the number of people who l i v e i n Mexico,

180,000,000 > 2m. 9. If r i s the number of years i n Norma's age,

r + (r + 5 ) < 23.

pages 176-173: 5-5

10. I f h i s t h e

h - < 4 . 11. If k i s t h e 1 2 . If c i s t h e

1 3 . If m i s t h e and m < 7 .

number of hours on t h e Job, h > 2 and

number k i l l e d , k > 250 and k < 300. speed of t h e c u r r e n t , c + 1 2 < 30. number of minutes of a d v e r t i s i n g , m > 3

1 4 . If x i s t h e l e n g t h of a s i d e of t h e square,

x + x + x + x = ( x + 5 ) + ( x + 5 ) + ( x + 5 ) 15. If r i s t h e number of s t u d e n t s who remained and n i s

1 t h e number of s t u d e n t s e n r o l l e d , then r < 7 n - 152. This problem i s s i m i l a r t o Exerc ise 6 of Oral Exerc ises 5-5b, i n t h a t a sentence f o r i t must be expressed i n terms of two v a r i a b l e s . There w i l l doub t l e ss be con- s i d e r a b l e d i s c u s s i o n of both e x e r c i s e s .

Summary The f irst p a r t of t h e summary i s a p a r t i n g e f f o r t t o

s t r e n g t h e n t h e i d e a of t r a n s l a t i o n back and f o r t h between a p h y s i c a l s i t u a t i o n and a mathematical ( o r numerical) one.

The l a t t e r p a r t of t h e summary reviews, by means of examples, t h e k inds of t r a n s l a t i o n s t h a t have come up i n t h i s chap te r .

Answers - t o Review Problem S e t ; pages 178-183:

t be t h e number of marbles I n one j a r . The number of marbles i n 2 j a r s 3 more than t h e number of marbles i n one J a r The number of marbles i n 3 j a r s , each holding a s many as t h e f i r s t , a f t e r two marbles have been removed The number of marbles i n one j a r a f t e r one has been removed A f t e r one marble i s removed from one j a r , 5 marbles a r e l e f t i n t h e jar. I f w e take ou t one h a l f of t h e marbles i n one J a r , we

w i l l t a k e out less than 4 marbles. I f we count t h e marbles i n two jars t h e number of marbles i s g r e a t e r than 6 . There a r e a t l e a s t 6 marbles i n one j a r .

pages 179-180

2. (a) 7w, w is the number of weeks.

(b) x(2x), x is the number.

(c) 3x + 5, x is the number of students.

(d) 7(x - 5), x is the number from which 5 is to be

subtracted.

(e) -$(x(2x)), x is the number of units in the length of the

shorter side of the rectangle.

(f) 5x + 10(2x), x is the number of nickels.

(g) 1.40 x -1- .30(x + l), x is the number of pounds of

chocolates.

(h) $(x)(x + 3$), x is the number of units in the base.

(1) T x2(x + $), x is the number of inches In the radius of the base.

( J ) (x + 21) 8.9 - , x is the number.

(k) .20 x, x is the number of gallons of salt solution.

(1) 2x - 4, x years is Mary's age now. (m) 25x + 32(x + 2), x is the number of loaves of bread.

(n) 25x -I- 10(x J - 3) + 5(x - 2), x is the number of quarters. 1

(0) $(x) (^- x - 2), x is the length of the base.

( p ) + 2x, x is the original number. - 2

(q) $(x + (x + 3$) ) (x - l), x 1s the number of inches in the length of the shorter base.

(r) x + 2x + 2(2x) x is the number of units in the length of the shortest side.

3. (a) x + 3x = 45, x is the number.

(b) x + (x + 1) = 45, x is the first number.

(c) Insufficient information

(d) x + 4 = 16, x years is Mary's sister's age.

(e) x > 14,000, x feet is the height of Pike's Peak. (f) x + (x + 2) = 75, x is the first odd number. This

problem has no solution since the sum of two odd numbers

is even, but we can still write the open sentence.

(g) 3x 2 x + 26, x is the number of students in the class.

pages 180-181

x 4- (2x + 20) -I- 70 = 180, x is t h e number of degrees in the measure of t he smal les t angle. Here the var iable x does - not represent t he number of degrees i n the measure of t h e l a r g e s t angle which appeared i n t he quest ion. x + (x 1-48) = 1124, x i s t h e smaller even number. x = 18 + 22, x i s t h e sum of t he numbers of years i n the ages of each 6 years from now.

2 (x + I ) ~ - x = 27, x u n i t s i s the s i d e of the smaller square. 8x = 12(5 - x ) , x hours i s t h e time spent r id ing i n t o t h e country.

2(7x + x) = 150, x inches i s t h e width of the rectangle . 2 1 7 x + 32 = 385-, x is t he number.

. lox = .025(x 4- 2 0 ) , x i s the number of pounds of the

o r i g i n a l so lu t ion . x - .2Qx = 29.95, x d o l l a r s is the o r i g i n a l p r ice .

1 62-x - 3 9 . 7 ~ = 125& x hours i s the required time. 2

x + (x + 4) + ((x + 4) + 6) = 50, x is the number of

pennies.

a l l numbers g r e a t e r than ze ro o I I I I I

0

a l l numbers l e s s than 1 5

a l l numbers equal t o o r I I I I A I ^ -

g r e a t e r than 4 0 1 2 3 4

pages 181 -182

( k ) all numbers greater than 7

or equal to $ (1) all numbers

4 4 1 4 1 (d) ax(-+') or a(x( ) + x ( ~ ) ) or x(a(.) + a(,.)) 5 5

4 1 (? + ^Â¥)a or other answers as in (d) above.

2 2 2 a b + T a b (h) 8 . 2 5 ~ + 3.96

No. Yes. No. No. No. Yes. No. No. No. No.

1 + 3 = 4. 4 is not an element of the set. 2 + 2 = 4 , 2 + 4 = 6 , 2 + 6 = 8 , etc. 1 + 5 = 6. 6 is not an element of the set. 30 + 2 = 32. 32 is not an element of the set. 30 + 5 = 35. 35 is not an element of the set.

o + o = o

1 + 1 = 2. 2 is not an element of the set. 1 + 3 = 4. 4 is not an element of the set. 5 + 10 = 15. 15 is not an element of the set. 10 + 10 = 20. 20 is not an element of the set.

pages 182-163

8. ( a ) 3

Suggested Test Items -- 1. Transla te the following word phrases i n t o open phrases.

( a ) t h r ee l e s s than twice t he number n ( b ) the product of x and the number which i s 7 times x ( c ) the sum sf 5 times a number d and a number 4 grea te r

than d

( d ) the number of marbles J immy has i f he had m marbles and

was given 1 0 more

( e ) the value i n cents of n n icke ls and 4 pennies

2. Transla te each o f the following i n t o an open phrase o r i n t o an open sentence, using a s ing le var iab le i n each. F i r s t t e l l

what the va r i ab l e represents .

( a ) $30 more than Jim's weekly salary

( b ) Tom's weekly salary i s more than $30.

( c ) Tom's weekly s a l a r y i s $30 more than J i m ' s . Together they

earn $140 p e r week.

(d) Tom's weekly s a l a r y Is $30 more than Jim's. Together they

earn more than $140 pe r week.

3. Write a word t r a n s l a t i o n f o r each of these phrases. Make your

word phrase as meaningful as possible .

(d) lorn + 25m

(4 4(a - 3 ) ( f ) x i - 3(x - 2 )

Write an open phrase which describes the following statement:

Choose a number and then add 4 t o it. Multiply

t h i s sum by 3. Subtract 5 from t h i s product.

If p i s the f i r s t of three consecutive odd numbers, then the second odd number i s

the t h i r d odd number is

the sum of the f i r s t and t h i r d numbers i s Â

Complete the following two problems so tha t each problem corre-

sponds t o the given open sentence.

6. open sentence: a + 4a + 25 = 180 Problem: The perimeter of a t r i ang le i s 180 inches.

7. Open Sentence: 5(x + 4) + l 0x = 125

Problem: John found a b i l l f o l d containing $125.

Write an open sentence or phrase f o r each of the following: 8. The area A i n square f e e t of a rectangle whose length i s

x yards and width i s y f e e t .

9. In an orchard containing 2800 t rees , the number of t r e e s i n

each row is 10 l e s s than twice the number of rows. How many

rows are there?

10. B i l l weighs 10 pounds more than Dave. Find Dave's weight i f

the combined weight of the two men i s 430 pounds.

11. Jack i s 3 years older than Ann, and the sum of t h e i r ages

i s l e s s than 27 years. How old i s Ann?

12. The number of cents Paul has i f he has d dimes and three times a s many quarters as dimes.

13. If a boy has 250 yards of chicken fence wire, how long and

how wide can he make h i s chicken yard, i f he would l i k e t o have the length 25 yards grea ter than the width?

14 . There a re f ive large packages and three small ones. Each

large package weighs 4 times a s much a s each small one, and

the eight packages together weigh 34 pounds 8 ounces. What

i s the weight of each package?

15. Separate $38 i n t o two p a r t s such t h a t one p a r t i s $19 more than t he other .

16. The th ickness of a c e r t a i n number of pages of a book i f each 1 page i s 7p?7- of an inch th ick.

17. The product of a whole number and i t s successor i s 342. What Is the number?

18. A f a t h e r earns twice as much pe r hour a s h i s son. I f the f a t h e r works f o r 8 hours and t he son f o r 5 hours, they earn l e s s than $30. How much does the son earn per hour?

Answers - t o Suggested Test Items --

2. ( a ) Let x be Jim's weekly salary i n do l l a r s . The t r a n s l a t i o n of t h e phrase: x + 30

( b ) If Tom's weekly s a l a r y i s x d o l l a r s , then t he t r a n s l a t i o n is: x > 30

( c ) If J i m ' s weekly s a l a r y i s x d o l l a r s , then Tom's weekly salary i s (x + 30) d o l l a r s . The t r a n s l a t i o n : x + (x + 30) = 140

( d ) If J i m ' s weekly s a l a r y i s x d o l l a r s , Tom's weekly s a l a r y i s (x + 30) d o l l a r s . The t r ans l a t i on : x + (x + 30) > 140

3. (Poss ible t r a n s l a t i o n s )

( a ) 5 more than twice t he number of pennies Jimmy has

( b ) 8 times a s many r a iny days a s i n June

( c ) t he a r ea i n square f e e t of a rec tang le whose width i s

7 f e e t l e s s than i t s leng th

( d ) t he t o t a l cos t i n cen ts of a c e r t a i n number of i c e cream cones a t 10$ each and the same number of sodas a t 25$

each

( e ) t he perimeter i n inches of a square whose s ide i s 3

inches sho r t e r than t h e s i d e of a given square

( 7 ) t he perimeter i n inches of a q u a d r i l a t e r a l th ree of

whose s i d e s a r e of equal length and the four th s ide 2 inches longer

P + 2

P + 4 p + (p + 4 ) which i s 2p + 4

One s i d e i s f o u r t imes a s long a s ano the r s i d e and t h e t h i r d

s i d e i s 25 inches i n l e n g t h . Find t h e l e n g t h of each s i d e .

There were 4 more $5 b i l l s i n t h e b i l l f o l d than $10 b i l l s .

How many $10 b i l l s d i d John f i n d ?

Let n be t h e number of rows. The number of t r e e s i n each row i s 2n - 10 .

n(2n - 10) = 2800

If Dave weighs x pounds,

x + (x + 1 0 ) = 430.

I f Ann i s x yea r s of age,

x + ( x + 3 ) < 2 7

Let t h e width of t h e yard be x y a r d s .

2x + 2(x + 25) = 250

Let t h e weight of each small package be x pounds. Then each

l a r g e package weighs 4x pounds.

3x + 5(4x) = 34$

x + (x + 19) = 38, where x i s t h e number of d o l l a r s i n t h e

smal ler p a r t .

x which can be w r i t t e n

n (n + 1) = 342 where n Is t h e smal le r whole number

Let rn be t h e number of d o l l a r s t h e son e a r n s p e r hour .

Then t h e f a t h e r ea rns 2rn d o l l a r s per hour .

page l o 4

CHALLENGE PROBLEMS

I n the event tha t you should have an exceptionally in teres ted and eager student we have t r i e d t o include a few problems of vary- ing d i f f i c u l t y b u t usually requiring more perseverance and insight than most problems i n the t e x t . We do not recommend these fo r c lass discussion or" as assigned problems f o r the e n t i r e c l a s s . There are , of course, many other resources f o r challenge problems. We recommend publications of the National Council of Teachers of Mathematics and Dover Publications among o thers .

Answers - t o Challenge Problems; pages 184-190:

This I s an In teres t ing study i n arrangements. The 8, 3 , and 2 a re fixed. The f i r s t of the signs may be x, +, or -, and f o r each of these the second of the signs may be x, + or -. Then there a re two ways i n which parentheses may be inser ted, grouping e i t h e r the f i r s t two terms or the l a s t two. After

a l l t h i s Is done, it is in te res t ing t o note which expressions

are names f o r the same number. For instance:

page 184

distributive property


associative property of addit ion

commutative property of multiplication and distributive property

This is really a trick, although it has an algebraic explana- tion. It may be that pupils will accept it and use it to

simplify mental multiplications. This is good, but not a requirement. In any discussion with a student it should be made clear that the development hinges on the use of 10 as a factor and thus the procedure should be used only for numbers

between 10 and 20.

3. 10 -r a is the first number 10 i- b is the second number

(10 + a)(10 + b) = 100 + 10a + lob + ab by the distributive property

= 10(10 + a + b) + ab by the distributive property

= 10((10 + a) + b) + ab by the associative pro erty of add?tion

page 164



associative property of addition

commutative property of mu1 t ipl icat ion and distributive property

This is really a trick, although it has an algebraic explana-

tion. It may be that pupils will accept it and use It to

simplify mental multiplications. This is good, but not a requirement. In any discussion with a student it should be made clear that the development hinges on the use of 10 as a factor and thus the procedure should be used only for numbers

between 10 and 20.

3. 10 -r a Is the first number 10 + b is the second number (10 + a)(10 + b) = 100 + 10a + lob + ab by the distributive

property

= 10(10 + a + b) + ab by the distributive property

= lO((l0 + a) + b) + ab by the associative pro erty of addition

pages 184-185

1 0 + a i s the first number, and b i s the u n i t s d i g i t of the second number, so (10 i a ) + b i s the sum of the f irst number and the u n i t s d i g i t of t he second number. 10 ( (10 + a ) + b ) i s the r e s u l t of mult iplying the sum of the f irst number and the u n i t s d i g i t of the second number by 10. l O ( ( l 0 + a ) + b ) + ab i s the complete t r a n s l a t i o n of the r u l e .

Since (3 + 5 + 8 + 7 + 4) = 27 = 3(9 ) , we see t h a t 35874 i s

d i v i s i b l e by 9. The general r u l e t o be formulated i s "A

number i s d i v i s i b l e by 9 i f the sum of i t s d i g i t s I s d iv i s - i b l e by 9. I'

Since we hope t o teach pupi l s t o general ize , i t would be well t o take t h i s opportunity t o do j u s t t h a t : Let t he thousand's d i g i t of a four d i g i t number be represented by a , the hundred's d i g i t by b, the t e n ' s d i g i t by c, and the u n i t ' s d i g i t by d. Then the number is:

lOOOa + lOOb + lOc + d = (999 + l ) a + (99 + l ) b + (9 + l ) c + d

9 9 9 a + a + 9 9 b + b + g c + c + d = 9 9 9 a + 9 9 b + 9 c + a + b + c + d = 9 ( l l l a + l l b + c ) + ( a + b + c + d)

Now, 9 ( l l l a + l l b + c ) i s d i v i s i b l e by 9, s ince 9 i s a f ac to r . Therefore, i f ( a + b + c -I- d ) i s a l s o d i v i s i b l e by 9, the e n t i r e number i s d i v i s i b l e by 9, as can be shown by the d i s t r i b u t i v e property. Hence our r u l e t h a t any number i s d i v i s i b l e by 9 i f the sum of i t s d i g i t s i s d i v i s i b l e by 9.

5. (a) 2x belongs t o t he s e t with graph

x + 1 belongs t o the s e t with graph

pages 185-186

2x belongs t o the s e t of numbers between 1 and 10.

x + 1 belongs t o the s e t of numbers between 7 and 6.

x belongs t o the s e t with graph

belongs to the s e t with graph b c x 0-L I 2 3 4 4

( a ) BLACK RED (b) mm BLACK

(c) No; none t o the r i g h t of 0, since each red coordinate i s

four-thirds the corresponding black coordinate. 4 3 ( d ) r = -yb, or b = ~ r .

cmBN 0 I 2 3 4

BLACK 0 I 2 3 4

(a) The black coordinate of the point with green coordinate 1 1 3 i s 1 + + = 6. Yes; every whole number i s the

green coordinate of a point.

page 186

(b) There i s no green coordinate f o r the point with black coordinate 3 . There a r e no green coordinates f o r points t o the r i g h t of black 2. Moreover, black 2 has no corresponding green coordinate.

( c ) The point would have black coordinate I l l 1 1 ' = ^ . l + Â ¥ ^ + Â ¥ ^ Â ¥ + ^ + E + ' ^ +

4 From 1 t o 2 : 7

3 From 1 t o 8 :

1 From 4 . I 3 * 9

b - a 2a + b ( b ) c = a + - , o r c = , ~ r .

9. This problem reviews sums of p a i r s of elements of a set; it i s not a problem s e t up primarily t o get an answer. The pupi l who t r i e s t o write an open sentence w i l l f i nd he i s wasting h i s time. Instead he should observe t h a t the man has a s e t of four elements: (1.69, 1.95, 2.65, 3.15) and that he should examine the s e t of a l l possible sums of p a i r s of elements of the se t .

pages 186-187

his is the set: (3.38, 3.64, 3.90, 4.34, 4.60, 4-84> 5.10, 5.30, 5.80, 6.301

From this we see:

(a) The smallest amount of change he could have is 5.00 - 4.84, or 1.6 cents.

(b) The greatest amount of change possible is 5.00 - 3.38, or $1.62.

(c) There are four pairs of two boxes he cannot afford: one

of $1.95 and one of $3.15; one of $2.65 and one of $3.15;

two of $2.65; two of $3.15.

10. We can write a numeral in powers other than 10, and 8 is as good as any other. For the "8 scale" we need the set of digits (0, 1, 2, 3 , - - - , 7). "8" would be written "10".

(~ead this "one - oh. ")

= eight In case the pupils wish more practice in changing bases we suggest the following:

11. The set of all girls with 2 heads is the null set. We cannot add the number 1 to the null set since it is not 0.

12. This problem was inserted to provide pleasurable experience in reading directions and in translating. Each of the numerals from 0 to 9 represents several letters of the

alphabet. The pupil translates first from numbers to letters, then from letters to numbers. Each translation, and in particular the second kind, involves a choice based on reasoning.

Probably the pupils will all accept "~ane is home" for "9034 = 7424. I' This is a correct translation. However, in case some pupil protests that "9034 = 7424" is not a true sentence, and hence Jane -- is not home, accept the suggestion as a possibility. Nevertheless, make it clear that the translation of "=" is "is" and not "is not." Some excep-

tionally eager students can be encouraged to devise their own codes or problems using letters for numbers and vice versa.

A possible translation is 'he is hungry. I 1

3(1(10) + 2) = 4(10) + 6 3(10 + 2) = 40 + 6

36 = 46 This sentence is not true. 7 + 2 = 2 + 7. This sentence is true. This points toward the commutative property of addition, whose truth, as it applies here to numbers, the pupil will probably accept readily. p(h) indicates the multiplication of h by p, hence

5 ( 7 ) = 7 ( 5 ) . This is a true sentence. (This points toward the commutative property of multiplication. ) Therefore, the sentence p(h) > r(f) is not a true sentence.

6 + 5 # 5 ( 6 ) is true. 8(m) indicates that m is to be multiplied by 8. Hence, we have 8(2) = 2(8). This is a true sentence.

(~ere again we use the commutative property of multiplication. )

pages 187-188

13. I does not r e f e r back t o the code used f o r the previous -

AM problems. Here we a r e t o s e l e c t numbers f o r the THE BOSS l e t t e r s which w i l l make the addi t ion cor rec t , being

care fu l t h a t no number i s represented by two d i f f e r en t l e t t e r s . There a r e many solut ions , such as

SEND has a unique so lu t ion . M must be 1, hence 0 must be zero. S must be e i t h e r 8 o r 9, but Inspection shows t h a t 8 i s impossible, so S Is 9. Consider-

a t i o n of t he second and t h i r d columns shows t h a t R i s 8. Then N must be E + 1. Since 0, 1, 8 and 9 have been used,

N # 3 f o r E # 2 N # 4 f o r E # 3 N # 5 f o r E # 4

so we l e t N be 6 and E be 5. Now we have used 0, 1,

5, 6, 8, 9. D and Y may be chosen from 2, 3, 4, 7. Since t h e sum of D and 5 must be more than 10, D # 2, D # 3, and D # 4. Therefore D i s 7 and Y i s 2.

T h i s problem involves q u i t e a b i t of reasoning f o r n in th graders but t he re a r e always some of them who w i l l work a t it u n t i l the problem is solved. Please do not s p o i l t h e i r p leasure , bu t l e t them reason out t he so lu t ion under t h e i r own power. We hope t o give them many opportuni t ies a t t h i s l eve l .

14. 8432 + 1567 = 9999

2765 + 7234 = 9999 3961 + 6038 = 9999 3(9999) = 30,000 - 3 30,000 + 4028 = 34028 30,000 - 3 + 4028 = 34025. The teacher was cor rec t .

pages 186-189

Another way of writing the second problem might be:

8025 Here again v r e have 3 sums which t o t a l 30,000 - 3. 4567 It i s easy t o add 5,678 t o 30,000 and subtract 3. 3902 5678 1974 5432

There i s a nice extension of t h i s t o 5 and 6 numbers which

the pupil might t ry . The game may a l s o be played with numbers

having more or l e s s than 4 d i g i t s . Pupils might l i k e t o try t h i s a lso.

- --

With the f a c t s already given, the t ab le can be completed by

performing a s ingle mult ipl icat ion: 10(10) = 100. The other

spaces can be f i l l e d a s follows: Use the commutative property

t o complete the l a s t two rows and columns. F i l l i n the row

and the column f o r 2 by use of the d i s t r ibu t ive property. 5 3 (For example: 12(2) = 12(T + T ) = 15 + 9 = 24. Now comes

10(10) = 100. Then 12(10)= 10(10) + 2(10) = 100 + 20 = 120.

Finally, 12(12) = (10 + 2)(10 + 2) = 10(10 + 2 ) + 2(10 + 2 ) =

100 + 20 + 20 + 4 = 144.

16. The s e t S includes at l e a s t the s e t of whole numbers g rea te r

than o r equal t o 2. Since 2 i s not specif ied a s the

smallest element of the s e t we cannot be ce r t a in of i t s lower

bound.

17. ( a ) a 0 b = 2a + b and b o a = 2 b + a . If a = 0 and b = 1, then 2a + b = 1

while 2b + a = 2 so we can see

t h a t 2a + b and 2b + a do not name the same number f o r a l l a and b . The operation i s not commutative.

( b ) a o b = b + a a n d b o a = , - , 7 2 - Since a + b = b + a, we can see t h a t

b + a a + and 7 name the same number f o r a l l 2

a and b . The operation I s commutative.

( c ) a o b = ( a - a ) b and b 0 a = (b - b ) a Since ( a - a )b and ( b - b ) a name the same number (0) f o r a l l a and b, the operation i s commutative.

(d) a o b = a + 3jb and

1 If a = 6 and b = 3 , then a + + - 7 while 1 1 1 b + fa = 5; so we can see t h a t a + $ and b + p

do not name the same number f o r every a and every b.

The operat ion i s not commutative.

( e ) a 0 b = ( a + l)(b + 1 ) and b o a = (b + l ) ( a + 1 )

Since (a + l ) ( b + 1 ) and (b + l ) ( a + 1) name the same

number f o r a l l a and b, t h e operation i s commutative.

18. ( a ) ( a 0 b ) o c = 2 ( 2 a + b ) + c = 4a + 2b + c . a o ( b o c ) = 2a + 2b + c.

If a = 1, b = 1, c = 2, then 4a + 2b + c = 8 while 2a + 2b + c = 6; so we can see t h a t 4a + 2b + c and 2a + 2b + c do not name the same number f o r a l l a, b, and a . The operation I s not a s soc i a t i ve .

pages 189-190

(b) ( a 0 b ) 0 c = a + b + 2 c and

a + b + 2 c So i f a = 1, b = 0, c = 2, then = 3 2 a + b + c while - = 1 and we can see tha t

a + b + 2 c 2 a + b + c - and Ã‘Ã‘Ã‘J, do not name the same number

f o r a l l 9.) b, and c . The operation i s not associat ive.

( c ) (a o b ) o c = 0 and a o ( b o c ) = 0 Since the same number (0) i s named f o r a l l a, b, and c, the operation i s associa t ive .

(d ) ( a o b ) 0 c = a + $ + and 1 a o ( b o c ) = a + 9 + 7

2 It i s c lea r these always d i f f e r by -c so 9 1 1 a + 3 + ?jc and a + 9 + 30 do not name the same

number f o r a l l a, b, and c . The operation i s not

associat ive.

( e ) (a o b ) o c = (ab + b + a + 2)(c + 1) and a 0 (b o c ) = ( a + l ) ( b c + c + b + 2). So if a = 0,

b = 1, c = 1, the f i r s t expression is 6 while the second i s 5. Thus we can see tha t the expressions do not name the same number f o r a l l a,b and c. The operation i s not associat ive.

19. Let x be the number of days it would take the two men 1 together t o paint the house. The f i r s t man can paint

of the house i n one day. The second man can paint if the house i n one day. Together the two men can paint of the house i n one day. The open sentence is I l l ^ + ^ = ?

x = ^ The t r u t h s e t i s [# .

The two men save f irst man working '3c

1 STT days by working together ins tead of the alone.

t h e time saved each day the two men work

toge ther ins tead of t he slower man working alone. For example, i f a job would take t h e two men working together 3 days, t h e first man could do it in 8 days. The saving i n

5 time i s 3 of 3 o r 5 days.

20. Let x be t he number of hours it would take the combination of t he of p ipes t o f i l l t he tank. One pipe can f i l l 5

7

tank i n one hour. The second can f i l l $- of t he tank i n one hour. And t h e t h i r d can d r a i n of t h e tank i n one hour.

-, Working toge ther t h e pipes can f i l l of t he tank i n one hour. The open sentence i s

The t r u t h s e t

The tank w i l l

open. After

1 $ + + + = - x (12 + 20 - 15)x = 60

60 x = - 17

60 i s fw). 6o hours If all pipes a r e l e f t he f i l l e d i n -

- 17 hours t he tank w i l l s tar t t o overflow.

17

Chapter 6

THE REAL NUMBERS

I n Chapters 1 t o 5 t h e s tuden t has been d i scover ing and applying p r o p e r t i e s of opera t ions on a s e t of numbers. This s e t c o n s i s t s of zero and t h e numbers ass igned t o t h e p o i n t s t o t h e r i g h t of zero on t h e number l i n e . H i s work wi th f a m i l i a r num- b e r s gave him s e c u r i t y wi th such concepts a s t h e a s s o c i a t i v e , commutative, and d i s t r i b u t i v e p r o p e r t i e s , open sen tences , t r u t h s e t s , e t c .

With t h i s background, he i s now ready t o g ive names t o num- be r s which we a s s i g n t o p o i n t s t o t h e l e f t of zero on t h e number l i n e . The t o t a l s e t of numbers corresponding t o a l l p o i n t s of t h e l i n e , t h e s e t of r e a l numbers, i s now h i s f i e l d of a c t i v i t y .

I n Chapter 6 we a t tempt t o f a m i l i a r i z e t h e s t u d e n t wi th t h e t o t a l s e t of r e a l numbers. This inc ludes t h e o r d e r of real num- be r s , comparison of r e a l numbers, and t h e opera t ion of de tennin- i n g t h e opposi te of a r e a l number. The f i n a l s e c t i o n i s devoted t o a d e f i n i t i o n and d i scuss ion of t h e a b s o l u t e va lue of a r e a l number.

I n genera l a system of numbers I s a set of numbers and t h e opera t ions on t h e s e numbers. Hence, we do not have t h e r e a l num- be r system u n t i l we de f ine t h e opera t ions of a d d l t l o n and mul t i - p l i c a t i o n f o r r e a l numbers. This i s done i n Chapter 7 ( a d d i t i o n ) and Chapter 8 ( m u l t i p l i c a t i o n ) . Our p o i n t of view i s t h a t t h e opera t ions must be extended from t h e non-negative r e a l numbers t o a l l r e a l numbers. Thus t h e d e f i n i t i o n s of a d d i t i o n and m u l t i p l l - c a t i o n must be formulated e x c l u s i v e l y i n terms of non-negative numbers and opera t ions ( i n c l u d i n g t a k i n g o p p o s i t e s ) on them. It

i s e s s e n t i a l , of course , t h a t t h e fundamental p r o p e r t i e s of t h e s e opera t ions be preserved i n t h i s so -ca l l ed ex tens ion p rocess .

Order i n t h e r e a l numbers i s in t roduced i n Chapter 6. I n Chapter 9 we r e t u r n t o o r d e r , but wi th an important s h i f t i n our po in t of view. Previous ly we have tended t o use o r d e r as a convenient way t o d i s c u s s c e r t a i n a s p e c t s of numbers. I n t h i s sense "<" and ">" were simply fragments of language. I n Chapter 9 we treat "<" a s an o r d e r r e l a t i o n having s p e c i f i c mathematical p r o p e r t i e s I n i t s own r i g h t .

page 195: 6-1

Chapter 10 d e a l s wi th s u b t r a c t i o n and d i v i s i o n . These opera t ions are def ined i n terms of a d d i t i o n and m u l t i p l i c a t i o n . I n t h i s sense we r e t a i n t h e no t ion t h a t t h e r e a l number system is a s t r u c t u r e which may be developed i n terms of - two b a s i c opera- t l o n s .

It should be mentioned t h a t i n t h i s course we have chosen t o approach t h e negat ive numbers i n a manner d i f f e r e n t from some w r i t e r s . Ins tead of p r e s e n t i n g a - new s e t of numbers ( t h e r e a l numbers) and then i d e n t i f y i n g a p a r t i c u l a r subse t of these ( t h e non-negat ive) wi th t h e o r i g i n a l s e t ( t h e numbers of a r i t h m e t i c ) , we have chosen t h e fo l lowing approach. We extend t h e numbers of a r i t h m e t i c t o t h e set of r e a l numbers by a t t a c h i n g - t h e negat ive numbers t o t h e f a m i l i a r numbers of a r i t h m e t i c . This has severa l advantages: F i r s t , w e do not need t o d i s t i n g u i s h between "signedTT and "unsigned" numbers; t o us t h e non-negative r e a l numbers - a r e t h e numbers of a r i t h m e t i c . Second, it is not necessary f o r us t o prove t h a t t h e f a m i l i a r p r o p e r t i e s hold f o r t h e non-negatives, f o r t h e s e p r o p e r t i e s a r e c a r r i e d over i n t a c t a long with t h e num- b e r s of a r i t h m e t i c . I n t h i s manner, we avoid t h e confusion of e s t a b l i s h i n g an "isomorphism" between p o s i t i v e numbers and u n - signed numbers". Notice t h a t we have no need whatsoever f o r t h e ambiguous word s i g n " .

I n g e n e r a l , we have taken t h e p o i n t of view t h a t a n i n t h grade s tuden t r e a l l y - has some experience wi th negat ive numbers. He i s q u i t e ready t o l a b e l t h e p o i n t s t o t h e l e f t of 0 and, i n s o doing, make t h e extens ion t o which we r e f e r r e d .

The t rea tment of abso lu te va lue i n t h i s chap te r exemplif ies what has been r e f e r r e d t o a s t h e ' s p i r a l t e c h n i q u e . The i n t r o - duct ion t o abso lu te va lue i s followed i n each succeeding chapter by more and more uses a t d i f f e r e n t l e v e l s of a b s t r a c t i o n . Thus t h e t e a c h e r need not g ive a f u l l development of t h i s t o p i c I n Chapter 6 s i n c e i t w i l l reappear r e g u l a r l y i n l a t e r p o r t i o n s of t h e book.

6-1. The Real Numbers. -- We in t roduce t h e negat ive numbers I n much t h e same way t h a t

we l a b e l e d t h e p o i n t s on t h e r i g h t s i d e of t h e number l i n e , which correspond t o t h e p o s i t i v e r e a l numbers. O u r no ta t ion f o r negat ive

pages 195-196: 6-1 -

four , f o r example, i s 4, and we d e f i n i t e l y in tend t h a t t h e dash 1 1 - 1 1 be w r i t t e n i n a r a i s e d p o s i t i o n . A t t h i s s t a g e , we do not want t h e s tuden t t o t h i n k t h a t something has been done t o t h e - - - number 4 t o g e t t h e number 4 , but r a t h e r t h a t 4 i s a name of t h e number which i s assigned t o t h e p o i n t 4 u n i t s t o t h e l e f t of 0 on t h e number l i n e . -

I n Sect ion 6-3, t h e s tuden t w i l l be a b l e t o t h i n k of 4 a s t h e number obta ined from 4 by an opera t ion c a l l e d "opposl t ing" . The opposi te of 4 w i l l be symbolized a s -4, t h e dash being w r i t t e n I n a lowered p o s i t i o n , and - 4 w i l l t u r n out t o be a more convenient name f o r -4 .

Since each number of a r i t h m e t i c has many names, s o does each - negative r e a l number. For example, t h e number 7 has t h e names - 14

'(7 x 1 ) , e t c .

I n drawing t h e graph of r e a l numbers, t h e s t u d e n t should be aware t h a t t h e number l i n e p i c t u r e is on ly an approximation t o t h e t r u e number l i n e . Consequently, any Information which he deduces from h i s number l i n e p i c t u r e i s on ly a s a icura te a s h i s drawing.

Once t h e negat ive numbers have been in t roduced, we in t roduce i n t e g e r s and t h e e n t i r e s e t of r a t i o n a l numbers. We in t roduce i r r a t i o n a l numbers only s o t h a t we can t a l k about r e a l numbers and t h e r e a l number l i n e .

We could c a l l t h i s s e t t h e " s e t of numbers", bu t some s t u - den t s may l e a r n about complex numbers l a t e r on. We do not want t h e t eacher t o d i s c u s s t h e s e complex numbers now but t h e s t u d e n t s should be aware t h a t t h e r e a r e numbers o t h e r than those w e have c a l l e d r e a l .

A common misunderstanding Is t h a t some numbers on t h e l i n e a r e r e a l and o t h e r s a r e I r r a t i o n a l . The s tuden t should be encouraged t o say , a t l e a s t f o r t h e time being, t h a t "-2" i s a r e a l

i s number which i s a r a t i o n a l number and a negat ive i n t e g e r ; a r e a l number which is a r a t i o n a l number; - 2 i s a r e a l number which i s a negat ive I r r a t i o n a l number".

We want t h e s tuden t t o be very much aware t h a t there are

i n f i n i t e l y many p o i n t s on t h e number l i n e which a r e not r a t i o n a l numbers. H e w i l l e v e n t u a l l y l e a r n how t o name many of t h e s e , but he should not be concerned about t h i s a t p r e s e n t . I n o rde r t h a t he does not Jump t o t h e conclus ion t h a t a l l t h e s e new numbers a r e

pages 196-190: 6-1

simply var iants of A, a, e t c . , we introduce an i n t u i t i v e

method f o r determining IT on the number l i n e I n Exercise 2 of Problem Set 6- lc .

The idea of r o l l i n g a c i r c l e along the number l i n e t o determine can, of course, be used t o "locate" numbers l i k e ^/2 TT

by considering the c i r c l e t o have diameter &. The number IT i s qui te d i f ferent i n character from numbers

l i k e ,/?, f i , ,/ T, ,/7, e t c . A l l these l a t t e r numbers a re solut ions t o equations of the form

i n which ao, al, a2, . . ., am are i n t e e r s . For number -fi s a t i s f i e s the equatlon *= 0, and i s a solut ion of

However, Tf s a t i s f i e s - no such equation. It i s an example of what

i s ca l led a transcendental number, with numbers l i k e f i , f i, e t c . , being cal led algebraic numbers.

It might be pointed out t o the student who i s inquis i t ive about i r r a t i o n a l numbers t h a t these numbers d i f f e r i n an in te r - e s t ing way from ra t ional numbers i n t h e i r decimal representation. Any ra t ional number can be represented by a repeating decimal. Some examples a re :

(usually writ ten .25)

The decimal representation of any i r r a t i o n a l number, such as tr , */?, Â¥v^ , e t c . , i s an i n f i n i t e non-repeating decimal.

Answers t o Oral Exercises 6-la; page 198: -- - Answers may vary f o r questions 1 - 4 . Any f ive elements of the l i s t e d s e t a re sa t i s fac to ry .

pages 198-199: 6-1

5 . (0, 1, 2, 5 , 4 , -.. 1 4 . (1, 2 , 3 , 4, 5, - * - 1 5. The empty se t i s the se t which has no elements.

Answers - t o Problem -- Set 6-la; pages 198-199:

1. (a) W = (0, 1, 2, 3, m e - I P = (1, 2, 3, 4, ... I L = (0, 1, 2, 3, ... ) - I = ( ..., -2, 1 , 0 , 1 , 2 , * * * 1 N = (1, 2, 3, 4, ... 1 - Q = (0, -1, -2, 3 , . . . I S = ("1, -2, -3, -4, ... 1

( b ) W and L a r e t h e s a m e . P and N are the same.

( c ) A l l are subsets of I.

Q and S are subsets of Q.

L, W, P and N are subsets of L.

P and N a re subsets of P.

the empty s e t

the empty s e t

4. ( a ) If x i s the number of pigeons B i l l had 3 years ago,

then the number he has now i s 2x + 25. 2x + 25 = 77

pages 199-201: 6-1

( b ) If x i s t h e r a t e i n mi les p e r hour of t h e f i r s t

t r a i n , then t h e r a t e I n mi les p e r hour of t h e second

t r a i n i s 2x + 1 0 .

4x + 4 ( 2 x + 1 0 ) = 340

( c ) Let w be t h e width i n inches .

2~ + 2 ( 6 2 ) - 196 o r 2w + 124 = 196

Answers -- t o Oral Exerc ises 6-111; pages 200-201: -

1. I = ( - 5 , - 4 , 1 , 0 , 1 , 2 , 6 )

2. w = {0, 1, 2, 6 )

3 . A = (1, 2, 6 ) 4 . P = ("5, -4, '1)

5. N = (1, 2, 61 1 3 7 6. G = (q-, 137, 2 , ~ , 61

- 1 0 3 7. L = (-5, -4 , -(?), -11 8. Y = (0, 1, 2, 6 )

Answers Problem -- Set 6-lb; page 201:

1- (a ) 1 1 1 - A 1 A 1 1

-4 -3 -2 - 1 -1-1 - - 0 i I 2 3 4 -

2

2 . ( a ) 5 i s t o the r i g h t of 4 . Here we a r e b u i l d i n g toward

( b ) 0 I 1 -

5. t h e o rde r ing of numbers again,

( c ) -4 II 7 . which we w i l l develop f u r t h e r

( d l 1 II - 1 i n t h e next s e c t i o n of t h i s

( e ) Same p o i n t c h a p t e r .

( f ) "(v) is t o t h e r i g h t ot -4.

pages 201-204: 6-1

3 . If w i s t h e number of inches i n t h e width, then 4w i s t h e number of Inches In t h e l e n g t h . The pe r imete r Is 24 i n c h e s .

w + 4 w + w + 4 w = 2 4 o r l O w = 24 s e t ( 2 . 4 )

Answers - t o Problem -- S e t 6-lc; pages 203-204: -

1. ( a ) 2 i s an i n t e g e r , r a t i o n a l , real.

( b ) ' (y) i s r a t i o n a l , real.

( c ) ^/2 i s a r e a l number.

( d ) 0 i s whole, a real number, an i n t e g e r , a r a t i o n a l number.

2 . ( a ) Fa l se

( b ) True ( c ) True ( d ) False - -

3 . '", V . TY i s between 5 and 4. Tf i s between 3 and "4 .

4 . ( a ) Three

( b ) Seven

5. ( a ) If s i s t h e number of y e a r s i n sister 's age, then 2s

I s t h e number i n b r o t h e r ' s age and Mary i s 2 ( 2 s ) yea r s

o l d . 2 ( 2 s ) + 2s + s = 1 5

o r t r u t h s e t i s [?I

( b ) I f q r e p r e s e n t s t h e r a t e of t r a v e l of one boy, 2q + 2(2q) = go.

( c ) If x r e p r e s e n t s t h e first i n t e g e r , x + (x + 2 ) = 86.

pages 205-206: 6-2

6-2. Order on t h e Real Number Line . ---- - We b e l i e v e t h a t t h e s tuden t w i l l expect t h e r e l a t i o n i s

g r e a t e r than" f o r t h e r e a l numbers t o have t h e same meaning a s it

d i d f o r what we now c a l l t h e non-negative r e a l numbers.

Although " is g r e a t e r than" was def ined t o mean "is t o t h e

r i g h t o f T on t h e number l i n e f o r t h e p o s i t i v e numbers, i t could

c l e a r l y be i n t e r p r e t e d a s " is f a r t h e r from zero than" . It i s then p l a u s i b l e tha t">" f o r t h e r e a l numbers might wel l have t h i s

l a t t e r meaning. On t h e o t h e r hand, t h e example of t h e thermo-

meter does not agree wi th t h i s i n t e r p r e t a t i o n , nor would such

f a m i l i a r t h i n g s a s t h e v a r i a t i o n i n t h e he igh t of t i d e s o r eleva-

t i o n s above and below s e a l e v e l .

There Is a l s o a good mathematical reason f o r r e j e c t i n g t h i s

p l a u s i b l e i n t e r p r e t a t i o n . The mathematician i s never r e a l l y

i n t e r e s t e d i n a r e l a t i o n a s such, but r a t h e r i n t h e p r o p e r t i e s it

en joys . Whatever meaning i s a t t a c h e d t o " is g r e a t e r than" we

want t o be a b l e t o say, f o r example, t h a t p r e c i s e l y one of t h e

sentences "3 > 3 " and " 3 > 5" i s t r u e . This p l a u s i b l e i n t e r - - p r e t a t i o n does not permit t h i s comparison, s ince n e i t h e r 3 nor

3 i s f a r t h e r from ze ro than t h e o t h e r . Here we choose t o r e t a i n

t h e i n t e r p r e t a t i o n " > I t t o mean "is t o t h e r i g h t of" on t h e num-

b e r l i n e .

The comparison p roper ty he re given i s a l s o c a l l e d t h e

t r ichotomy proper ty of <. Notice t h a t i t i s a p roper ty of <; t h a t

i s , given any two numbers, they can be ordered s o t h a t one i s l e s s than t h e o t h e r . When t h e p roper ty i s s t a t e d us ing numerals,

we must inc lude t h e t h i r d p o s s i b i l i t y t h a t t h e numerals name t h e

same number. Hence, t h e name "tr ichotomyT' . Although "a < bT1 and "b > a" involve d i f f e r e n t o rde r s ,

t h e s e sentences s a y e x a c t l y t h e same t h i n g about t h e numbers a

and b . Thus, we can s t a t e a t r ichotomy proper ty of > as :

For any number a and any number b,

e x a c t l y one of t h e s e is t r u e : > b , a = b , b > a .

I f i n s t e a d of c o n c e n t r a t i n g a t t e n t i o n on t h e o r d e r r e l a t i o n , we concen t ra te on t h e two numbers, then e i t h e r "a > b o r

a < b i s t r u e , b u t not both . Here we f i x t h e numbers a and

pages 206-207: 6-2

b and then make a decision as t o which order r e l a t ion appl ies . It i s purely a matter of which we are in teres ted in : t h e numbers or the order. The comparison property I s concerned with an order .

Answers -- t o Oral Exercises 6-2a; page 206: - 1. By " i s l e s s than" w e s h a l l mean " I s t o the l e f t of" on the

number l l n e .

2 . This, 'I>", - means "Is t o the r igh t of o r equal to" on the number l i n e .

3 . This, "<", means "is t o the l e f t of o r equal to" on the number l i n e .

Answers - t o Problem Set 6-2a* pages 206-207: - -' 1. ( a ) False ( f ) False

( b ) True (g) True ( c ) True ( h ) False ( d ) False (1) True ( e ) True ( J ) True

4. ( a ) 6 degrees Do these on the number l i n e a s preparation (b) t o "5 degrees f o r the addition of r e a l numbers t h a t i s

(c ) 50 degrees developed i n the next chapter.

pages 207-209: 6-2 and 6-3

5. (a) n > - 1 8 ( e ) n + 5 < n + 8 (b) a > b (f) n - 5 = 2 n - 4 ( c ) n = x + 5 (g) 3 n = 8 o r n - 4 > 2 (d) n - 7 < n + 4 ( h ) n = -4 - 6

Answers - t o Problem -- Set 6-2b;

1. ( a ) > (b) > ( 4 > ( a ) =

( 4 < 2. ( a ) 5 < 6

( b ) -3 < 0

( 4 -(Î < '($1

pages 208-209:

4 . ( a ) I f t he o r ig ina l p r i ce was p do l l a r s , then the discount 1 1 was -5- p and the s a l e p r i ce was p - -5- p.

So 1 2 P - 3 p = 33 o r 3 p = 33. ( ~ n s w e r : the o r ig ina l p r ice was $49.50)

(b) If x represents t h e bro ther ' s age, 2x - 4 = 12. (x " 8)

( c ) If d represents t he height of t he box, 8 - 1 2 - d = 864. ( d = 9 )

6-3. Opposites.

Your s tudents have observed by now t h a t , except f o r zero, the r e a l numbers occur i n p a i r s , t he two numbers of each p a i r being equid i s tan t from zero on the r e a l number l i n e . Each number i n such a p a i r i s ca l led the opposite of t he other . To complete the p i c tu re , zero i s defined t o be I ts own opposite.

pages 209-210: 6-3

On l o c a t i n g t h e opposi te of a given number on t h e number l i n e , you may want t o use a compass t o emphasize t h a t t h e number

and i t s opposi te a r e e q u i d i s t a n t from ze ro .

It i s c l e a r l y much t o o t ed ious t o have t o w r i t e " t h e oppo- s i t e of 2", " t h e opposi te of - 1 e t c . By having t h e s t u d e n t s

wr i t e down a few such phrases , we hope t o suggest t o them t h a t a

shorthand i s needed. The lower dash - which we use i s pe r -

haps t h e most sugges t ive device t o i n d i c a t e t h e opposi te of a

given number and we a r e ve ry quick t o observe t h a t , f o r example - 2 and -2 a r e two d i f f e r e n t names f o r t h e same number.

Having observed t h a t each negat ive number i s a l s o t h e

opposi te of a p o s i t i v e number, it i s apparent t h a t we have no

need f o r two symbolisms t o denote t h e negat ive numbers. Since

t h e lower dash " - i s a p p l i c a b l e t o numerals f o r - all r e a l num-

bers while t h e upper dash " - ' I has s i g n i f i c a n c e only when a t t ached

t o numerals f o r p o s i t i v e numbers, we n a t u r a l l y r e t a i n t h e lower dash. There a r e o t h e r l e s s important reasons f o r dropping t h e

upper dash i n f avor of t h e lower: i t i s e a s i e r t o w r i t e , say , - -5 than 5; more c a r e must be used i n denot ing negat ive f r a c - -

12 t i o n s with t h e upper dash than with t h e lower ( f o r example, - -12 5 could be misread a s -); t h e lower dash i s u n i v e r s a l l y used, 5 - - -5 -

e t c . Henceforth, then , negat ive numbers l i k e 3 , (7), 2,

3 -2, e t c . e t c . w i l l be w r i t t e n a s -5, - 7, The s tuden t must l e a r n t o des igna te t h e opposi te of a given

number by means of t h e d e f i n i t i o n . The s tuden t should not be permit ted t o say, "TO f i n d t h e oppos i t e of a number, change i t s

s i g n " . This i s very imprecise ( i n f a c t , we have never a t t a c h e d

a "sign" t o t h e p o s i t i v e numbers) and w i l l l e a d t o a pure ly

manipulative a lgebra which we want t o avoid a t a l l c o s t s .

The s tuden t i s wel l aware t h a t t h e lower dash I' - i s read

"minus" i n t h e case of s u b t r a c t i o n . We p r e f e r t o r e t a i n t h e word

"minus" f o r t h e opera t ion of s u b t r a c t i o n and - not use i t a s an

a l t e r n a t e word f o r "opposi te o f " . Thus t h e dash a t t ached t o a

v a r i a b l e , such a s l l-x' t w i l l be read "opposi te of" . The opposi te of t h e oppos i t e of t h e oppos i t e of a number i s

the opposi te of t h a t number. What i s t h e opposi te of t h e oppos i t e

of a negat ive number? The ( n e g a t i v e ) number, of course!

pages 210-211: 6-3

If x i s a p o s i t i v e number, then -x i s a negat ive number. The oppos i t e of any negat ive number x is a p o s i t i v e number -x.

And -0 = 0. Thus, t h e s tuden t should - not Jump t o t h e conclusion t h a t when n i s a r e a l number, then -n is a negat ive number; t h i s i s t r u e on ly when n i s a p o s i t i v e number. Note t h e emphasis he re on t h e use of I' - I t as " t h e opposi te" . Because of corn- p l i c a t i o n s t h a t w i l l a r i s e i n l a t e r work regard ing - a t which t h e s t u d e n t s i n s i s t on c a l l i n g a "negat ive number a" , i t i s worthwhile t o reemphasize t h e meaning of t h e upper dash ( read "nega- t i v e " ) a s meaning " t o t h e l e f t of zero" o r " l e s s than 0 " while t h e middle dash ( r e a d "opposi te o f " ) means "on t h e opposi te s i d e of zerof ' .

There a r e no o r a l e x e r c i s e s f o r 6-3a s i n c e t h e problem s e t given might J u s t a s wel l be done a l l o r i n p a r t o r a l l y .

Answers - t o Problem S e t 6-3a; page 211: -

- (c ) 33.5, nega t ive 33.5 ( j) "1,000,000,000, negat ive 1

b i l l i o n

( d l 0

( e ) 100

(k) 1,000,000,000

(1) -8, negat ive 8 -

(m) 9, negat ive 9 -

(n) 16, nega t ive 1 6

The only true s ta tement i s h he oppos i t e of a p o s i t i v e number i s a nega t ive number".

The only t r u e s ta tement i s he oppos i t e o f a negat ive number i s a p o s i t i v e number".

The oppos i t e of ze ro is zero.

" ~ e g a t i v e 9" and " t h e oppos i t e o f 9" - a r e names f o r t h e same number. The first says "9 u n i t s t o t h e l e f t o f o", t h e second says " the number which corresponds t o t h e p o i n t which i s t h e same d i s t a n c e from 0 a s 9 is , bu t on t h e opposi te s i d e of 0".

pages 211-216: 6-5

Answers t o Oral Exercises 6-3b; page 213: -- 1. (a ) 20 (a 5

( b ) -20 (e) 210 - 1 (4 ( f ) -37.5

Answers - t o Problem Set 6-3b; page 213: - 1. (a ) 40 (4 -8 (4 0

( b ) - ( I ) ( d l 8 ( f ) -9

2. If y i s a posi t ive number, then -y i s a negative number.

3. I f y i s a negative number, then -y is a posi t ive number.

5. If -y i s posi t ive, then y is negative.

6. I f -y i s negative, then y i s posi t ive.

7. If -y i s 0, then y I s 0.

( t r u t h s e t (T} Answer: 8Jr f e e t i s width of walk.)

8. If x i s the number of f e e t i n - 3 0 + 2 X -7 the width of the walk, then the

Answers t o O ~ a l Exercises 6-3c; page 216: -- 1. ( a ) 2.97 > -2.97 2.97 > -2.97

( b ) 2 > -12 12 > -2

( c ) -358 > -762 762 > 358 (4 1 > -1 1 > -1

(e ) -121 > -370 370 > 121 ( f ) .24 > .12 -.I2 > -.24

( g ) 0 = -0 Zero i s the only number w i t h the property

length i s 30 + 2x, the width i s 20 + x. The perimeter i s the sum

of the s ides .

(20 + x) + (20 + x) + (30 + 2x) i-

(30 + 2x) = 150

r 20+X

t ha t It i s equal t o i t s opposite.

LÃ‘soÃ‘ or 100 + 6x = 150

-4 q X v v

+X->

pages 216-217; 6-5

2 . This means t h a t x > 3 o r x < 3 .

Answers - t o Problem Se t 6-3c; - pages 217-218:

1. (a) -1 < 3 , -3 < 1

( h ) t h e same number

2 . ( a ) -("7.2) ( f ) . O 1

Here we a r e b u i l d i n g toward t h e meaning of "absolute value of a number" and t h a t t h e g r e a t e r of a number and i t s opposi te i s always t h e p o s i t i v e va lue .

4.. ( a ) , . , , b

0 1 2 3 4

pages 217-219: 6-3 and 6-4

5. (a) The set of all numbers except "3

(b) The set of all numbers except 3

( c ) The set of all numbers less than 0

(d) The set of all numbers greater than 0.

*(e) The set of all numbers equal to or less than 0

*(f) The set of all numbers equal to or greater than 0

6. (a) If John scored n points, then n > "100. (b) If he has n dollars, then n - <0 and n - > "200. (c) If the original bill was d dollars, then d - 10 > 25.

6-4. Absolute Value.

The concept of the absolute value of a number is one of the most useful ideas in mathematics. We will find an Immediate application of absolute value when we define addition and multi-

plication of real numbers in Chapters 7 and 8. In Chapter 10 it is used to define distance between points; in Chapter 12 we define

as 1x1; in Chapter 15 it will provide a good example of an equation with extraneous solutions. Through Chapters 16 to 18 absolute values are Involved in open sentences in two variables

and in Chapter 19 it gives us interesting examples of functions. In later mathematics courses, in particular. in the calculus and In approximation theory, the idea of absolute value is indlspens- able.

pages 219-222: 6-4

The usual d e f i n i t i o n o f t h e a b s o l u t e va lue o f t h e r e a l number n i s t h a t it i s t h e number In1 f o r which

Qui te l i k e l y it i s t h i s form o f t h e d e f i n i t i o n t h a t i s t h e o r i g i n o f t h e d i f f i c u l t y which t h e s t u d e n t s sometimes have when they f i r s t encounter abso lu te va lue . We have t r i e d t o circumvent t h i s d i f f i -

c u l t y by d e f i n i n g the a b s o l u t e va lue of a number i n such a way tha t i t can be p i c t u r e d on t h e r e a l number l i n e : The abso lu te va lue of 0 i s 0 and of any o t h e r r e a l number i s t h e g r e a t e r of t h a t number and i t s oppos i t e . This impl ies t h a t t h e abso lu te value of a number is 0 o r a p o s i t i v e number.

By observing t h a t t h i s "grea ter t1 o f a number and i t s opposi te i s j u s t t h e d i s t a n c e between t h e number and 0 on t h e r e a l number l i n e , we a r e a b l e t o i n t e r p r e t t h e abso lu te va lue "geometrically1' .

Avoid at a l l c o s t s al lowing t h e s t u d e n t t o t h i n k of absolu te va lue a s t h e number obta ined by d r o p p i n g t h e s ign". Such a h a b i t l e a d s t o e n d l e s s t r o u b l e when v a r i a b l e s a r e involved.

It i s q u i t e apparent t h a t t h e g r e a t e r o f a p o s i t i v e number and i t s oppos i t e i s j u s t t h e number i t s e l f . Furthermore, 101

i s def ined o u t r i g h t t o be 0. These two s ta tements can be expressed symbol ica l ly a s :

I f x > 0, then 1x1 = x.

For nega t ive numbers, t h e number l i n e p i c t u r e should convince 1 t h e s t u d e n t s t h a t t h e g r e a t e r o f , f o r example, -5, -(?), -3.1, and

1 -467 and t h e i r oppos i t e s 5, 7, 3.1, and 467 a r e , r e s p e c t i v e l y , 1 5, 7, 3.1, and 467. This same p i c t u r e cannot h e l p but t e l l them

that the g r e a t e r o f any nega t ive number and i t s oppos i t e Is t h e o p p o s i t e o f t h e (nega t ive ) number. Symbolically i f x < 0, then 1x1 = -X.

We have t h e r e f o r e a r r i v e d a t t h e u s u a l d e f i n i t i o n of absolu te va lue .

For a l l r e a l numbers x,

pages 220-223: 6-4

3 . A non-negative number

4. A pos i t i ve number

5. Yes

6 - 1x1

8. When x = O

Answers -- t o Oral Exercises 6-4b; pages 2P2-223:

1. -X

2. (a ) False (d) True (g) True

(b) True (e) True (h) True (c) False (f) True

Answers - t o Problem -- Set 6-4; page 223:

1. (a) (-1, 1 1 ( c ) (-3, 31 (b) ( -3 , 3 ) ( d l ( -3 , 31

pages 219-222: 6-4

The usua l d e f i n i t i o n o f t h e a b s o l u t e va lue o f t h e r e a l number n i s t h a t it i s t h e number In1 f o r which

Qui te l i k e l y it is t h i s form o f t h e d e f i n i t i o n t h a t i s t h e o r i g i n o f t h e d i f f i c u l t y which t h e s t u d e n t s sometimes have when they first encounter a b s o l u t e va lue . We have t r i e d t o circumvent t h i s d i f f i - c u l t y by d e f i n i n g t h e abso lu te va lue of a number i n such a way t h a t it can be p i c t u r e d on t h e r e a l number l i n e : The abso lu te va lue of 0 is 0 and o f any o t h e r r e a l number i s t h e g r e a t e r o f t h a t number and i t s oppos i t e . This Impl ies t h a t t h e abso lu te value of

a number is 0 o r a p o s i t i v e number. observing t h a t t h i s "g rea te r " o f a number and i t s opposi te

i s j u s t t h e d i s t a n c e between t h e number and 0 on t h e r e a l number l i n e , w e a r e a b l e t o i n t e r p r e t the abso lu te va lue "geometrically".

Avoid a t a l l c o s t s al lowing t h e s t u d e n t t o t h i n k of abso lu te va lue a s t h e number obta ined by "dropping t h e sign". Such a h a b i t l e a d s t o e n d l e s s t r o u b l e when v a r i a b l e s a r e involved.

It i s q u i t e apparent t h a t t h e g r e a t e r o f a p o s i t i v e number and i t s oppos i t e i s J u s t t h e number i t s e l f . Furthermore, 101

i s def ined o u t r i g h t t o be 0. These two s ta tements can be expressed symbol ica l ly a s :

I f x > 0, then 1x1 = x. - For nega t ive numbers, t h e number l i n e p i c t u r e should convince

1 t h e s t u d e n t s t h a t t h e g r e a t e r o f , f o r example, -5, -(Â¥?) -3.1, and 1 -467 and t h e i r oppos i t e s 5, -y, 3.1, and 467 a r e , r e spec t ive ly ,

1 5, 7, 3.1, and 467. This same p i c t u r e cannot h e l p but t e l l them t h a t t h e g r e a t e r o f any nega t ive number and i t s oppos i t e is t h e o p p o s i t e of t h e (nega t ive ) number. Symbolically i f x < 0, then 1x1 = -X.

We have t h e r e f o r e a r r i v e d a t t h e u s u a l d e f i n i t i o n of absolu te va lue .

For a l l r e a l numbers x ,

page 2 2 3 : 6-4

The s i m i l a r i t i e s of t h e s e two p a i r s of graphs a r e worth

p o i n t i n g out s i n c e they provide a c l u e f o r a procedure f o r so lv ing

equat ions and i n e q u a l i t i e s wi th abso lu te value of the v a r i a b l e .

That i s , t o w r i t e as two sentences with conjunct ion and/or a s i s a p p r o p r i a t e .

/

I I I I l a - I e t c . - - a - -5 -4 -3 -2 - 1 0 1 2 3 4 5

-5 belongs t o t h e s e t , 0 does no t , -10 does, 4 does.

1 5 11 4 - ( a ) 7 9 -p ( i n f i n i t e l y many p o s s i b l e answers)

3 ( b ) -1, -- 2 , -5 ( i n f i n i t e l y many p o s s i b l e answers)

( c ) -- I -- l1 -2 ( i n f i n i t e l y many p o s s i b l e answers) 2' 19' 12

( d ) There a r e none. ( e . g . , t h e s e t of numbers i n P but

not i n R i s t h e empty s e t . )

5 . The t r u t h s e t o f x The t r u t h s e t o f lx

*6. ~f Pete i s x y e a r s

Bob i s 2x years o ld

= 0 i s ( 0 ) .

= -1 is t h e empty s e t .

o ld , then Sam i s x + 3 years o ld and

The f a t h e r being more than twice t h e sum of t h e i r ages g ives

t h e sentence:

45 > 2 ( x + ( x + 3 ) + 2x) his i s t h e expected response. )

Answers: Pete i s l e s s than 6 years o l d .

Sam i s l e s s than 7; y e a r s o l d . Bob i s l e s s than 4 years o l d .

It might be worth p o i n t i n g o u t t h a t we s t a r t e d with t h e smal les t

number i n d e s c r i b i n g a v a r i a b l e and show t h e s t u d e n t s what it would look l i k e i f we s t a r t e d wi th Bob's age .

A If x = number y e a r s of Bob's age, than 73- i s the number of cÃ

years of P e t e ' s age and Sam i s ($ + 3 ) years o l d , 80 t h e sentence

pages 223-226: 6-4

becomes:

45 > 2($ + ($ + 3 ) + x ) . ('This could be an expected response . )

4 5 > x + x + 6 + 2 x 45 > 4x + 6 4x < 39

x < 4, b u t t h i s i s not t h e same x a s b e f o r e .

Answers - t o Review Problem S e t ; pages 225-227: - ( a ) W i s a subse t of I . ( f ) P and Q a r e not r e l a t e d as

s u b s e t s ~ t h e y have no elements i n common.

( b ) N i s a subse t of W . (g) J i s a subset of R*.

( c ) N i s a subse t of R . ( h ) I and J have no elements i n common.

( d ) R i s a subse t of R*. (1) W i s a subse t of R*. ( e ) I i s a subse t of R. ( J ) Neither; P does not con ta in

ze ro which Is i n W .

a > b means "a i s g r e a t e r than b" o r "a i s t o t h e r i g h t of b " .

IT, -/7r, 6 e t c . Vl6 i s no t i r r a t i o n a l -/^Â i s not i r r a t i o n a l 6 is i r r a t i o n a l

page 227

If n i s the number o f pages i n t h e smal le r volume, then t h e l a r g e r volume has (n + 310) pages and t h e sentence is

n + (n + 310) > 1000. ( ~ n s w e r : One book has more than 345 pages and t h e o t h e r has more than 655 pages . )

I f t h e second plane was f l y i n g a t an average speed o f x mi les O 0 = 4 i s t h e number of hours flown by t h e p e r hour, and

f i r s t p lane , then t h e second plane f lew 3 hours and t h e sentence i s :

3x = 800. ( ~ n s w e r : 26% mph. i s t h e average speed. )

Suggested Tes t Items -- Determine which of t h e fo l lowing sentences a r e t r u e :

( a ) 1-71 = 7 ( d ) 1-51 + 1-71 = 1 2 2 2

( b ) - T # - I - ~ I ( e ) -1-21 = 2

Rearrange t h e fo l lowing numbers i n o r d e r from t h e l e a s t t o t h e g r e a t e s t :

1 1 1 - 8 , -2, -3, 0, , 5 - - 2 '

I n each of t h e fol lowing w r i t e one of t h e symbols <, >, o r = i n t h e p lace Ind ica ted s o t h a t a t r u e sentence r e s u l t s .

If a < b , where a and b a r e r e a l numbers, w r i t e a t r u e sentence express ing t h e o r d e r of -a and -b.

I f a < b, i s it p o s s i b l e t o t e l l whether -a < b, -a = b, o r -a > b? Give i l l u s t r a t i o n s t o suppor t your answer.

Write an open sentence whose t r u t h set i s

7. If b i s a negative number,indicate which of t he following numbers a r e pos i t i ve and which a re negative.

8. Draw the graph of t he t r u t h s e t of each of the following open sentences.

9. Describe t h e t r u t h s e t of each of the open sentences.

10. Describe t he var iab le and t r a n s l a t e i n t o an open sentence: Peter l i v e s one mile c lo se r t o school than Ralph. Peter l a more than $ miles from school. What dis tance is the school from Ralph's home?

11. Consider the s e t of r e a l numbers 1 l3 1.42, 182,,/?). w = (-4, %? 7 9 0, - T,

Which elements of t h i s s e t a r e ( a ) i n t e g e r s ?

(b) r a t i o n a l numbers but not in tegers? ( c ) negative r a t i o n a l numbers? ( d) i r r a t i o n a l numbers? ( e ) non-negative r e a l numbers? ( f ) r a t i o n a l numbers that a r e g rea t e r than -4 and l e s s

than 2?

Answers - t o Suffg ested Test Items -- 1. ( a ) True ( d ) True

( b ) F a l s e (e) F a l s e ( c ) True

1 1 1 = -2, - 3, - 8, 0, 2* - 2,

3 . ( a ) 2 > -1-31 14 22. (4 - > - 17 ( b ) -4 > -7 ( 4 18 + 51 = I81 + I51

5. It i s impossible t o say whether -a < b, -a = b, o r -a > b. The answer depends on t h e abso lu te va lues o f a and b .

The graph8 below i l l u s t r a t e some p o s s i b i l i t i e s .

Numerical e x e r c i s e s such a s the fo l lowing can be used. -2 < 5 and -(-2) < 5 -7 < 5 and -(-?I > 5

6. (a) x > - 2 and x < 3 - (b ) x < -2 o r x > 4

7. ( a ) p o s i t i v e ( b ) p o s i t i v e ( c ) p o s i t i v e

(d ) nega t ive ( e ) nega t ive ( f ) nega t ive

(a I I the empty s e t

0 1 2

(e) 4 4 all poin ts except 0

9 . ( a ) t he set of negative r e a l numbers

( b ) f! (c) t he set of non-negative r e a l numbers

( d ) t he s e t of negative r e a l s and zero Note: 1 0 = -0

10. I f Ralph l i v e s x miles from school, then x > s. 11. ( a ) -4, 0, 182

13 (b) 43 - TI-, 1.42 13 ( 4 -4, -

( d l 7, y?"

(e) 4, 7, 0, 1.42, 182, ./2" 1 3 (f) 0, -T , 1.42

Chapter 7

ADDITION OF REAL NUMBERS

I n t h i s c h a p t e r we t a k e up t h e s tudy of a d d i t i o n . Our problem i s e s s e n t i a l l y t h a t of d e f i n i n g t h i s opera t ion on t h e l a r g e r s e t which inc ludes t h e n e g a t i v e s . Though most s t u d e n t s can achieve s a t i s f a c t o r y competence i n a c t u a l computations wi th these numbers through va r ious i n t u i t i v e dev ices , a formal d e f l - n i t i o n is a necessary mathematical t o o l f o r t h e es tabl i shment of p r o p e r t i e s and a genuine understanding of t h e n a t u r e and s t r u c - t u r e of t h e r e a l numbers.

We f i r s t cons ide r some examples u s i n g ga ins and l o s s e s t o suggest how a d d l t i o n invo lv ing negat ive numbers m i g h t be de f ined . The number l i n e i s a l s o used t o p i c t u r e t h i s . F i n a l l y , as an outgrowth of t h e s e experiments , a formal and p r e c i s e d e f i n i t i o n i s formulated.

The p r o p e r t i e s of a d d i t i o n a r e then p resen ted , wi th s t r e s s on t h e f a c t t h a t our d e f i n i t i o n of a d d i t i o n of r e a l numbers permits t h e f a m i l i a r p r o p e r t i e s of a d d i t i o n of t h e numbers of a r i thmet ic t o hold .

Very e a r l y I n t h e c h a p t e r t h e s t u d e n t should l e a r n how t o f i n d sums invo lv ing nega t ive numbers. Th i s i s easy and i s suggested completely by t h e p r o f i t and l o s s examples, and by t h e number l i n e . However, our immediate o b j e c t i v e i s more ambit ious t h a t j u s t t each ing t h e a r i t h m e t i c of negat ive numbers. We want t o b r i n g out t h e important f a c t t h a t what i s r e a l l y involved h e r e i s an extens ion of t h e opera t ion of a d d i t i o n from t h e numbers of a r i thmet ic (where t h e opera t ion i s f a m i l i a r ) t o a l l r e a l numbers In such a way t h a t t h e b a s i c p r o p e r t i e s of a d d i t i o n a r e p rese rved . This means t h a t we must d e f i n e a d d i t i o n i n terms of on ly t h e non- negat ive numbers and t h e f a m i l i a r o p e r a t i o n s on them. The r e s u l t i n t h e language of a l g e b r a is a formula f o r a + b invo lv ing t h e f a m i l i a r opera t ions of a d d i t i o n , s u b t r a c t i o n , and t a k i n g opposi tes app l i ed t o t h e non-negative numbers, la 1 and Ib 1. The complete formula appears formidable because of t h e v a r i e t y of c a s e s . However t h e i d e a Is simple and i s nothing more than a general d e s c r i p t i o n of e x a c t l y what w e always do i n o b t a i n i n g sums which Involve one o r more nega t ive numbers.

pages 229-232: 7-1

The main problem i s t o l e a d up t o t h e genera l d e f i n i t i o n of a + b i n a p l a u s i b l e way. We have chosen t o make f u l l use of t h e number l i n e and e s p e c i a l l y t o make use of abso lu te va lue .

7-1. Using t h e Real Numbers i n Addition. -- - The p r o f i t and l o s s approach t o a d d i t i o n of p o s i t i v e and

negat ive numbers seems t o be a n a t u r a l one. The only t h i n g which may seem new t o t h e s t u d e n t i s t h e r e p r e s e n t a t i o n i n terms of p o s i t i v e and negat ive numbers.

2 . ( a ) a = 3

( b ) a = 7 (0) a = - 7 ( d ) b = -5

( e ) c = 6

( f ) m = - 3 (g) n = 0

( h ) n = = O

pages 232-234: 7-1 and 7-2

(1) m = - 1

( J ) c = 6 ( k ) b = -11

7-2. Addition and t h e Number Line . -- - Recall t h a t t h e main purpose of a d d i t i o n on t h e number l i n e

i s t o l e a d up t o t h e d e f i n i t i o n of a d d i t i o n given on pages 239-240. By t h i s time t h e s t u d e n t s a r e f a m i l i a r wi th t h e number l i n e , and i t i s hoped t h a t i l l u s t r a t i n g a d d i t i o n on It w i l l seem n a t u r a l . Note a l s o t h a t t h e concept of abso lu te va lue , in t roduced i n t h e l a s t chapter , i s used ex tens ive ly ; i t i s c e n t r a l t o t h e d e f i n i - t i o n of a d d i t i o n developed h e r e .

Although some of t h e e x e r c i s e s i n t h i s c h a p t e r , which a r e designed t o s t r eng then understanding, w i l l c a l l f o r s p e c i f i c a p p l i c a t i o n of t h e formal d e f i n i t i o n , t h e s t u d e n t s w i l l not be expected t o use t h i s on a l l occasions a s a r u l e by which t o add r e a l numbers. The po in t of view here i s t h a t t h e s tuden t now has a d e s c r i p t i o n of t h e process he has a l r e a d y l ea rned how t o do . In genera l a s tuden t should be encouraged t o app ly any i n t u i t i v e process f o r a d d i t i o n of r e a l numbers which he f i n d s r e l i a b l e .

Answers t o Oral Exerc ises 7-2a; pages 234-235: 7-

- 5O F

4O F

- 3

lo c 0 -1 c

( a ) S t a r t a t zero . Move 5 u n i t s t o t h e l e f t , then 2

u n i t s t o t h e r i g h t . The sum i s -3 .

( b ) S t a r t a t z e r o . Move 5 u n i t s t o t h e l e f t , then 2

more u n i t s t o t h e l e f t . The sum i s -7.

( c ) S t a r t a t ze ro . Move 5 u n i t s t o t h e r i g h t , then 2

more u n i t s t o t h e r i g h t . The sum i s 7 .

pages 234-235: 7-2 ( d ) S t a r t a t ze ro . Move 5 u n i t s t o t h e r i g h t , then 2

u n i t s t o t h e l e f t . The sum i s 3 .

( e ) S t a r t a t ze ro . Move 6 u n i t s t o t h e l e f t , then 7 more u n i t s t o t h e l e f t . The sum Is -13.

( f ) S t a r t a t ze ro . Move 11 u n i t s t o t h e l e f t , then 15 u n i t s t o t h e r i g h t . The sum i s 4 .

(g) S t a r t a t ze ro . Move 4 u n i t s t o t h e r i g h t , then 12

more u n i t s t o t h e r i g h t . The sum i s 1 6 .

( h ) S t a r t a t ze ro . Move 6 u n i t s t o t h e r i g h t , then 7 u n i t s t o t h e l e f t . The sum i s -1.

(1) S t a r t a t z e r o . Move 6 u n i t s t o t h e r i g h t , then 6 u n i t s t o t h e l e f t . The sum Is 0.

( j ) S t a r t a t z e r o . Move 7 u n i t s t o t h e l e f t , then no u n i t s e i t h e r way. The sum i s -7.

( k ) S t a r t a t ze ro . Move 4 u n i t s t o the r i g h t , then 6 u n i t s t o t h e l e f t . The sum i s -2 . Then move 8 more u n i t s t o t h e l e f t . The sum i s -10.

(1) S t a r t a t ze ro . Move 5 u n i t s t o t h e l e f t , then 2

more u n i t s t o t h e l e f t . The sum i s -7. Then move 7 more u n i t s t o t h e l e f t . The sum i s - 1 4 .

( m ) S t a r t a t ze ro . Move 4 u n i t s t o the l e f t , then 8 u n i t s t o t h e r i g h t . The sum i s 4. Then move 4 u n i t s t o t h e l e f t . The sum i s 0.

( n ) S t a r t a t ze ro . Move no u n i t i n e i t h e r d i r e c t i o n , then move 2 u n i t s t o t h e l e f t . The sum is -2. Then

move 2 u n i t s t o t h e r i g h t . The sum Is 0.

( 0 ) S t a r t a t zero . Move 7 u n i t s t o t h e r i g h t , then 2

u n i t s t o t h e l e f t . The sum is 5 . Then move 3 u n i t s

t o t h e r i g h t . The sum i s 8.

pages 235-241: 7-2

( d ) True (g) False ( J ) T r u e

( e ) False ( h ) True ( k ) False

( f ) True (1) True (1) False

( c ) True ( e ) False ( g ) False

( d ) True ( f ) True ( h ) True

( j ) the s e t of numbers grea ter than 4

( k ) the s e t of numbers grea ter than 0 ( the posi t ive numbers)

(1) the s e t of numbers l e s s than ( -2 )

( m ) the s e t of r e a l numbers

Answers - t o Problem Set 7-2b; page 241: -- 1. ( 7 ) + ( -3 ) = 171 - 1-51 4. 3 + ( -7) = -(1-71 - 131 )

7 - 3 = - ( 7 - 3 ) = 4 = - 4

pages 241-243: 7-2 and 7-3

1 0 . True

11. Fa l se

1 2 . True

13. True 1 4 . Fa l se

15. True 16. False

17. True 18. Fa l se

1 9 . Fa l se

7-3 . Addition Proper ty -- of Zero; Addition Proper ty - of Opposites.

Note t h a t t h e a d d i t i o n p roper ty of zero and t h e a d d i t i o n

p roper ty of oppos i t e s a r e obta ined d i r e c t l y from t h e d e f i n i t i o n

of a d d i t i o n . Note a l s o t h a t t h e a d d i t i o n p roper ty of opposi tes says t h a t t h e sum of a and (-a) i s z e r o . It does not say -- t h a t i f t h e sum of a and ano the r number i s zero, t h e o the r number i s ( - a ) . This fact i s proved l a t e r .

Answers t o Oral Ekerc ises u; -- 1. True

2 . Fa l se

3 . True

4. True

5. Fa l se 6. True

7 . True

page 242:

8. Fa l se

9 . True 1 0 . Fa l se 11. Fa l se

1 2 . Fa l se

13. True 1 4 . True

Answers - t o Problem - Set m; pages 242-243:

1. 1 4 4. -8 2 . 0 5 -9 3 O 6 . -45

pages 243-244:

7 . 0 8. Any number 9 . Any number

10 . Any number 11. Any number

7-3 and 7-4

g r e a t e r than 3 l e s s than ( - 3 ) g r e a t e r than 6 l e s s than ( - 4 )

7-4. P roper t i e s - of Addit ion.

The d e f i n i t i o n of t h e a d d i t i o n of r e a l numbers has been made i n terms of t h e non-negative numbers and t h e f a m i l i a r opera t ions upon them. We have seen t h a t i t agrees wi th our i n t u i t i v e f e e l i n g f o r t h e opera t ion of a d d i t i o n of r e a l numbers a s shown i n working with ga ins o r l o s s e s and with t h e number l i n e . It i s f u r t h e r requi red t h a t a d d i t i o n of r e a l numbers have t h e same bas ic p r o p e r t i e s t h a t we observed f o r a d d i t i o n of numbers of a r i t h m e t i c . It would be awkward, f o r i n s t a n c e , t o have a d d i t i o n of numbers of a r i t h m e t i c commutative and a d d i t i o n of r e a l numbers not commutative.

Notice t h a t , while we d i d not c a l l them such f o r t h e s t u d e n t s , t h e commutative and a s s o c i a t i v e p r o p e r t i e s were, f o r a l l i n t e n t s and purposes, regarded a s axioms f o r t h e numbers of a r i t h m e t i c , and t h e opera t ion of a d d i t i o n was regarded e s s e n t i a l l y a s an undefined opera t ion . For t h e real numbers, however, we have made a d e f i n i t i o n of a d d i t i o n i n terms of e a r l i e r concepts . I f our d e f i n i t i o n has been p roper ly chosen, we should f i n d t h a t t he

p r o p e r t i e s can be proved a s theorems. While most s t u d e n t s w i l l not f u l l y a p p r e c i a t e a l l t h i s , t h e t eacher should have i t i n mind as background.

We have t r i e d t o g ive t h e s t u d e n t s a f e e l i n g f o r t h e prov- a b i l i t y of these p r o p e r t i e s , but very few of them w i l l be ready t o fo l low through t h e d e t a i l s . However, f o r t h e occas ional s tuden t who is a b l e and i n t e r e s t e d , we have l e f t t h e way open f o r him t o s a t i s f y himself f u l l y t h a t t h e p r o p e r t i e s hold i n cases , not j u s t i n some p a r t i c u l a r c a s e s he might t ry.

Answers -- t o Oral Exerc ises 7-4; pages 244-245:

1. Yes 2 . Commutative

pages 244-241:): 7-4

3 . (a) -2

(b) -2

(c) Yes (d ) Associative

4. Two real numbers may be added in either order. The sum will be the same in both cases. The commutative property may a l so be more b r i e f l y stated: For any real numbers a and

b, a + b = b + a .

5. For any real numbers a, b, and c, (a + b) + c = a + (b + c). A precise word statement of this property becomes quite Involved.

(a) Commutative

(b ) Associative

(c ) Associative

(d) Commutative

(e) Commutative and associative

(f ) Associative

(g ) Commutative (h) Commutative and associative

(I) Commutative and associative

(j) Commutative and associative

Answers - to Problem -- Set 7-4; pages 245-246:

1. (a) ( - 3 ) + 7 + 5 + 3 + (-5) - ( ( - 3 ) + 3) + 6+ (-5)) + 7 = 7

(b) 14 + 6 + (-7) + 4 + 5 = (14 + 6) + (-7 + ( 4 + 3 ) ) = 20

(c) 5 + (-8) + 6 + (-3) + 2 = 5 + (-3) + ( - 8 ) + ( 6 + 2 )

= 5 + ( -3 ) = 2

(dl ( - 9 ) + 5 + 6 + (-3) = 6-91 + 5) + (Â + (-3)) - 4 + 3 = -1

Here there is no particularly easy grouping. The

student may want to add from left to right mentally,

getting first (-4), then 2, and then (-1).

(e) 11 + (-17) + 9 + ( - 3 ) + 4 - (11 + 9 ) + (-17) + (-3)) + 4 = 20 + (-20) + 4 = 4

pages 245-248: 7-4 and 7-5

True True

True False

( e ) Fa l se ( f ) F a l s e ( g ) True ( h ) F a l s e

( f ) 8 ( g ) 10 ( h ) a l l r e a l numbers ( i ) a l l r e a l numbers

7-5. Addition Proper ty - of E q u a l i t y .

You may recognize t h e "Addition Proper ty of ~ q u a l i t y " as t h e t r a d i t i o n a l s ta tement , "If equa l s a r e added t o equa l s , t h e sums a r e equal ." While w e s h a l l have f requen t occasion t o u s e t h i s Idea, we p r e f e r not t o t r e a t i t a s a p r o p e r t y of r e a l numbers because i t i s r e a l l y J u s t an outgrowth of two names f o r t h e same number. The name " ~ d d i t i o n Proper ty o f ~ q u a l i t y " w i l l be a convenient way t o r e f e r t o t h i s Idea when we need t o use i t .

From another p o i n t of view, t h e a d d i t i o n p roper ty of e q u a l i t y can a l s o be thought of a s be ing a way of say ing t h a t t h e opera- t i o n of a d d i t i o n i s s i n g l e valued; t h a t Is, t h e r e s u l t of adding two given numbers i s a s i n g l e number. I n o t h e r words, whenever we add two given numbers we always g e t t h e same r e s u l t . There- f o r e , i f a , b and c a r e r e a l numbers and a Â¥ b, then t h e s tatement "a + c = b + c" can be thought of a s say ing t h a t t h e r e s u l t of adding t h e two given numbers was t h e same when t h e y had t h e names a " and "c" a s when t h e y had t h e names "b" and "c" .

pages 248-249: 7-5

Though t h i s proper ty has more t o do with the language we

use i n t a l k i n g about numbers than with the numbers o r t he opera-

t i o n s upon them, i t i s c l e a r l y a useful t oo l In f inding the t r u t h

s e t s of sentences, and i t w i l l be put t o use i n t h i s way in the

next sec t ion of the t e x t .

Answers -- t o Oral Exercises - 7-5; pages 248-249:

The r e s u l t i n g sentence i s t r u e .

The r e s u l t i n g sentence w i l l not be t r u e . The number repre-

sented by t he right s ide w i l l be l a r g e r than the number

represented by t h e l e f t s i d e .

The r e s u l t i n g sentence w i l l not be t r u e . The same order r e l a t i onsh ip a s i n quest ion 2 w i l l e x i s t .

( a ) and ( c ) a r e statements of t he proper ty .

( a ) a l l r e a l numbers

( b ) a l l r e a l numbers

( c ) a l l r e a l numbers

( a ) -5 ( b ) 6 (4 11 ( d l -8 (4 -9

( a ) -5 ( b ) 4

(4 16 ( d l 8 (4 9

( 1 ) No number need be added, s ince by the use of the assoc ia t ive property of add i t ion , t he addi t ion pro- pe r ty of opposites, and the addi t ion property of zero, the var iab le i s i so l a t ed on t he l e f t s i d e .

Answers - t o Problem - Set E; pages 249-250:

1. ( a ) True ( c ) False ( e ) True (g) TnIe ( b ) True ( d ) True ( f ) False ( h ) True

pages 250-254: 7-5 and 7-6

2 . In t h i s problem we a r e not i n t e r e s t e d I n t h e s t u d e n t s * f i n d i n g t h e t r u t h s e t of an open sen tence . The Important t h i n g here i s f o r him t o l e a r n how t o use t h e a d d i t i o n proper ty of e q u a l i t y t o ob ta in an equ iva len t open sen tence . The s tudent should not t a k e shor t -cu t s now.

( a ) add ( - 4 ) ( f ) add ( - 6 ) ( b ) add 6 and (-4), o r 2 ( g ) add ( - 4 ) and 6 , o r 2

( c ) add ( -5) ( h ) add 30 and 10, or 40 ( d ) add 2 ( I ) add 3 ( 4 + 2 ) , o r 18 ( e ) add 2 and 7, o r 9 ( j ) No number need be added.

7-6. ruth sets of Open Sentences . --7

Late r i n Chapter 8 we s h a l l l e a r n about e q u i v a l e n t s e n t e n c e s and t h e permiss ib le opera t ions which keep sentences e q u i v a l e n t . For t h e p resen t , however, n o t i c e t h a t a l l we a r e c la iming when we apply t h e a d d i t i o n p roper ty of e q u a l i t y i s t h a t - i f a number makes t h e o r i g i n a l sentence t r u e , i t w i l l make t h e new sentence t r u e . We then have a cnance t o t e s t each number of t h e t r u t h s e t of t h e new sentence and s e e whether i t makes t h e o r i g i n a l sentence t r u e . It i s necessary t o make t h i s check every t ime, u n t i l we have t h e more complete reasoning of Chapter 8.

At tent ion i s a l s o focused on t h e f a c t t h a t t h e use of t h e add i t ion p roper ty of e q u a l i t y wi th a subsequent " c h e c k i n g by s u b s t i t u t i o n i s more than a convenient a l t e r n a t i v e t o guess ing . It does, i n f a c t , g ive us t h e complete t r u t h s e t . I n o t h e r words, t h e ques t ion concerning the p o s s i b i l i t y of a d d i t i o n a l t r u t h num- be r s i s d e f i n i t e l y answered, i n t h e nega t ive . S tudents may have some d i f f i c u l t y i n grasping t h i s , but should be encouraged t o t ry .

It i s a l s o necessary t o reexamine a t t h i s p o i n t t h e genera l quest ion of t h e domain of t h e v a r i a b l e s i n c e our bas ic s e t has been enlarged t o inc lude nega t ive numbers. I n Chapter 3 a

page 254: 7-6

covering statement was made t o the e f fec t tha t unless otherwise specified the domain of the variable was t o be assumed t o be a l l numbers - of ari thmetic f o r which the given sentence had meaning. A s imi lar statement i s made i n t h i s sect ion. Since we do not wish t o labor the point a t t h i s time as f a r a s the student is concerned, no fur ther discussion of domain i s presented i n the t e x t . However, the teacher should be aware tha t u n t i l m u l t i p l i -

cat ion i s defined f o r negative numbers i n Chapter 8, such expressions as 3x or 5x a re , theore t ica l ly , without meaning. Hence, i n constructing the exercises and examples care has been taken t o avoid at taching a coeff ic ient t o the variable f o r any sentence having a negative t r u t h number.

Answers t o Oral Exercises 7-6; page 254: -- - 2

1. ( - 6 . 1 . 5

1 2 . b7;) 7. 3 + 2, o r 5

3 - 9 8. ( - 3 ) 4. ( -3 ) 9. 2

5 . ( - 6 ) 10. ( -2 )

Answers - t o Problem -- Set 7-6; pages 254-255:

1. The form of the student 's answer, i f he does not guess the t r u t h number d i rec t ly , i s suggested i n the examples In t h i s sect ion of the t e x t . Par ts ( a ) and ( h ) are written out i n t h i s manner below.

( a ) I f x + 5 = -3 i s t rue f o r some x, then (x + 5 ) + (-5) = (-3) + (-5) i s t r u e f o r the

same x

x + (5 + (-5)) = ( - 3 ) + (-5) x + 0 = ( - 3 ) + (-5)

x = ( - 5 ) + ( -5) and x = -8 i s t rue f o r the same x .

(-8) i s the t r u t h se t of the l a s t sentence and since (-8) + 5 = -3 i s t rue , (-8) i s the t r u t h s e t fo r x + 5 = - 3 .

pages 254-255: 7-6

( h ) I f ( -5) + 3x + (-8) = 15 + (-20) + 1 i s t rue fo r some x.

then ((-5) + 3x) + (-8) = (15 + (-20)) + 1 i s t rue fo r the same x

(N + (-5)) + (-8) = (15 + (-20-9 + 1 >x + ((-5) + (-8)) = 6 5 + (-20)) + 1

and 3x = 9 i s t rue for the same x.

( 3 ) i s the t r u t h se t of 3x = 9 , and since (-5) + 3 ( 3 ) + (-8) = 15 + (-20) + 1 is t rue,

(31 i s the t r u t h se t of (-5) + 3x + (-8) = 15 + (-20) + 1.

1 ( f ) -I? (g) 3.1

(h) se t of all rea l numbers

3 . ( a ) If 2x + ( -5) = -3 i s t rue fo r some x,

then (sx + (-5)) + 5 ": -3 + 5 Is true fo r the same x

1 i s the t r u t h number of 2x = 2,

and since 2 (1 ) + (-5) = -3 i s t rue ,

1 i s the t r u t h number of 2x + (-5) = -3.

pages 255-256: 7-6 and 7-7

7-7. Addit ive I n v e r s e .

A t t h e end of t h i s s e c t i o n t h e s tuden t may be having h i s

f i r s t exper ience a t anything approaching a formal p roof . H i s

c h i e f d i f f i c u l t y he re i s s e e i n g t h e need f o r such a p roof . We

ask t h e s tuden t t o e x t r a c t from h i s experience t h e f a c t t h a t f o r every number t h e r e i s ano the r number such t h a t t h e i r sum i s zero. -- A t t h e same time t h e s tuden t can e q u a l l y well e x t r a c t from h i s

experience t h a t t h e r e i s only - one such number. Why then, do we

accept t h e f i r s t i d e a from experience but prove t h e second? The

reason i s t h a t we - can prove t h e second. The two i d e a s d i f f e r i n

t h a t one must be e x t r a c t e d from exper ience while t h e o t h e r need

not be . The e x i s t e n c e of t h e a d d i t i v e i n v e r s e i s i n t h i a sense - a more bas ic Idea than t h e i d e a t h a t t h e r e is only one such num-

b e r . Speaking more formal ly , t h e ex i s t ence of t h e a d d i t i v e inverse i s an assumption; t h e uniqueness of t h e a d d i t i v e inverse

i s a theorem. You a r e r e f e r r e d t o Haag, S tud ies - i n Mathematics,

Volume 111, S t r u c t u r e of Elementary Algebra, Chapter 2, Sect ion

3 , f o r f u r t h e r r ead ing .

A t t h i s po in t we a r e s t i l l q u i t e informal about proofs and

t r y t o l e a d i n t o t h i s k ind of t h i n k i n g g radua l ly and c a r e f u l l y .

The viewpoint about p roofs i n t h i s course i s not t h a t w e a r e t r y i n g t o prove r i g o r o u s l y every th ing we say - we cannot a t t h i s

s t a g e - but t h a t we a r e t r y i n g t o g ive t h e s t u d e n t s a l i t t l e

exper ience , wi th in t h e i r a b i l i t y , with t h e kind of th ink ing we c a l l "proof" . Don't f r i g h t e n them by making a b i g i s s u e of it, and don' t be discouraged i f some s t u d e n t s do not immediately ge t

t h e p o i n t . Discuss t h e proofs wi th them a s c l e a r l y and simply as you can . We hope t h a t by t h e end of t h e yea r they w i l l have some

f e e l i n g f o r deduct ive reasoning, a b e t t e r i d e a of t h e na tu re of mathematics, and perhaps a g r e a t e r i n t e r e s t i n a lgebra because of t h e bea r ing of proof on t h e s t r u c t u r e . For background reading on

proofs t h e t e a c h e r i s again r e f e r r e d t o Haag, S tud ies - i n

pages 256-257: 7-7

Mathematics, Volume - 111, Struc ture - of Elementary Algebra, Chapter

2, Section 3 . The pr inc ipa l emphasis i n t h i s sec t ion has been on i n t r o -

ducing the student t o formal proof . Notice, however, t h a t t he theorem through which t h i s f i r s t experience i n proof was given

i s I t s e l f a s ign i f i can t s t r u c t u r a l property of the r e a l numbers.

Answers t o Oral Exercises 7-7a; pages 256-257: --

6 . Each i s the addi t ive inverse of t he o the r .

(1) Ei the r -8 o r - ( 3 + 5 ) o r (-3) + ( -5 )

( J ) - (x + 5)

(4 - ( a + b )

(n) 4% + n) ( 0 ) - ( 4 y - x + 2 )

4. (a) (0) (g) If 3x + ( - 1 ) = 24 + ( - 4 ) i s t r u e f o r some x

(b) [ 2 ) t h e n 3x + ( - 1 ) + 1 = 24 + ( - 4 ) + 1

is. t r u e f o r t h e same x .

( 7 ) i s t h e t r u t h set o f 3x = 21.

S ince 5 ( 7 ) + ( - 1 ) = 24 + ( - 4 ) i s t r u e

(71 i s a l s o t h e t r u t h set of t h e

o r i g i n a l open sen tence .

5 . ( a ) a + ( - a ) = 0

1 ( c ) 5 x + (-,.) = - x

6 . If x i s t h e number of n i c k e l s John had, t h e n 4x i s t h e

number of penn ie s , and 2x i s t h e number of dimes. The

s e n t e n c e Is 5x + 4x + 20x + 7 = 94.

7 . I f t h e s m a l l e s t a n g l e has n d e g r e e s , t h e n t h e l a r g e s t

a n g l e h a s 2n + 20 deg rees , and n + (2n + 2 0 ) + 70 = 180.

8. x + x + 6 x + 6 x = 1 1 2

where x is the number o f i n c h e s i n t h e wid th

pages 259-264: 7-7

Answers t o Problem S e t 7-7b; page 262: - -- Addition p roper ty of oppos i t e s

commutative p roper ty of a d d i t i o n

' a s s o c i a t i v e p roper ty of a d d i t i o n

a s s o c i a t i v e p roper ty of a d d i t i o n

a d d i t i o n p r o p e r t y of oppos i t e s ( o r d e f i n i t i o n of a d d i t i o n )

= 3 + (-3) a d d i t i o n p r o p e r t y of ze ro

= 0

a d d i t i o n p r o p e r t y of oppos i t e s

We can conclude t h a t -(3 + (-4)) = ([-?) + 4) because of uniqueness of a d d i t i v e i n v e r s e s .

The a d d i t i o n p r o p e r t y of oppos i t e s

The a d d i t i o n p roper ty of oppos i t e s

The theorem on t h e uniqueness of t h e a d d i t i v e i n v e r s e of a r e a l number

pages 264-265

(b) Since both are negative, (-5) + (-11) = - ( 1-51 + 1-111 )

= - (5 + 11) = -16

( f ) Since Ill1 = 1 - v \ , (-'rr) + - M = 0

( h ) Since both are negative,

- 3 5 ) + (-65) = -(I-351 + 1-651) = -(35 + 65) = -100

( i) Since 12 and 7 1 2 + 7 = 19

( j ) Since 110 1 > 1-6

(-6) + 10 = (

(k) Since 1 21 > 11 1

are numbers of arithmetic,

(1 ) Since 1-201 1 > 1200 l J 200 + (-201) = -(I-2011 - 12001) = -(201 - 200) = -1

( a ) True ( d ) True (g) False ( b ) True ( e ) True ( h ) False ( c ) ~ a l s e ( f ) False (1) True

pages 265-267

5 . ( a ) True

( b ) False

( c ) True

( d ) False f o r a l l x except x = 0

( e ) True

( f ) True

8. ( a ) a s s o c i a t i v e p roper ty of a d d i t i o n and a d d i t i o n p roper ty of oppos i t e s

( b ) commutative p roper ty of a d d i t i o n

( c ) a d d i t i o n p roper ty of oppos i t e s and a d d i t i o n p roper ty of zero

( d ) a s s o c i a t i v e p roper ty of a d d i t i o n and a d d i t i o n p r o p e r t y of oppos i t e s and a d d i t i o n p roper ty of zero

( e ) a s s o c i a t i v e p roper ty of a d d i t i o n

( f ) commutative p roper ty of a d d i t i o n

( g ) a s s o c i a t i v e p roper ty of a d d i t i o n ( h ) commutative p roper ty of a d d i t i o n

( I ) a d d i t i o n p r o p e r t y of oppos i t e s and a d d i t i o n p roper ty of zero

( j ) a d d i t i o n p roper ty of oppos i t e s , commutative p roper ty of a d d l t i o n and a d d i t i o n p roper ty of ze ro

9 . The fol lowing a r e merely suggested methods. The s tuden t

should use t h e p r o p e r t i e s i n t h e way t h a t makes t h e computa-

t i o n e a s i e s t f o r him. However, t h e t e a c h e r should d i s c u s s

with t h e s tuden t s t h e d i f f e r e n t ways of commuting and

pages 267-268

associat ing the numbers.

( d ) (-3) + (-5)) + 8 + (11 + 12) + ((-4) + (-3)) = 16

( f ) ( ~ - ~ l + (4) + 21 + (-8) + ( - 7 ) = 6

10. ( a ) I f t i s the number of f e e t above sea leve l tha t the

t i d e regis tered, t - (-0.6) + 5.1.

( b ) I f b i s the number of Inches tha t Dave shot above the

center of the t a rge t on the second shot, b = 10 + ( - 3 ) .

( c ) I f f i s the number of f e e t tha t the submarine cruised below sea leve l a f t e r the change of posit ion,

f = 254 + (-78) .

( d ) If x i s the number of do l l a r s tha t h i s daughter

received, then 2x is the number of dol la rs tha t h i s

son received, and 3x I s the number of dol la rs tha t

the widow received, so x + 2x + 3x - 50,000.

( e ) If x i s the t o t a l number of dol la rs tha t M r . ~ o h n s o n owed the bank, x > 200.

Suggested Test Items --

2. Write a common name f o r each of the following, doing the addition i n the eas ies t way. In each case t e l l what propert ies you used t o make your work eas ier .

( a ) (-17) + (30 + (-83)) ( c ) (23 + (-12)) + (-11)

(b) ((-19) + 183) + 19 (d) ( (-98) + 102) + (-63)

3. If a and b a r e two r e a l numbers,determine whether the sum

of the following

4. Consider the sentences:

D. If a = b , then a + 2 = b + 2 E. -(-4) = 4 F. o + (-6) = -6 G. (-4) + (-3) = -(I-41 + 1-31)

Which of the sentences i l l u s t r a t e :

(a ) the commutative property of mult ipl icat ion

(b) the addition property of zero

the addition property of equality the fact that the sum of two negative numbers is the opposite of the sum of their absolute values

the opposite of the opposite of a number is the number itself the associative property of addition the addition property of opposites the associative property of multiplication

Find the truth set of each of the following open sentences.

(a) x + 2 = 7 (d) (-6) + 7 = (-8) + a (b) 0 = 7 + n (e) 1x1 + (-2) = 1 (c) rn + (-6) = o (f) x + 1x1 = O

When a certain number is added to 99, the result is 287.

(a) Write an open sentence to find the number

(b) Find the number by finding the truth set of the sentence

Which of the following sets of numbers is closed under addition?

(a) (-3, -2, -1, 0, 1, 2, 31 (b) ( 0 - * , -12, -9, -6, -3, 0)

( c ) the set of all negative real numbers

Describe in terms of operations with numbers of arithmetic.

(a) the sum of 8 and (-3) (b) the sum of (-11) and 5 (c) the sum of (-12) and (-5)

Which of the following sentences are true? Which are false?

10. ( a ) A number is three more than its additive inverse. What is the number? Find the answer to this question by finding the truth set of an open sentence. (Hint: If there is a number n such that n = 3 + (-n), then n + n = 3 + (-n) + n (why?), and n + n = 3 (why?).)

(b) A number is equal to its additive inverse. For what

numbers is this sentence true? Answer this question by finding the truth set of an open sentence. (Hint: If there is a number n such that n = -n, then n -I- n = (-n) + n. Why?)

2. (a) (-17)+(30+(-83)) = (-17)+ ((-83)+30) commutative property of addition

= ((-17)+ (-83))+30 associative property of addition

(b) ((-19) + l83)+19 = (183 + (-19))+19 commutative property of addition

= i83+ ((-19)+19 J) associative property of addition

addition property of opposites

= 183 addition property of zero

(c) (23 + (-12))+(-11) = 23+ ((-12)+(-11 j) associative property of addition

addition property of opposites

(d) ((-98) + 102) + (-63) = 4 + (-63)

= -59 There i s no eas ies t way t o do t h i s .

3. (a ) pos i t ive (b) pos i t ive (c) negative (d) negative

( e ) zero ( f ) negative (g) posi t ive

5. ( a ) (51 (e) 1x1 + (-2) + 2 = 3

(b) (-71 1x1 = 3 ruth se t : (-3, 3)

( 4 (6) ( f ) the s e t consisting of a l l negative r ea l

(d) (9) numbers and zero

For such numbers x 1 = -x, and so x + 1x1 = x + (-x)

= 0.

6. (a) 99 + n = 287

(b) n + 99 = 287 n + 99 + (-99) = 287 + (-99) n = 188 The t r u t h s e t : (188)

The number i s 3.88.

7. ( a ) not closed under addi t ion (-3) + (-2) = -5, and -5 i s not an element of the s e t

(b) closed under addi t ion

( c ) closed under addition

8. (a) 8 + ( - 3 ) = 181 - 1-31 , 8 - 3

= 5

(-11) + 5 = -(!-I11 - 151)

= -(I1 - 5 )

= -6

( -12) + (-5) = -(1-121 + 1-51 )

= - (12 + 5 )

= -17

False ( d ) True True ( e ) Fa l se True ( f ) True

t h e r e i s a number n such t h a t

n = 3 + ( -n ) ,

then n + n = 3 + ( - n ) + n

then -n = - n = 2, 3 an, 2' 3 2 = 3 + ( - - ) I1

2 2 Is t h e requ i red number. t r u e . Hence, - 2

t h e r e i s a number n such t h a t

n = -n,

then

Since 0 i s i t s own a d d i t i v e Inverse , we have shown t h a t 0 i s t h e on ly number with t h i s p r o p e r t y .

page 269: 8-1

Chapter 8

MULTIPLICATION OF REAL NUMBERS

This i s t h e second of t h r e e chap te r s i n which t h e opera t ions

with t h e numbers of a r i t h m e t i c a r e extended t o t h e r e a l numbers

and t h e p r o p e r t i e s of t h e s e opera t ions a r e brought o u t . You may

want t o r e f e r t o t h e s tatement a t t h e beginning of t h e commentary

f o r Chapter 6 t o have another look a t t h e o v e r a l l p lan of t h e s e

th ree c h a p t e r s .

Background reading f o r t h e mathematics of t h i s c h a p t e r i s a v a i l a b l e i n S tudies - i n Mathematics, m, Chapter 3, Sec t ions 2

and 4 .

8-1 . Products ., A s i n t h e case of a d d i t i o n , t h e po in t of view here i s t h a t we

extend t h e opera t ion of m u l t i p l i c a t i o n from t h e numbers of a r i t h -

metic t o a l l r e a l numbers s o a s t o preserve t h e fundamental

p r o p e r t i e s . This a c t u a l l y f o r c e s us t o d e f i n e m u l t i p l i c a t i o n i n the way we do. In o t h e r words, it could not be done i n any o t h e r

way without g iv ing up some of t h e p r o p e r t i e s .

The general d e f i n i t i o n of m u l t i p l i c a t i o n f o r r e a l numbers i s s t a t e d i n terms of abso lu te va lues because la1 and l ' b l a r e

numbers of a r i t h m e t i c . The only problem f o r r e a l numbers i s t o

determine whether t h e product i s p o s i t i v e o r nega t ive .

There a r e s e v e r a l ways of making m u l t i p l i c a t i o n of r e a l

numbers seem p l a u s i b l e . It seems b e s t t o l e t t h e choice of d e f i -

n i t i o n of m u l t i p l i c a t i o n be a necessary outgrowth of a d e s i r e t o

r e t a i n t h e d i s t r i b u t i v e p roper ty f o r r e a l numbers. A t two p o i n t s

i n Sect ion 8-1, p r i o r t o t h e use of t h e d i s t r i b u t i v e p roper ty t o

discover t h e na ture of t h e products , t h e r e appear p a r t i a l mul t i - p l i c a t i o n t a b l e s , included simply t o h e l p e s t a b l i s h t h e p l a u s i -

b i l i t y of t h e d e f i n i t i o n of m u l t i p l i c a t i o n by p e r m i t t i n g t h e s tudent t o see t h a t t h e r e s u l t s obta ined us ing t h e d i s t r i b u t i v e proper ty a r e t h e same a s those seen i n t h e extended m u l t i p l i c a t i o n

t a b l e . Note, however, t h a t i f t h e d e f i n i t i o n of m u l t i p l i c a t i o n i s

based on cons ide ra t ions of mathematical s t r u c t u r e , t h e Implied

extension of t h e symmetry of a m u l t i p l i c a t i o n t a b l e must be

pages 270-274: 8-1

regarded only as supporting evidence f o r the def in i t ion .

Answers -- t o Oral Exercises 8-la; pages 270-271: -

2 . ( a ) True ( d ) False

( b ) False ( e ) False

( c ) False

3 . ( a ) t rue f o r a l l values of a

( b ) t rue f o r a l l values of a

( c ) not t rue f o r a l l values of n ( i n f a c t , t rue f o r no values of n )

( d ) t rue f o r a l l values of m

( e ) not t rue f o r a l l values of m ( t rue fo r no values of m )

( f ) not t rue f o r a l l values of a ( t rue f o r no values of a ) ( g ) not t rue f o r a l l values of x and y ( t rue f o r no

values of x and Y)

Answers t o Oral Exercises 8-lb; -- 1. ( a ) 18

(b) 0

( c ) 18 ( d l 0 ( e ) -20

( f ) -24

(43) -3

page 274:

( i) -12

( J ) -3.6 (k) - .32 (1) 0

(m) 0

(n) -6 ( 0 ) -5

2 . The operations i n ( b ) and ( c ) can be so performed, b u t

not the operations i n ( a ) and ( d ) .

pages 274-278: 8-1

3 . ( a ) -6 ( b ) Y e s

c ) 6 ( d ) " ( 3 ) ( - 2 ) " names the number -6.

" 13 1 1-2 1 I' names the number 6 .

Hence "- ( 13 1 1-21 ) " names the number

4. ( a ) False

( b ) False

( c ) True

( d ) True

Answers - t o Problem Set 8- lb ; page 275: -- 1. ( a ) True ( h ) False

( b ) False (1) True

( c ) True ( , j ) False ( d ) False ( k ) False

( e ) True (1) True

( f ) True ( m ) True

(g) False ( n ) False

5 (0 - 3 e ) 0

( d ) -20 ( f ) -20

Answers t o Oral Exercises 8-lc; page 278: -- 50

-12 -12

12

1 2

0

0

6

Yes 6

The expressions

name 6 . 1-21 1-51 and ( -2) ( -3) both

pages 278-280: 8-1

3 . ( a ) False

( b ) True

( c ) True

( d ) True

( e ) True

( f ) True

Answers - t o Problem Set 8-lc; pages 279-281: --

( a ) True ( f ) True ( k ) False

( b ) False ( g ) True (1) True

( c ) False ( h ) False ( m ) False

( d ) True (1) True ( n ) True

( e ) True ( j ) False ( 0 ) False

(P ) True ( q ) False

( r ) False

a ) (-31 ( d ) the s e t of r ea l numbers l e s s than

( b ) (-31 6 and greater than -6 ( c ) ( -1 , -2) ( e ) the s e t of a l l r ea l numbers except

zero

One Integer i s n . The other integer i s n + 4 . Their product i s 5 . The open sentence i s n ( n + 4 ) = 5. Possible pa i r s of integers whose product i s 5 are

1 and 5 ; -5 and -1.

I n e i t h e r of the above cases, the second Integer i s 4 more

than the f i r s t .

Therefore the in tegers a re 1 and 5 o r -5 and -1.

( a ) True

( b ) False

( c ) True

( d ) False

( e ) False

( f ) False

(I31 True ( h ) True

pages 280-284: 8-1

True True Fa1 s e True False e ere t h e s tuden t should no te t h a t t h e l e f t s i d e

of t h e equat ion i s a p o s i t i v e number and t h e r i g h t s i d e a negat ive number, and t h u s it i s not necessary t o s impl i fy e i t h e r s i d e f u r t h e r t o show t h a t t h e sentence i s f a l s e . )

Fa1 se True Fa l se True

( c ) no. The product of any two negat ive numbers i s p o s i t i v e .

Answers t o Oral I --- 1. ( 6 ) ( - 5 ) - 2. (-8)(-3) =

5 . ( 2 ) ( 4 ) = 12

4. ( 5 ) ( 0 ) - 1 5

pages 284-286: 8-1 and 8-2

Answers - t o Problem S e t 8-ld; page 285: -- 1. ( a ) I f a < O and b < 0 , then ab = la1 . b l .

( d ) If a > 0 and b > 0, then ab = la1 l b l .

( e ) If a = 0 and b = 0, then ab = la 1 . lb I . ( f ) I f a = 0 and b # 0, then ab = la 1 lb I .

2 . ( a ) 45 ( f ) 10,000 ( k ) -4.14

( b ) -45 (g ). -100,000 (1) - 4 . 1 4

( c ) 45 ( h ) 8 ( m ) -4.14 ( d ) 100 (1) -8 * ( n ) 2

( e ) -1000 ( J ) 8 ( 0 ) 2

8-2. P r o p e r t i e s - of M u l t i p l i c a t i o n .

Once t h e d e f i n i t i o n of m u l t i p l i c a t i o n of r e a l numbers i s formulated, i t can be proved t h a t t h e p r o p e r t i e s of m u l t i p l i c a t i o n

which h e l d f o r t h e numbers of a r i t h m e t i c a l s o hold f o r t h e e n t i r e

s e t of r e a l numbers.

The proof of t h e m u l t i p l i c a t i o n p roper ty of one i s included

i n t h e s t u d e n t ' s t e x t . Though it i s probably t h e e a s i e s t

p r o p e r t y t o prove among t h e p r o p e r t i e s d e a l t with i n t h i s sec t ion

of t h e t e x t , it w i l l doub t l e ss be d i f f i c u l t f o r slower s t u d e n t s .

The t e a c h e r should not expect mastery of t h e proof , but it i s

hoped t h a t t h e proof can be followed by t h e s tuden t t o t h e ex ten t

t h a t It w i l l provide an experience t o g ive meaning t o t h e asse r -

t i o n i n h i s text t h a t t h e p r o p e r t i e s of m u l t i p l i c a t i o n f o r t h e

r e a l numbers can be proved from our d e f i n i t i o n of m u l t i p l i c a t i o n .

pages 286-289: 8-2

The proofs of t h e a s s o c i a t i v e p r o p e r t y and t h e d i s t r i b u t i v e proper ty a r e not d i f f i c u l t , but a r e l eng thy and t e d i o u s . The proof of t h e commutative p r o p e r t y i s q u i t e b r i e f and i s given below:

If one o r both t h e numbers a, b a r e ze ro , then a b = ba, by t h e multiplication p r o p e r t y of ze ro . If a and b a r e both p o s i t i v e o r both negat ive , then

Since la 1 and Ib 1 a r e numbers of a r i t h m e t i c , and t h e commutative p r o p e r t y holds f o r t h e m u l -

t i p l i c a t i o n of t h e numbers of a r i t h m e t i c ,

Hence, ab -Â¡ ba

f o r these two c a s e s . If one of a and b i s p o s i t i v e o r 0 and t h e

o t h e r i s nega t ive , then

a b = - ( l a 1 l b l ) and b a a - ( l b l la l ) .

Since

and s i n c e i f numbers a r e equal t h e i r oppos i t e s a r e equa l ,

Hence, ab = ba

f o r t h i s case a l s o . Since a l l p o s s i b l e c a s e s have been cons idered , then

ab = ba, f o r any r e a l numbers a and b .

Answers t o O r a l Exerc ises 8-2a- page 289: -- -' 1. (a) 4 ( d ) -10

( b ) -5 (4 -9 ( 0 ) 1 2 ( f ) -14

pages 289-290: 8-2

Answers - to Problem Set 8-2a; pages 289-291: -- 1. (a) commutative property of multiplication

(b) commutative property of multiplication

(c) associative property of multiplication

(d) associative and commutative properties of multiplication

(e) multiplication property of one

(f) associative property of multiplication and multipli-

cation property of one

(g) associative property of multiplication and multipli-

cation property of one

(h) associative property of multiplication and commutative

property of multiplication, or the associative property

of multiplication and the multiplication property of

zero

(1) associative property of multiplication and the commu-

tative property of multiplication, or the associative

property of multiplication and the multiplication

property of zero

( j ) distributive property

(k) distributive property and commutative property of

multiplication

(1) commutative property of multiplication

Answers to Oral Exercises 8-2b; -- - 1. -2

2 . 3 3 . -a 4. a

5. -x + ( -5) 6 . x + ( - 5 ) 7. - x + 5 8. x + 5 9 . -a + (-2) 10. -a + (-2)

The distributive property

is not useful to simplify

this problem unless a

and y name the same

number.

Answers - to Problem Set 8-2b; pages 294-295: -- 1- (a) m check: (-m) + (m) = 0

(b) m + (-4) check: -m + 4 + m + (-4) = (-m) + m + 4+ (-4)

= 0

(c) 2x + 3 check: -2x+ (-3) + 2x + 3

= (-2x) +a+ (-3)+ 3 = 0

pages 294-2523 8-2

(d) -57 + (-7x) check: s imi lar t o above

(4 Y + X check: s imilar t o above

( f ) 3m + ( -4 ) check: s imi lar t o above

(h) 8x2 + l6x check: (-8x2 + (-l6x) + (8x2 + l6x) 2 2 = - 8 ~ + 8~ + ( - i 6 ~ ) + 1 6 ~ = o

( i ) (-3y2) + 7y check: 3y2 + (-7y) + (-3y2) + 7y 2

= 3y2 + (-3y ) + (-7y) + 7y = 0

(d) the s e t of all r e a l numbers

4. If x Is the smaller of the numbers, the l a r g e r is x + 5, and

If i s t r u e f o r some x, then i s t rue f o r the same x

-2x + (-5) = 17 i s t r u e f o r the same x -2x + (-5) + 5 = I 7 + 5 i s t rue f o r the same x

-2x = 22 Is t rue f o r the same x x Ã‡ -11 i s t r u e f o r the same x.

If x i s -11, the l e f t s ide of the first sentence i s ( - 1 1 + ((-11) + 5). This i s 17, the same number a s the r ight

pages 295-297: 8-2 and 8-3

s i d e , s o [-11) i s t h e t r u t h set of t h e f irst s e n t e n c e ,

and t h e two numbers r e q u i r e d i n t h e problem a r e -11

and -6.

Le t d be t h e number o f d o l l a r s s p e n t by t h e daughter . ;

then

d + (3d + 5 ) = 49.

The daugh te r s p e n t $11, and t h e mother $5.5.

Let n be one number,

then 3(-n) i s t h e o t h e r number, and

n + 5 ( - n ) = -86.

The numbers a r e 43 and -129.

8-3. Using - t h e M u l t i p l i c a t i o n P r o p e r t i e s .

I n t h i s s e c t i o n , t h e r e i s a s e r i e s of " s u b s e c t i o n s " , each of

which i n t r o d u c e s o r emphasizes a p a r t i c u l a r k i n d of s i m p l i f i c a t i o n

o r change i n t h e form of a p h r a s e . A l l of t h e p r o c e s s e s a r e

d i r e c t consequences of t h e p r o p e r t i e s of m u l t i p l i c a t i o n J u s t

developed. We wish t o g i v e s u f f i c i e n t p r a c t i c e w i t h t h e s e t e c h -

n iques , bu t we wish a l s o t o keep them c l o s e l y a s s o c i a t e d w i t h t h e

i d e a s on which t h e y depend. We have t o walk a narrow p a t h between,

on t h e one hand, becoming e n t i r e l y mechanical and l o s i n g s i g h t o f

t h e i d e a s and, on t h e o t h e r hand, d w e l l i n g on t h e i d e a s t o t h e e x t e n t t h a t t h e s t u d e n t becomes s low and clumsy i n t h e a l g e b r a i c

manipula t ion . A good s logan t o f o l l o w h e r e i s t h a t man ipu la t ion

must be based on unde r s t and ing . We stress h e r e a g a i n t h a t t h e

s t u d e n t must -- e a r n t h e right t o "push symbols" ( s k i p p i n g s t e p s ,

computing wi thout g i v i n g r e a s o n s , e t c . ) by f i r s t m a s t e r i n g t h e

i d e a s which l i e behind and g i v e meaning t o t h e man ipu la t ion of t h e

symbols.

I n c o l l e c t i n g terms we want t h e d i r e c t a p p l i c a t i o n of t h e d i s t r i b u t i v e p r o p e r t y t o be t h e main t h o u g h t . Don't g i v e t h e

impression t h a t c o l l e c t i n g te rms i s a new pr0ces.s. We are avoid-

i n g t h e ph rases " l i k e terms" and "similar terms" because t h e y are

unnecessary and t e n d t o encourage man ipu la t ion wi thou t under-

s t and ing .

pages 297-298: 8-3

Answers t o Oral Exercises 8-3a; pages 297-298:

1. (a) -12 + 4b (d ) -am + (-cm)

(b) 6 + 2b (e ) 28 + 8a 2 (c) -4m + 5 m ( f ) 0

2. (a ) 2(a + b ) (d) Ei ther ((-4) + (-5)) m, o r -9

(b) (-5)(a + b) ( 4 m(l + m)

(c) 3(3x + ( - 4 ) ~ ) ( f ) The following a r e a l l correct:

(-2 + 4)x, 2x, -2x(l + (-2))

Answers - t o Problem - Set 8-3a; pages 298-300:

1. (a) 15 + 5a ( f ) (-2a) + 2b

(b) (-12) + 4b (g) 5 m + 5n

(c) 6 + 2 b (h) 8.4a + (-4.8b)

(d l (-8) + ( - 4 4 (1) -am + (-cm)

(e ) (-3a) + (-3b)

m ( l + m)

a + (-2))

The d i s t r ibu t ive property

does not apply.

page 299: 8-3

'3. In some of t h e problems it i s suggested t h a t t h e t e a c h e r i n s i s t t h a t t h e s tuden t fo l low t h e suggested s t e p s i n t h e s o l u t i o n s given f o r ( k ) and (1). Perhaps i t would be

h e l p f u l t o ask them t o i d e n t i f y t h e p r o p e r t y t h a t t h e y have appl ied i n each s t e p .

(f) 4a + 3b. Call a t t e n t i o n t o t h e f a c t t h a t s i n c e t h e d i s t r i b u t i v e p roper ty does not apply he re , t h e terms cannot be c o l l e c t e d .

( g ) ( 4 + l ) a = 5 a

( h ) ( -5 )x + 2y. The terms cannot be c o l l e c t e d .

2 t + 3t + (-4)w + (-2)w commutative p roper ty of a d d i t i o n

( 2 + 3 ) t + ((-4) + (-2))w d i s t r i b u t i v e p roper ty 5t + ( - 6 ) w

( -5)a + 6a + (-2)b + 5b commutative p roper ty of a d d i t i o n

^(-5) + 6) a + ((-2) + 5) b d i s t r i b u t i v e p r o p e r t y

( r ) This i s a l ready i n i t s s imples t form.

pages 299-300: 8-5

4. a ( b + c + d ) = a(b + ( c + d)) Be s u r e t h i s problem i s not overlooked. Though t h e s t u -

= a ( b ) + a ( c + d ) dent might do t h e next prob- = ab + a ( c ) + a ( d ) lem c o r r e c t l y without doing . . . .

= ab + ac + ad Problem 4, t h i s problem shows him t h a t i t i s t h e same fami- l i a r d i s t r i b u t i v e proper ty which J u s t i f i e s t h e work of s i m p l i f i c a t i o n i n exe rc i ses such a s those i n Problem 5 .

5 . ( a ) 2 k ( a ) + 2k(m) + 2k(5)

( b ) ( - 6 ) a + ( - 6 ) ( - b ) + (-6) ( -7)

(4 6 - 3 ) + (-7) + 10)x

( d ) (-6) ( c + b + a )

(4 5 4 4 + 5a( -2 ) + 5 4 - 4

(f) 5a(a + s + (-1)) ; o r , 5(a2 + 2a + ( - a ) ) ;

o r , a(5a + 10 + (-59

6 . ( a ) 0 (b) (51 ( c ) t h e s e t of a l l r e a l numbers

( d ) t h e s e t of a l l r e a l numbers

( e ) t h e s e t of a l l r e a l numbers

( f ) (-31

7 . Let x be t h e smal les t of t h e numbers. Then

The numbers a r e 49, 51, 53 .

8 . If n I s t h e number of inches i n t h e width,

2n + 2(n + 1) = 24

and t h e domain of n i s t h e se t of p o s i t i v e I n t e g e r s .

23 23 The t r u t h s e t Is (-r-), but .T i s not an i n t e g e r .

Thus, It i s not p o s s i b l e t o f i n d an I n t e g r a l l eng th and

width f o r t h i s r e c t a n g l e .

pages 300-303: 8-3

Answers - to Problem - Set -3 8-3b* page 302:

When the student has worked step by step through a number of exercises of this sort well enough to convince the teacher that he understands the process, then he certainly should be permitted to

take short cuts in doing this work. The teacher should be ready,

however, with occasional questions to be sure that the ideas behind

the manipulation are always on call.

1. (a) 8a2m 2 2 (J) a b CXY

(b) -lay2 2 2

(k) -9m x

(c) -6ab2 (1) 72a 2 m 2 n 2

(d) 24cd 2 2 2 (m) -m n

2 (e) -3c d

(f) -6aw

(g) 4abcd

(h) -12abxy 2 2

(1) 21am n

Answers - to Problem Set 8-3c; page 303: - 1. 6x2 + 12xz 2 3 . -2rn + 6mn 2. -18ax + 12bx 4 . 3mx + 3my

pages 305-304: 8-3

2 2 9 . b + (-c) . This exercise and 15. 3m x + 4 . 5 m y

some of those which follow 16. may also be done by use of

the property that the oppo- 17. (-a) + (-b) + (-c) site of the s u m of two real 18. (-b) + c + ( - m )

numbers is the sum of their 19. -b + d + t opposites.

20. 4mx + (-6my) + 8 m z 10. -4x + (-3y)

Answers - to Problem Set 8-3d; page 304: - 1. a 2 + 5 a + 6 17. b2 + 7b + (-8)

pages 304-306: 8-3 and 8-4

8-4. M u l t i p l i c a t i v e Inverse .

By a s e r i e s of examples and ques t ions , and wi th t h e a i d of t h e number l i n e , t h e e x i s t e n c e and uniqueness of t h e m u l t l p l l - c a t i v e inverse are s e t before t h e s t u d e n t .

The s t u d e n t ' s first oppor tun i ty t o d i scover t h a t ze ro has no m u l t i p l i c a t i v e i n v e r s e comes i n Oral Exerc i ses 8-4a, Problem 23. This po in t i s emphasized aga in i n t h e t e x t i n t h e s e c t i o n fol lowing these e x e r c i s e s . The s t u d e n t should understand not only - t h a t ze ro has no m u l t i p l i c a t i v e inverse , but a l s o a It does n o t .

The word " r e c i p r o c a l " is not Introduced u n t i l Chapter 10,

where It i s given as an a l t e r n a t i v e f o r m u l t i p l i c a t i v e I n v e r s e . 1 A t t h a t po in t t he s ta tement i s made t h a t t h e symbol Is used

t o r epresen t t h e r e c i p r o c a l of t h e number x . Such a postponement i s expedient s i n c e t h e s t u d e n t w i l l no t have encountered d i v i s i o n with negat ive numbers i n t h e p resen t c h a p t e r . Hence, t h e symbol - f o r x < 0 might cause t r o u b l e a t t h i s s t a g e . I n Chapter 10 x the f u l l connection between d i v i s i o n and r e c i p r o c a l can be e s t a b l i s h e d on a l o g i c a l b a s i s .

Some d i scuss ion should b r i n g ou t t h e i d e a t h a t t h e m u l t i p l i - c a t i v e inverse i s unique, J u s t as t h e a d d i t i v e i n v e r s e i s unique. The uniqueness of t h e m u l t i p l i c a t i v e Inverse w i l l be used i n subsequent work.

pages 306-310: 8-4

Answers to Oral Exercises 8-4b; page 309: -- - 1. 1 7. 1 and -1: (1) (1) = 1

and (-!)(-I) = J -

8. -a 3. Zero has no multiplicative

inverse 9 . a

4. There is no number n such 10. no, zero does not have a that n(0) - 1. multiplicative Inverse.

Answers to Problem Set 8-4b:

1. (a) True

(b) False

(c) True

(d) False

(e) True

(f) False

11. The product of the two numbers will be one.

pages 309-310:

(g) False

(h) True

(i) False

( J ) True

(k) False

(1) True

pages 310-312: 8-5

8-5. Mult ipl icat ion Property o f - Equal i ty .

Both the addi t ion property of equa l i t y - comments f o r which the teacher may want t o review a t t h i s point - and the mul t ip l i - ca t ion property of e q u a l i t y a r e concerned with t he language with which we work r a t h e r than the a lgebra ic s t r u c t u r e . If a , b , and c a r e r e a l numbers and a = b, then t h e statement 'lac = bc" can be thought of a s saying t h a t the r e s u l t of mult iplying two given numbers was the same when they had the names " a 1 and " c ' ~ as when they had the names b t l and I1ctt.

A s i n the case of t he addi t ion property of equa l i ty , i t i s the usefulness of the mul t ip l ica t ion property of e q u a l i t y i n f inding the t r u t h s e t s of sentences t h a t J u s t i f i e s i t s e l e v a t i o n t 1 t o the s t a t u s of a proper ty .

Answers t o Oral Exercises 8-5; page 312: -- 1. Using the mul t ip l ica t ion property of equa l i ty , mul t ip ly each

1 and 2 a r e inverses . s ide of the sentence by F, s ince - 2 L

multiply each s ide by 1 5 9

I t I I I1 1, 1 IT 10.

mult iply each s i d e by 2

I t I t I1 T I -3

Answers - t o Problem - Set a; page 312:

A few of t he exerc i ses i n t h i s s e t w i l l be worked out i n various degrees of d e t a i l . For t h e o the r exerc i ses only the t r u t h s e t w i l l be given. I n ass igning exerc i ses f o r s tudents t o work, it probably would be unwise t o expect them t o work out i n d e t a i l more than four o r f i v e of these exerc i ses . After a l l , i t

i s a l s o a worthwhile ob jec t ive t o ge t s tudents t o t he point where

pages 312-313: 8-5 and 8-6

they can determine the t r u t h s e t s of open sentences of t h i s type inspec t ion .

If the re i s an a such that 10. (101

2a = 12 i s t r u e , 11. (6)

then the same a makes 1 2 . (-12)

and 13 . (:I

1 7 and 14 . [ - -) 3

and 5 15. t r u e .

16. [ - y

8-6. ~ o l u t i o n s - of open sentences.

Equivalent sentences w i l l be discussed I n more d e t a i l i n Chapter 15. You may wish t o r e f e r t o t h i s l a t e r discussion, i n both t e x t and commentary, before tak ing it up a t t h i s po in t . The

228

pages 313-314: 8-6

idea is introduced here f o r l i n e a r equations because the s tudent i s probably beginning t o be aware of i t by now and su re ly i s growing impatient with the checking rou t ine . It i s not our i n - tent ion t o do away with checking a l t oge the r f o r these equations, but r a the r t o put i t i n i t s proper perspect ive - a check f o r e r ro r s i n a r i thmet ic .

It i s i m p o r t a n t t h a t t he teacher note, and help t he s tudent note, t h a t i n the process of solving equations, not a l l s t e p s involve d i r e c t l y the equivalence of two equat ions . Those s t e p s In which the addi t ion property and mul t ip l ica t ion proper ty of equa l i ty a r e used must r a i s e t h e question of equivalence, but on the other hand the re may be s t eps taken with t he s o l e purpose of simplifying one member o r both members of an equation.

Thus i n going from

equivalence i s an i s sue because, f o r example, the phrase on the l e f t names a number d i f f e r e n t from t h a t named by the l e f t member -

of the o r ig ina l equation, a s t he addi t ion property f o r equa l i t y has been used. But i n going from

(3x + 7) + ((4 + (-7)) - (x + 15) + ((-4 + (-7))

t o 2x = 8,

the question of equivalence does not e n t e r the p i c tu re because a l l t h a t i s happening i s t h a t each member of the equation is being wri t ten i n simpler form. Both types of s t e p s a r e important, of course, and s tudents should be ab l e t o give reasons f o r them.

A prolonged discussion i n t he t e x t of t he d i f fe rence between these s teps could have been a d i s t r a c t i o n t o t he main idea , and so the t a sk of emphasizing the d i s t i n c t i o n i s l a r g e l y t h e teacher ' s . This i s probably appropr ia te , because many na tu ra l opportunit ies t o point t h i s out w i l l a r i s e i n c l a s s discuss ion throughout the course .

In connection with t he work on equivalent equations, some teachers repor t t h a t c l a s se s have found good p rac t i ce and enjoy- ment a s well In t he process of bui lding complicated equations from simple ones by use of equivalent equations. For example,

pages 314-316: 8-6

One of t he p r inc ipa l reasons f o r Introducing the idea of equivalent sentences a t t h i s time is the need f o r them i n study- i n g t r u t h s e t s of i nequa l i t i e s , coming i n Chapter 9 . It i s impossible, f o r example, t o "check" the t r u t h s e t of "x + 8 > 10" I n t h e sense t h a t one can check the t r u t h s e t of "x + 8 = 10".

It i s important i n t he former case t o know t h a t "x + 8 > 10" and ' x > 2" a r e equivalent sentences and so have Iden t i ca l t r u t h s e t s . Therefore, no ' c h e c k i n g need be done i n t he o r ig ina l sentence (again, assuming no a r i thmet ic e r r o r s ) ; the t r u t h s e t of "x > 2" t h e t r u t h s e t of "x + 8 > 10" .

I n t he f i rs t example i n t h i s sec t ion of the t e x t It i s pointed out t h a t t h e s teps used i n going from the o r ig ina l sentences t o t he simple sentence a r e r eve r s ib l e . Thus, i f t he re i s an x such t h a t 2x + 5 = 27 i s t r u e , then x = 11 is t r u e f o r the same x; and, conversely, i f there i s an x such t h a t x = 11 is t r u e , then 2x + 5 = 27 Is t r u e f o r t he same x . Although It Is not c a l l e d by t h i s name o r s t r e s sed i n t he t e x t , t h i s is the f i rs t s i t u a t i o n involving "if and only i f " , and it may be a good place f o r t h e teacher t o begin bui lding f o r t h i s important concept, e spec i a l l y s ince the notion is perhaps more e a s i l y v i s - ua l ized I n terms of equations and t r u t h s e t s than i n t he more s u b t l e proofs which the s tudent may l a t e r encounter i n o ther courses .

Although t h e Idea i s not d i f f i c u l t , " i f and only i f " often gives r i s e t o confusion. The form always i s "A i f and only i f

B", where A and B a r e sentences. We are a c t u a l l y deal ing w i t h t h e compound sentence, "A - i f B and A only - i f B" . The sentence " A i f B" i s a compact way t o wr i t e "If B then A" ,

and " A only i f B" i s a way of wr i t i ng "If A then B".

These condi t iona l sentences a r e sometimes wr i t t en "B implies A"

and " A implies B". Some w r i t e r s abbrevia te " i f and only i f " t o " i f f " . The compound sentence then reduces t o " A I f f B".

The confusion with " i f and only i f " comes from t r y i n g t o remember which statement i s t h e " i f " statement and which i s the "only

pages 316-317: 8-6

i f statement. Everyone has t h i s t rouble but i t i s fo r tuna t e ly not an important mat ter . What - i s important i s that t he compound sentence "A i f and only If B means "If A then B - and i f B then A " .

The preceding remarks are f o r the benef i t of t he teacher only. It i s probably not w i s e t o introduce " i f and only i f 1 ' notat ion t o t he s tudents a t t h i s time.

Answers - t o Problem Set 8-6; page 317: -- Since t h e s tudent has been shown procedures which assure t he

formation of equivalent sentences, it w i l l no longer be necessary, i n general , f o r him t o "go t h e o the r way", i .e . , c a r r y out t h e reverse operations. In t he first fou r problems, however, we give him t h i s experience, which may, a s suggested e a r l i e r , he lp s e t the s tage f o r an understanding of i f and only i f " .

Going t h e o the r way:

The t r u t h s e t is (7) .

20. any r e a l number

21. (0)

pages 318-320: 8-7

8-7. Products and t h e Number Zero. -- The theorem on products and the number zero i s presented i n

t h i s s ec t ion i n two p a r t s . The first, s ince it i s a d i r e c t consequence of t h e mu l t i p l i ca t ion property of zero, requires very l i t t l e e luc ida t ion . The second p a r t Is f a r l e s s obvious. It i s proved here i n d e t a i l f o r two reasons: one, t o d i spe l i n the s tuden t ' s mind t h e erroneous notion that t h e first r e s u l t implies t he second, a common e r r o r ; two, because of the s ignif icance of t he second r e s u l t i n determining complete t r u t h s e t s of ce r t a in types of equat ions . For example, without t he second property we could not a s s e r t that 3 and 4 a r e t he only t r u t h numbers of t he sentence (x - 3 ) ( x - 4) = 0.

Our theorem can be s t a t e d i n one piece as an " i f and only i f "

statement a s follows:

For any r e a l numbers x and y, xy = 0 If and only i f x = 0 o r y = 0. (The use of "or" here includes the case when both x = 0 and y = 0.)

A s before, t h i s form i s not given i n the t e x t s ince the two par t approach seems a t t h i s point t o make f o r g r e a t e r c l a r i t y .

Answers t o O r a l Exercises 8-7; page 320: -- - 1. (a) True ( e ) False

(b ) False ( f ) True ( c ) True (g) False ( d ) True (h ) True

Answers t o Problem Set 8-7; pages 320-323: - -- 1. ( a ) (01 (4 (0)

(b) (0) ( f ) (01 ( 4 (01 (g) t h e s e t of a l l r e a l numbers (d l (01 (h ) (0)

pages 321-322: 8-7

4. Be sure t h a t t he s tudents wr i t e out t h e s t eps c a r e f u l l y i n the so lu t ions . The following method i s suggested:

( a ) I f x i s the number of cen ts t h a t Mr. Johnson paid for each foo t of wire,

then 3Ox i s t he number of cen ts t h a t M r . Johnson paid f o r t he f i r s t purchase of wire,

and 55x i s the number of cen t s t h a t M r . Johnson paid f o r t he l a t e r purchase of wire;

page 322: 8-7

25x i s the number of cen t s t h a t the neighbor paid f o r t he wire t h a t he purchased.

Then t h e open sentence i s

Checking: A t 7# per foo t ,

30 f e e t of wire c o s t s (30 ) (7 ) o r 210#; 55 f e e t of wire c o s t s (55 ) (7 ) o r 385k Mr. Johnsonts wire c o s t s (210 + 385) o r 595$. The neighbor's wire c o s t s (25) (7) o r 175$ f o r

25 f e e t of w i r e . Mr. Johnson's t o t a l cos t Is (175 + 420) o r 595#. Thus 7# p e r foo t is the cos t of the wire.

( b ) If n i s t h e i n t ege r , (n + 1) i s the successor of t h a t In t ege r . The open sentence I s

If the number i s 6, fou r times the number is 24.

If t h e number Is 6, i t s successor i s 7, twice the successor i s 14, and 10 more than 14 i s 2 4 .

Therefore 6 i s t h e required in t ege r .

page 322: 8-7

(c) If rn is the number of miles per hour that the first man drove,

then 5m is the number of miles that the first man drove in 5 hours

and 3m is the number of miles that the second man drove In 3 hours.

A diagram similar to the one below may be helpful. 1-;--5 m T,--!--y--120 m i . - - 1

--- --

3m--*1<---- - 250 mi.-- - ------ ------

Start Finish

The open sentence is

Check: If each man drove at the rate of 65 miles per hour, then

the first man drove 325 miles in 5 hours, the second man drove 195 miles in 3 hours; 325 + 120 = 445, 195 + 250 = 445.

From here on the solutions are In more compact

form and the check is not given.

(d) Let -/' be the number of units In the length of the third side.

Open sentence :

pages 322-325: 8-7

( e ) n: the Integer

The more a l e r t students w i l l observe t h a t t h i s sentence i s t r u e f o r any integer .

( f ) a: the number of p igs

4a + 2(a + 16) = 74 6a + 32 = 74

6a = 42

a = 7

(g) b: the number of h i t s

lob + (-5)(b + 10) = -25 5b = 25 b = 5

( ~ a i n i s 10 if he h i t s and -5 i f he misses.)

Answers - t o Review Problem - Set; pages 324-331:

3. (a) 2a + 4b

(b) - 4 ~ + 16d

( c ) 42c + (-36d)

(d) -32am + (-24an)

(e ) Slab -I- 28ac

24am ( i ) 24ab

-5m 2 (j) -12bm + (k) -4%

2

TC (1) 0 2 ( m ) -60a mx

(h) 4am + (-2an)

(k) 10c2 + 20cd ( s ) 2 + (-25)

(1) 15bm + (-24b2) ( 6 ) W 2 + (- ,Ã‡ 3 + 2 16 ( m ) - k d + 4cd (u) a2 + 3a + 2.25

(n) a2 + 7a + 12 (v) b2 I- (-4.41) 2

(0) a + a + (-12) (4 6y2 + ( - 4 3 4 + (-834

(PI m2 + (-36) *(x) -6y2 + $yz + (-4y) + (-122) + 16

( q ) y2 + ( - 1 1 ~ ) + 18 *(y) 8m2 + Smn + 2m + n + (-1) 3 (4 y 2 + 2 y + v

4. (a) l3x

(b ) -13a

(k ) 12a + 3.c

(1) 6 a + 4 b + c

(m) 6P + l l q

(n) -2p + (-6r)

( 0 ) -9b

(P) 0

(q) 2 t + 5s

(4 a + g b

(s) -4m + n + a

(t) 7a

In Problems (h) through ( j ) s imilar a l te rnate answers are acceptable.

(h ) 2a(2m + 3n)

( a ) True ( b ) False ( c ) True (d ) False

(a) True ( b ) True ( c ) False (d ) False ( e ) True ( f ) True (g) False

( a ) False (b ) False ( c ) True ( d ) True (e ) False

( e ) False ( f ) False (g) True (h) False (1) True

(h) True (I) False ( j ) False

a( -1) # 1 i s fa l se when a = -1

(k) False when a Is 0

(1) True

( f ) True (j) False (g) False ( k ) True (h) False (1) False (1) False ( m ) True

(1) a l l rea l numbers greater than o r equal t o 5

(m) a l l numbers l e s s ( 9 ) 0 . . than 1

(n) fS . .

(0) a l l r e a l numbers g r e a t e r than 5 ( v ) a l l r e a l numbers and a l l r e a l numbers l e s s than (-5) (w) ( - 4 )

(p) all r e a l numbers (x) (0, 1)

11. Students should be encouraged t o check t h e i r answers. In verbal problems t h i s checking should be done f i r s t I n t he o r ig ina l statement of t h e problem, then, i f necessary, i n the open sentence. Here t he work is shown i n d e t a i l only f o r p a r t s ( a ) and ( b ) . After a while t h e s tudents should be able t o omit some of t he s t e p s .

(a) Let x be t h e number. Then the open sentence is

Check: If 21 i s the number, then twice t he number i s 42, and the sum of twice t h e number and 5 is 4 2 + 5 o r 47.

(b) Let b be the number of bushels of wheat each t ruck can hold. Then 3b is the number of bushels one truck hauled and 4b is the number of bushels t he o the r t ruck hauled.

Check: I f each truck holds 70 bushels, then the first truck hauled 3(70) o r 210 bushels, the second one hauled 4(70) or 280 bushels. Together they hauled 210 + 280 o r 490 bushels.

( c ) c,: number of cents tha t one can of peaches costs

(d) x: number of degrees i n the second angle

Check: 42 + 84 + 54 = 180

( e ) Let t be the number of hours t h a t the passenger t r a i n ran before overtaking the f r e i g h t t r a in . Then the f r e i g h t t r a i n ran ( t + 1) hours. 6ot i s the number of miles the passenger t r a i n traveled. 40( t + 1) i s the number of miles the f r e igh t t r a i n traveled .

60t = 4O(t + 1)

t = 2 t + 1 = 3

9:00 A.M.

120 miles

( f ) Let w be the number of f e e t i n the width.

The dimensions a re 30 f e e t by 68 f e e t .

(If the student should say o r write "w = 30 feet", remind him t h a t w represents a number, so tha t

w = 30 is a statement about numbers, whereas "the width is 30 feet" is a statement indicating how long a certain line is.)

12. (a) 4 ; 2 ; : ; ; ! ; ; ; ; c , w - 6 - 5 -4 -3 -2 - 1 0 I 2 3 4 5 6

, l * , l , ,

(b 1 , , . , I . . , .

0 (the null set)

Sueeested Test Items -- 1. Find the value of each of the following when x is -3,

1 y l a 2, a is -4, and b is y .

(a) 2ax + 3by (4 (a + x + (-Y))2

( d ) 2x + (-a) + (-y2) 2. Write these Indicated products as Indicated sums.

(a) (-7) (3x + 4 ~ ) (d l (5x + (-2))(3x + 7)

(b) 3 ~ ( y + ( - 2 x ~ ) + x2) ( 4 (x + (-4)) (5x + ( - 5 3

(4 (x + 6)(x + 7) (f) (-6x34

3. Write each of the following as an indicated product.

(a) 7a + 7b (dl 2xy + (-xy) + (-x)

(b) 3m + 1511 ( e ) (-4)a2 + (-4)x 2

Collect terms in the following.

(a) z + 32 (c) 4x + (-6) + 6x + 12y (b) (-15a) + a (d) x + 3y + 7x +

Write the multiplicative inverse of each of the numbers.

(a) 3 (4 0 (b) -5 (f) 15 + (-711

16 (4 (g) .23

The following sentences are true for every a, every c.

A. a b = b a B. (ab)c = a(bc)

C . a(1) = a

D. a(0) = O

E. (-a)(-b) = ab F. If a = b, then ac s= bc.

G. a(b + c) = ab + ac

(-2y) + 4~

following

every b, and

Which of the sentences expresses:

(a) the associative property of multiplication? (b ) the distributive property?

(c) the multiplication property of equality?

(d) the multiplication property of one?

Find the truth set of the following open sentences.

(a) + (-8) = 4

(b) 1-51 + 7 + (-5) + 2x = O

(4 -((-5)x + 7) = 5x + (-7)

(d) 1x1 = Q(-3) + (-2)(-8) (e) 2x + 3x = 8 - 3x

8. If a and b a re real numbers, s t a t e the property used i n each s tep of the following.

9. Find t r u t h s e t s f o r the following open sentences and draw t h e i r graphs.

( a ) 7 r + 4 + 3r = (-4r) + 18

LO. Write an open sentence f o r each of the following problems. State the t r u t h s e t s and answer the questions.

( a ) Two automobiles 360 miles apart start toward each other a t the same time and meet in 6 hours. If the r a t e of the first car i s twice t h a t of the second car, what i s the r a t e of each?

(b) Four times a cer ta in in teger i s t w o more than three times i t s successor. What i s the integer?

( c ) The perimeter of a t r iangle i s 40 inches. The second side i s 3 inches more than the first side, and the th i rd s ide i s one inch more than twice the first side. Find the length of each side.

11. Which of the following sentences a r e t rue f o r a l l values of the var iables? In each case t e l l what propert ies and defini t ions helped you decide.

(a) a + (-a) = 0 (d ) - (x + y ) = - l ( x + y )

(b) (73)(-^L.3)(0) = 0 ( e ) (-7)(-$) > (7)(67)

(g) 1-2611 < (130)W (h) x + x = 2 x

12. Write and solve an open sentence to answer each of the

following . (a) When a number and twice its opposite are added, the

3 result is p For what number is this sentence true? Write and solve an open sentence to answer this questioi Tell what properties you used in finding the solution of this sentence.

(b) Two numbers are multiplicative inverses, and one of the]

is one-fourth of the other. Find the pairs of inverses for which this sentence is true by writing and solving an open sentence.

( c ) The product of a certain number and its opposite is the opposite of the square of the number. Find the number for which this is true by writing an open sentence and finding its truth set.

Answers - to Suggested Test Items

(a) 27 (4 81 (b) -20 (dl -6

(a) (-21x) + (-28y) (d) 15x2 + 29x + (-14) 2 (b) 3y2 + (-6xy2) -I- 3x y (e) 5x2 + (-23x) + 12

(c) x2 + 13x + 42 (f) dy

(e) Zero has no multiplicative Inverse.

(h) Zero has no multiplicative inverse.

( c ) t he set of all real numbers

distributive property distributive property The product of one number and the opposite of another

number is the opposite of the product of the two numbers.

addition property of opposites addition property of zero

(The associative property of addition is also used in the latter steps, since it makes possible the grouping Implicit in these steps . )

10. (a) If the second car is t ravel ing r miles per hour, the f i r s t car i s t ravel ing 2 r miles per hour. In 6 hours, then, the second car t r ave l s 6r miles, and the f i r s t car 6(2r ) miles. Since the number of miles traveled by

both cars together i s 360, we have

Â¥Hi r a t e of the f

the second car is

rst car i s 20 m.p.h., and the r 40 m.p.h.

(b) If n i s the integer, n + 1 is i t s successor, and

The in teger i s 5 and i t s successor i s 6.

( c ) If the first s ide i s m inches long, the second side i s m + 3 inches long, and the th i rd side is 2m + 1

inches long. Then

The first s ide i s 9 inches long, the second side i s 12 inches long, and the th i rd s ide is 19 inches long.

1. (a) True. Addition property of opposites

(b) True. Multiplication property of zero

( c ) True. Multiplication property of one

(d) True. -a = (-1)a

( e ) False. (-a)(-b) = ab

(f ) True. ~ i s t r i b u t i v e property

(g) False. The absolute value of a number i s always a non-negative number, and the product of a negative number and a pos i t ive number Is a negative number.

(h) True. Distr ibut ive property

12. (a) Let x be the number, then -x i s i t s opposite. The open sentence is

3 x + 2(-x) =, T,

3 x(l + (-2)) = 7 Distr ibut ive property

x = - F Multiplication pro e r t y of equal i ty and a(-1 7 = -a

3 The t r u t h s e t is (-

1 (b) Let x be one number and y x be the other. The open sentence is

1 X ( ~ X ) = 1.

The t r u t h s e t i a (-2, 2 ) . The p a i r s of Inverses a re 1 1 -2, - - and 2, . 2

( c ) Let x be the number, then -x i s i t s opposite. The open sentence I s

2 x(-x) - -x.

The truth s e t is the s e t of a l l real numbers.

Chapter 9

PROPERTIES OF ORDER

In t h i s chapter t he p rope r t i e s of t h e order r e l a t i o n i s l e s s than" a r e systemat ical ly developed. Throughout t he discuss ion a l l order re la t ionsh ips a r e phrased i n terms of t he symbol < I 1 .

The motivation f o r t h i s i s twofold, l ) , t o keep the development as uncluttered as poss ib le , and 2 ) , t o emphasize the f a c t t h a t we a re e s s e n t i a l l y considering only one order r e l a t i o n among t h e r e a l numbers. To be sure , i n t a l k i n g about a given p a i r of numbers, we may, and f requent ly do, s h i f t from " l e s s than" t o ' g r e a t e r than" and back again without t roub le . However, t h i s tends t o obscure t he idea of order r e l a t i o n and i s - not permissible when we a r e studying the order r e l a t i o n "<" i t s e l f . We are making an Issue of t h i s mat ter because It i s mathematically i m - portant f o r the student t o begin th inking of order r e l a t i o n and not jus t o rder . It i s not e s s e n t i a l t h a t t hey be able t o explain i t . I f the teacher i s ca re fu l t o duscuss t he p rope r t i e s co r r ec t - l y , then the student w i l l automatically l e a r n t o th ink about an order r e l a t i o n a s a mathematical ob jec t r a t h e r than as a convenient way of discuss ing a p a i r of r e a l numbers.

It has been the custom i n t he pas t t o a s s e r t t h a t p rope r t i e s analogous t o those applying t o t h e order "is l e s s than" can a l s o be "proved" i n a similar way f o r t he order r e l a t i o n "is g r e a t e r t h a n . Rather than oppress t h e student wi th a host of add i t i ona l proper t ies , we make ins tead a simple statement t o t h e e f f e c t t h a t the expression a < b may be wr i t t en i n an a l t e r n a t i v e form b > a, both expressions conveying p rec i se ly t h e same r e l a t i onsh ip idea, namely t h a t the real number a is loca ted t o t h e l e f t of the r e a l number b on the number l i n e .

The student has a l ready been fami l ia r ized with t h e symbols of inequa l i ty and has used them i n connection with open sentences . Thus, c e r t a i n developments i n t h i s chapter may appear t o be r e p e t i t i v e . It i s hoped, however, t h a t t h e s tudent w i l l be ab l e t o grasp the d i s t i n c t i o n between the use of t h e symbols "<" and 'I>'' t o form mathematical sentences and a s tudy of t h e p rope r t i e s of an order r e l a t i o n . Again i n t h i s chapter we introduce some simple proofs. Considerable ca re should be taken t o prepare t h e

s tudents f o r t he presenta t ion s o t h a t they might understand the s ign i f icance of the proofs . The teacher may decide t o omit some of t h e proofs . This can be done without l o s s of con t inu i ty . It i s f e l t , however, t h a t a s t rong e f f o r t should be made t o present a t l e a s t one o r two of t h e proofs I n c l a s s .

9-1 . The Order Relat ion f o r Real Numbers . -- -- It Is l i k e l y t h a t a s tudent ' s unfavorable reac t ion t o many

of t h e p rope r t i e s presented i n t h i s chapter w i l l stem from a sense t h a t they a r e f o r t he most p a r t i n t u i t i v e l y obvious. The comparison and t r a n s i t i v e proper t ies , f o r example, may seem scarce ly worth mentioning. The teacher , however, should be aware of t he f a c t t h a t t h e r e a r e lumped toge ther i n our statement of t he comparison property two d i s t inguishable ideas: (1) a s t a t e - ment about t h e language of algebra and ( 2 ) a bas ic property of o rder . The first of these merely recognizes t h a t it i s pos- s i b l e f o r a and b t o represent t he same number. Then, of course, the order r e l a t i o n does not apply, s ince the re i s but one number involved. I f , on t h e o the r hand, a / b, then exac t ly one of t he following i s t rue : a < b y o r b < a . Some authors s t a t e t h e comparison proper ty f o r a b only, thus s t r e s s i n g the order r e l a t i on ; o the r s have termed t h e property t he trichotomy property, thereby tending t o s t r e s s t he Idea t h a t , i f a and b each represent any real number, it is always poss ible t o hang" exac t ly one of t h r e e symbols between them t o make a t r u e sentence. A s Indicated a t t he ou t se t i n our comments f o r t h i s chapter , we hope I n our approach t o play down t h e tendency t o focus on the numbers themselves, and t o emphasize t h e order r e l a t i o n .

I n connection with the t r a n s i t i v e property it m i g h t be help- f u l t o c i t e some examples, perhaps non-numerical ones, of a r e l a - t i o n which does not have the t r a n s i t i v e property. For instance, t he f a c t t h a t John i s the f a t h e r of Sam, and Sam i s t h e f a t h e r of Tom, does not imply that John i s the f a t h e r of Tom. Likewise i f

John loves Mary and Mary loves Joe, it w i l l not always follow t h a t John loves Joe! If the student i s familiar with some e le - mentary geometry, t he r e l a t i o n "is perpendicular t o w i l l provide a s i g n i f i c a n t example of a non- t ransi t ive r e l a t i onsh ip between l i n e s In t h e plane.

pages 337-341: 9-1 and 9-2

Answers t o Oral Exercises 9-1; page 337: -- - In doing Problems 9 and 10, t he s tudent w i l l very l i k e l y

use the verb phrase "is g r e a t e r than" i n reading t h e sentence t h a t i s h i s answer. Thus the s tudent w i l l be more l i k e l y t o c l inch the idea t h a t , given two d i f f e r e n t numbers, one i s always l e s s than the o ther .

8. b < 5, s ince b < -1 and -1 < 5

9. -a > -4; t h a t i s , -4 < -a 10. 0 > -c; t h a t is , -c < 0

Answers - t o Problem Set 9-1; pages 337-338: -- 1. (a) -5 < -2 (d) fr > .3124 ; t h a t I s .3124 < fp

4 ( e ) a > b ; t h a t i s b < a

( c ) -5 < .01 ( f ) x < x + 1

3 . ( a ) x + ( - 1 ) < 3 (d ) - ( a + b ) < b + (-a) (b ) 0 < z ( e ) - 1-31 < -2 s ince 2 < 1-51 ( c ) 2 < m ( f ) Since a = b, t he re i s but one num-

be r involved, and the re can be no order ing.

9-2. Addition Property -- of Order.

The second addi t ion proper ty i s introduced as an i l l u s t r a t i o n of a simple deduction based on two o the r p rope r t i e s . Once again the r e s u l t may seem " i n t u i t i v e l y obvious".

Answers t o Oral Exercises 9-2a; pages 340-342: -- - 1. ( a ) True (b ) True ( c ) True

pages 341-343: 9-2

(d) No decis ion can be reached. (g ) More information i s needed.

( e ) True (h ) True

( f ) True (1) False

2 . (a) The "=" r e l a t i o n does have the t r a n s i t i v e property. If 8 - 5 - 3 a n d 3 = 4 - 1 , then 8 - 5 - 4 - 1 .

(b) The r e l a t i o n ">" has t h e t r a n s i t i v e property. If 7 > 3 and 3 > - I , then 7 > -1.

( c ) The r e l a t i o n # does not have the t r a n s i t i v e property. 8 # 7, and 7 # 7 + 1, but 8 = 7 + 1 -

This p a r t i c u l a r exerc i se provides a good opportunity f o r the teacher t o point out t h a t it requi res only one, perhaps somewhat I so l a t ed counter-example t o prove t h a t a property does not hold. The s tudent may e a s i l y be l e d t o bel ieve t h a t t he r e l a t i o n "#" is t r a n s i t i v e s ince it would appear t o work i n a l l cases In which t h e o r i g i n a l choice of c was such t h a t c fi a t o begin w i t h .

He may a l s o be suspicious of the given answer on the ground t h a t t he hypotheses look l i k e a # b and b / a , with no c involved. Here again, It may be necessary t o reaff i rm the fact t h a t d i f - f e r e n t l e t t e r s may be names f o r t he same number.

(d) If a and b a r e any two d i f f e r e n t r e a l numbers, then one of t h e statement, "a < b" and "b < a" , i s t rue and t h e o the r i s false.

Answers t o Problem Se t 9-2a; pages 342-343: - -- 1. (a) <

(b) < ( 4 < (d l > ( e ) No decis ion can be made.

More information i s needed.

pages 343-346: 9-2

(f) < (g) < Since a + 2 < -1, -1 < 0, and O < b

(h) Can't t e l l s ince c could be pos i t i ve , 0, o r negative.

(1) < This problem a n t i c i p a t e s t he work a t t he c lo se of Section 9-2.

( J ) <

2. (a) False, 3 + 4 - " 4

(b) True, -6 < -3

c ) False, $J < 5

( e ) False, 3 + (-12) - (-18) + 9

( f ) True, 18 < 24

Pages 343-345. Here and i n Section 9-3 the concept of equivalent i n e q u a l i t i e s i s presented without an attempt a t r igor - ous j u s t i f i c a t i o n . A more de t a i l ed treatment of t h e same top ic i s given i n Chapter 15.

Answers t o Oral Exercises 9-2b; pages 345-346: -- - 1. (a) No (b) Yea ( c ) Yes

2. ( a ) Add (-3) t o each "sidett of t h e "<I1 statement. x < (-4) + (2 ) + (-3)

(b) Add 8 t o each s ide ; 2n < (-27) + 8

( c ) Add (-4) t o each s ide ; (-8) + 12 + (-4) < (-3n) 8 3 8

(d ) Add ( - -) t o each s ide ; 7 + ( - -) + 2 + ( - -) < 2x 1

3 2

3 ( e ) Add y t o each s ide ; .8 + 1 4 + ( - + 7 1 < 4~

pages 346-347: 9-2

3 . ( a ) Add -x + (-3) t o each s ide , o r add -x t o each s ide and (-3) t o each s i d e i n two separa te s teps .

(b ) Add 4 + (-3y) t o each s i d e .

( c ) Combine terms t o obta in -4n + 14 < -3n then add 4n t o each s i d e .

( d ) Add - - +, t o each s i d e . 3 ( e ) Combine terms t o obta in .3 + 3.237 < .3 + 2.2y

then add - .3 + (-2.237) t o each s i d e .

Answers t o Problem Set 9-2b- pages 347-348: - - -' 1. (a) t h e s e t of a l l numbers l e s s than 8

(b) t h e s e t of a l l numbers l e s s than 2

( c ) t he s e t of a l l numbers l e s s than 0

(d) the set of a l l numbers g r e a t e r than 0

11 ( e ) t he s e t of a l l numbers g rea t e r than yw 4

( f ) the s e t of a l l numbers g r e a t e r than - 3 (g) t he s e t of all numbers l e s s than o r equal t o -4

7 (h) t h e s e t of a l l numbers g rea t e r than o r equal t o -

3

19 ( J ) t he s e t of a l l numbers g rea t e r than

(k) t h e s e t o f a l l numbers g r e a t e r than 4

(1) t h e s e t of a l l numbers g r e a t e r than -7

(m) t h e s e t of a l l numbers g r e a t e r than -4

(n) t h e s e t of a l l numbers g rea t e r than o r equal t o -1

(0) (-3)

(PI (1)

(q) t he s e t of a l l numbers l e s s than 15

( r ) t h e s e t of a l l numbers

(9 ) (-4)

( t ) t h e s e t of a l l numbers l e s s than 5

pages 347-348: 9-2

(1 - ( t h e empty s e t ) I 0 I

4. Suppose n i s the number.

and n > 4.

then

All numbers g r e a t e r than 4.

The teacher should note t h a t we have asser ted here t h a t the " r e v e r s i b i l i t y " of add i t ion by a r e a l number assures equivalent sentences. Therefore t he re is no need from the point of view of mathematical theory t o reverse t h e s t eps i n t he process of solving the sentence. Nevertheless, going through t h e reverse s t e p s does a f ford the s tudent one means of checking h i s work f o r computational e r r o r s . He may p re fe r , ins tead, t o choose severa l numbers from what appears t o be the t r u t h s e t of t he sentence, and then check these i n t h e o r ig ina l sentence, but t h i s I s an incomplete s o r t of check, hardly more than an Ind ica t ion of t h e p l a u s i b i l i t y of h i s supposed so lu t ion s e t .

pages 348-351: 9-2

5 . Suppose x I s the t h i r d t e s t score.

If 82 + 91 + x > go, 3 Check: i f

He must score higher than 97.

Pages 348-350. The "two-way" connection between the order rela- t ion and addition w i l l play a leading ro le i n the development of the mult ipl icat ion property. It i s essen t i a l t o the proof.

A t t h i s point the words equation and inequality a re in t roduced as names f o r the two types of sentences under consideration. It i s qui te l i k e l y tha t these terms are already famil iar t o the students.

Answers -- t o Oral Exercises 9-2c;page 350: - 1. ( a ) 3 + 4 = 7 ( e ) -99 + .009 = .999

(b ) -2 + 6 = 4 (f) -.3999 = -.4000 + .OOO1

( c ) - 4 % - 5 + 1 (g) (x) + 2 = x + 2

9 (d ) -- = l2 + 2 5 - T " 5 (h) k + 1 = (k) + 1

2 . ( a ) x < 6

(b ) w < 9.5 (4 m > n ( f ) x + (-1) < y + 2 or

(x + (-1)) + 3 > Y (x - Y) (g) X < Y + 5 or X + 2 > Y

(h) k < 1

Answers - t o Problem - Set -' 9-2c- pages 350-351:

1. ( a ) -15 > -24 ; 9

pages 351-357: 9-2 and 9-3

True ( c ) True Fa1 se (d ) True

( e ) True ( f ) True

(g) False (h ) True

3 . Addition property of order Addition proper ty of zero a + c names the same number a s b .

9-3. Mult ip l icat ion Property -- of Order.

A deductive argument i s given t o show the p l a u s i b i l i t y of the mul t ip l ica t ion property of o rder . This argument does not cons t i t u t e a complete proof of the property but it does contain the e s s e n t i a l Ideas t h a t a r e involved I n t he proof .

A second mul t ip l ica t ion proper ty analogous t o t h e second addit ion property i s Included. The r e s u l t s of t h i s a r e f r u i t f u l I n the study of square roo t s . They should be noted even though the student may wish t o s ide-s tep t h e proof.

Answers t o O r a l Exercises 9-3a; page 357: -- 1. 2a < 10 6 . -3 < x + (-1) 2 . - 2 b < 6 7. 15 < -3(a + (-b)) 3. - p < O 8. 2 5 < 5 a + 5 b

4 . 3m < 3n 9. a < - ^

5. -2q < 10 10. -2 < x

Answers - t o Problem - Set 9-3a; pages 357-358:

1. ( a ) < ( c ) We c a n ' t say. (b1 > ( d l >

pages 357-360: 9-3

2 . (a) 15 < -3x

(b ) a < -1

(c) - 2 < z - 2

3. (a) x = 6

(b ) z = -3

(c) 2x < 2

An6wers t o Oral Exercises 3-3b; page 360: --

3 -2 8. Do not multiply. Add -5.

1 9. Add -4 ; multiply by

10. Add -3 ; mul t ip ly by 1 5

Answers - to Problem - Set 9-3b; pages 360-361:

1. (a) a l l numbers less than 3

(b ) a l l numbers less than 3 ( c ) a l l numbers greater than & (d) a l l numbers less than 1

(e) a l l numbers less than -3

( f ) a l l numbers o r greater 7 (g) a l l numbers greater than -18

pages 360-361: 9-3

(h) a l l numbers greater than -8

(1) (01

( J ) f6

(a ) a l l numbers less than (-3) 11

(b) a l l numbers greater than ( - T) 2

or greater ( c ) a l l numbers - - 2

( f ) a l l numbers greater than -12

(g) a l l numbers less than .31

(h) a l l numbers

19 (j) a l l numbers greater than - - 11

If Moe pays x dollars, then Joe pays (x + 130) dollars.

Joe : x + 130 < 255, Moe : x < 125; Total cost < 380.

Hence, Moe pays less than #125.

pages 561-363: 9-3

5. Suppose n i s the number.

The number Is greater than 2.

6. Suppose there are x students i n the c l a s s .

There are l e s s than 23 students.

7. If Norma i s x years old, then B i l l Is (x + 5) years old.

Norma i s l e s s than 9 years o ld .

Answers - t o Review Problem E; pages 363-365:

1. (a) False (f) Tme

(b) True (g) False (c) False (h) -e

(d) Txme (1) False

( 4 True ( J ) False

2 . (a) (-4) + 7 < ( - 2 ) x + (-51, 3 < (-a) + (-51, 8 < (4x19

and -4 < x a re all equivalent sentences.

Hence, the t r u t h s e t i s the s e t of numbers x such tha t x < -4.

( b ) 4~ + ( - 3 ) > 5 + (-21x9 4x > 8 + (-2)x,

6x > 8,

and X > 7 are a l l equ iva len t sen tences .

Hence, t h e t r u t h set Is t h e set of numbers x such t h a t

and 0 > x are a l l e q u i v a l e n t sen tences .

Hence, t h e t r u t h set l a t h e set of numbers x such t h a t x < 0.

and $ < x a r e a l l e q u i v a l e n t sen tences .

Hence, t h e t ru th set is t h e set of numbers x such t h a t

( 4 (-6)

5 o r less ( f ) t h e s e t of a l l numbers 7 (el 1-

1 7 (h) a l l numbers less than -g- (1) a l l numbers greater than 4 - '5 ( j ) a l l numbers greater than 2

*4. If the rectangle i s x Inches wide, then

it i s y l2 inches long. 12 If Ã < 6 (where x > 0 because the number of Inches in

width i s pos i t ive) , 12 < 6x,

2 < x.

Hence, the width i s grea ter than 2 inches.

*5. If the rectangle i s x inches wide, then

l2 Inches long. it i s y-

If 4 < - 12 l2 x and < 6 (x i s pos i t ive) ,

4x < 12 and 12 < 6x, x < 3 and 2 < x.

Hence, the width i s between 2 and 3 inches.

6. ~f x > o , x 2 > o . If x < 0, x2 > 0. If x # 0, x2 > 0.

If x is any non-zero number, x2 > 0. If x i s r ea l number, x2 > 0.

7. ( a ) 3a2 + (-6a2b) + 3ab

(b) + + y + (-1)) ; or, 2(sx2 + xy + (-x)) ;

or, x(4x + 2y + (-2))

( c ) 12a + 8a2 + (-4ax) (dl (x + 2) (a + (-41)

( e l x2 + x + ( - 2 )

( f ) 6 x 2 + 8 x + 2

(g) -14y2 + 32y + (-8)

(h) 2m(2x + 1 + 3a); or m(4x + 2 + 6a); or 2(2mx + m + 3am)

pages 364-365

8. (a) -1lx (b) 7a + 2b can ' t be s impl i f ied f u r t h e r .

(4 0 (d ) 2 r s t + (-6stm) can' t be s impl i f ied f u r t h e r . ( e l x + ( - 3 ~ )

9. If h i s t h e number of hours required, then 34h and 45h a r e t he d i s tances t rave led by the c a r s , g iving t h e sentence

2 The time required was 3= hours.

10. n i s the number of votes received by Charles. n + 30 Is t h e number of votes f o r Henry.

Charles received 243 vo tes . ( ~ o t e that the domain of t h e va r i ab l e f o r t h i s problem is t h e set of non- negative i n t e g e r s . )

11. a i s the number of d o l l a r s l e f t t o the son. 2a is the number of d o l l a r s lef t t o the daughter.

a + 2a + 5000 - 10,500. 3a + 5000 + (-5000) = 10,500 + (-5000)

3a - 5500 'Rie son received (1853.33.

12. (a) -3a + 2b (b) 2x + (^?a) + 7

( c ) -2a + 3 + (-2a) + (-3) ( d l 3a + (-2b)

2 (e) -2x + x + 1

*13. ( a ) Prove tha t -(a + b) = (-a) + (-b). Proof:

(-a) + (-b) + (a + b) = (-a) + a + (-b) + b commutative property of addition

= ((-a) + a) + ( ( A ) + b) associat ive property of addition

o + o addi t ion property of opposites

= 0

addi t ion property of zero This means t h a t (-a) + (-b) i s the addi t ive inverse of a + b. Therefore (-a) + (-b) I s equal t o -(a + b), s ince addi t ive inverses a r e unique.

(b) Prove: if a + c = b + c then a = b. Proof:

a + c = b + c given a + c + (-c) = b + c + (-c)

addi t ion property of equal i ty a + (c + (-"I) = b + (c + (-c))

associat ive property of addition a + O = b + O

addi t ion property of opposites a b

addi t ion property of zero

With those students f o r whom Problem 13 i s c l ea r ly too d i f f i c u l t as an independent exercise, the teacher may want t o go through the proof i n class , where students may be able t o work prof i tab ly with it as a jo in t enterprise. This procedure i s of considerable value t o the teacher a s well, f o r it gives him d i r e c t information regarding the degree of understanding and appreciation of formal proof t h a t h i s

students have a t this stage of the course.

Suggested Test I t e m s -- 1. We know tha t the sentence "4 < 7" i s t rue. What t rue sen-

tences r e s u l t when both numbers a re

( a ) increased by 5 (d) multiplied by (-5) (b) changed by adding -5 ( e ) multiplied by 0

( c ) multiplied by 5

2. Which of the following sentences a re t rue ? Which a re f a l s e ?

(a) If a + 2 = b, then b < a . (b) If a + ( -3 ) = b, then b < a. ( c ) If (a + 5) i- (-2) = b, then b < a . (d ) I f a < 4 and 4 > b , t h e n a < b . ( e ) If a + 2 < 7 and b + 2 > 7 , then a < b .

and n. In each p a r t of t h i s problem make a s 3 Given 9, 7, many statements Involving "<" about n and the given numbers a s you can, i f you know:

4. A man has three pieces of metal, each having the same volume. The sample of lead outweighs the sample of iron. The sample of gold outweighs the sample of lead. Which i s a heavier piece of metal, gold o r i ron? What property of r e a l numbers i s i l l u s - t ra ted here ?

5. Find the t r u t h s e t s of the following open sentences and draw t h e i r graphs.

6. If p, q and t a r e r e a l numbers and p < q, which of t he following sentences a r e t rue?

( a ) p + t < q + t , if t > o (b) p + t > q + t , if t < 0

( c ) p t < q t , if t > 0

(dl p t > q t , if t < 0

7. Write an open sentence f o r each of t h e following problems. Find out a l l you can about t h e numbers asked f o r i n t he quest ion.

( a ) Paul bought a Jigsaw puzzle and put it together, only t o discover t h a t t he re were 13 pieces missing. If t he l a b e l on the puzzle box sa id o v e r 350 pieces", how many pieces were i n the puzzle when Paul bought It?

(b) Tom has $12 more than B i l l . After Tom spends $3 f o r meals, t h e two boys together have l e s s than $60. How much money does B i l l have?

(c) If 13 i s added t o a number and the sum is mult ipl ied by 2, t he product I s more than 76. What is the number?

(d ) Tom works a t t h e r a t e of p d o l l a r s per day. After working 5 days he c o l l e c t s h i s pay and spends $6 of it. If he then has more than $20 l e f t , what was h i s r a t e of pay?

( e ) A farmer discovered t h a t l e s s than 70$ of a ce r t a in kind of seed grew i n t o p l an t s . If he has 245 plants , how many seeds d i d he p lan t?

8. If m i s any pos i t i ve r e a l number and n i s any negative r e a l number, which of t h e following sentences a r e t rue?

( a ) n < m (d ) n + m < 2 m

( c ) 2 n < m + n ( f ) -m < -n

Answers - t o Suggested Tes t Items --

2 . ( a ) Fa l se

(b ) T r u e

( c ) False

( d ) Fa l se ( e ) True

3. (a) No f u r t h e r s ta tement

4. Since t h e number measuring t h e weight o f i r o n Is l e s s than t h e number measuring t h e weight of l e a d , and t h e number measuring t h e weight of l e a d i s l e s s than t h e number measur- i n g t h e weight o f gold, by the t r a n s i t i v e p roper ty of o r d e r , t h e number measuring t h e weight of I r o n I s less than t h e num- b e r measuring t h e weight of go ld . Hence gold i s h e a v i e r than I r o n .

5. ( a ) The set of a l l real 4 ~ : : : : : : ! ! ! : numbers less than -5 -5 -4 -3 -2 -1 0 I 2 3 4 5

( b ) The set of a l l real . n n - - * . m - a -

numbers g r e a t e r than I I . ^ . . . . . . . ^

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 2

( c ) The s e t of a l l real . A e , . 4 n - e a ,

numbers g r e a t e r than - , . - . . . . . .

-5 -4 -3 -2 -1 0 1 2 3 4 5 -4

( d ) The set of a l l real . . , . . . a n . ,

numbers l e s s than 6 - e -5 -4 -3 -2 -i 0 ; 2 3 4 5 6

The se t of a l l real . numbers greater than 0 5 4

The set of a l l real 4 : : s i : ! ! ! ~ ! ! > numbers less than -3 -5 -4 -3 -2 -1 0 I 2 3 4 5 or greater than 3

True False

( c ) True (dl Wue

If Paul had p pieces i n h i s puzzle when he bought i t , then

P + 13 > 350. P > 337

Thus there were more than 337 pieces l e f t I n Paul's puzzle .

If B i l l had B dollars, Tom had B + 12.

B + (B + 9) < 60. 2B + 9 < 60

2B < 51 B < 25.50

Thus B i l l had lees than $25.50.

If n i s the number required,

2(n + 13) > 76. 2n + 26 > 76

2n > 50 > 25

The number l a greater than 25.

If Tom works a t the ra te of p dollars per day,

Tom's rate of pay i s more than $5.20 per day.

( e ) If the fanner p lanted p s eeds ,

The farmer p lanted more than 350 s e e d s .

8 . ( a ) True ( b ) True ( c ) True

( d ) True ( e ) F a l s e

( f ) True

Chapter 10

SUBTRACTION AND DIVISION OF REAL NUMBERS

The log ica l s t r u c t u r e of a r i thmet ic and a lgebra could be developed without even mentioning subtract ion o r d iv i s ion . How- ever, it i s convenient t o have the binary operat ions of subtrac- t i o n and divis ion, i f only f o r ease i n wr i t ing . Evidently, these operations must be defined d i r e c t l y i n terms of t he basic opera- t i ons of addit ion and mul t ip l ica t ion .

There a r e two equivalent ways of def in ing sub t rac t ion e i t h e r of which could have been used here . They a r e

(1) a - b = a + (-b)

(2 ) a - b i s t h e so lu t ion of t he open sentence i n x, a = b + x .

The wr i t e r s of t h i s book chose the f irst of these because it lends i t s e l f more r e a d i l y t o t he point of view t h a t sub t rac t ion of a number is a kind of inverse operation t o addi t ion of t h a t number, an operation which i s al ready known f o r numbers of a r l t h - metic and must be extended t o a l l r e a l numbers. Thus we have only t o i d e n t i f y subtract ion i n ar i thmet ic with a + (-b) I n order t o motivate t he d e f i n i t i o n f o r a l l r e a l numbers. This de f in i t i on a l s o bui lds on the work done previously with the addi t ive inverse, which i s important i n Its own r i g h t , and f i t s i n nicely with the p i c tu re of add i t ion and sub t rac t ion i n t he number l i n e .

There a r e a l s o two ways of def ining divis ion:

(2 ) 5. i s the so lu t ion of t he equation a = bx, b 0.

In t h i s case a l s o t he f irst method was chosen because it p a r a l l e l s the chosen de f in i t i on of sub t rac t ion and emphasizes the mul t lp l l - ca t lve inverse . It should a l s o be mentioned t h a t from these de f in i t i ons the various proper t ies of sub t rac t ion and d iv i s ion flow e a s i l y from e a r l i e r p roper t ies of add i t ion and mul t lp l lca- t i o n .

The second method of def in ing sub t rac t ion and d iv i s ion uses t lsolution of equationst1 a8 motivation. It has some advantage when t h e object ive is t o motivate extensions of t h e number system

pages 367-369: 10-1

by demanding t h a t c e r t a i n s imple equat ions always have s o l u t i o n s .

For example t h e equat ion a = b + x does not always have a s o l - u t i o n i n t h e p o s i t i v e i n t e g e r s (even i f a and b a r e p o s i t i v e i n t e g e r s ) bu t does always have a s o l u t i o n when t h e system is extended t o Include t h e nega t ive i n t e g e r s . S imi la r ly , t h e equa t ion a = bx (b # 0 ) does no t always have a s o l u t i o n i n t h e i n t e g e r s (even i f a and b a r e i n t e g e r s ) but does always have a s o l u t i o n when t h e system i s expanded t o inc lude t h e r a t i o n a l numbers. I n l a t e r courses t h e i n t r o d u c t i o n of t h e com- p l e x numbers i s motivated by t h e demand t h a t x2 + a = 0 ( i n p a r t i c u l a r x2 + 1 - 0) have a s o l u t i o n f o r every a .

The s t u d e n t i s motivated by being asked t o desc r ibe subtrac- t i o n of numbers of a r i t h m e t i c i n terms of what must be added t o t h e smal le r t o o b t a i n t h e larger. When it Is e s t a b l i s h e d t h a t we must add t h e oppos i t e of t h e smal le r , we immediately t ake t h i s a s t h e d e f i n i t i o n o f s u b t r a c t i o n f o r a l l r e a l numbers. A s i m i - - l a r motivat ion l e a d s t o t h e d e f i n i t i o n of d i v i s i o n .

Reference t o s u b t r a c t i o n and d i v i s i o n w i l l be found i n S t u d i e s - i n Mathematics, Volume 111, Sect ion 3.1.

10-1 . Meaning - of S u b t r a c t i o n .

We assume t h a t t h e s t u d e n t i s f a m i l i a r i n a r i t h m e t i c with s u b t r a c t i n g b from a by f i n d i n g how much must be added t o b t o o b t a i n a . From t h i s our bowledge of equivalent equat ions qu ick ly l e a d s t o adding t h e oppos i t e of b t o a .

For t h e s tuden t who has been s u b t r a c t i n g by t a k i n g awayf, we hope t h e I l l u s t r a t i o n of making change w i l l h e l p t h e t r ans i t101 t o an a d d i t i v e viewpoint .

Page 369. 2 0 - 9 = 10 - 15 =

(-8) - 6 =

(-10) + (-7) =

7 - ( - 4 ) =

(-5) - 2 =

5 - (-2) =

We read "5 - (-2)" as " f i v e minus the opposite of 2".

The first "-" Indica tes sub t rac t ion . The second ' I - " means "the opposite of1 ' . (of course I n t h i s case t he second could a l s o be read n e g a t i v e 2 . If a va r i ab l e were involved, however, the l1 - I f would have t o be read " the opposite1' . )

We s h a l l soon want our s tudents t o be able t o look a t a - b

and think of it as a - sum, t he sum of a and ( -b ) . This i s Jus- t i f l e d by our d e f i n i t i o n of sub t rac t ion .

You have, no doubt, noticed t h a t we a r e not using t h e word ' s i g n f o r the symbol 't-" o r 'I+". We f ind t h a t we do not r e a l l y need the word, and s ince i t s misuse i n the pas t has caused considerable l ack of understanding ( i n such th ings a s " g e t t i n g the absolute value of a number by tak ing o f f i t s s ign") we pre- f e r not t o use t he word "sign" i n any of our exposi t ion.

A r e l a t ed point t h a t w e should mention i s t h a t we do not wri te +5 f o r the number f i v e . The p o s i t i v e numbers a r e t h e numbers of a r i thmet ic . We there fore do not need a new symbol f o r them. Thus we wr i t e 5, not +5, and t h e symbol "+I i s used only t o ind ica te addi t ion .

Answers t o Oral Exercises 10-1; page 371: -- - 1. ( a ) 5 + ( -4 ) (1) (8 + (-12)) + (-2)

(b) 11 + (-12) ( ; ) I 2 + (-(a + (-12))) ( c ) -4 + (-8) (k) + (d ) -11 + 5 (1) 8k + l l k ( e ) 24 + 8 (m) 6x + (-2x)

( f ) 4a + ( - 3 4 (n) 0 + 3 m (g) - 2 x + 2 (0 ) 6 2 + ( -9 2)

(h) 737 + 2~

Answers - t o Problem Se t 10-1; pages 371-373: -- 1. ( a ) - ( a + 7 ) = ( - a ) + ( - 7 )

- a - 7

2 ( c ) -x - x - 2

page 372: 10-1

(b) 132 + 18 = 150 (g) 7m + (-m) + (-12)

= 6m + (-12) (c) -12 + 24 = 12 (h) - 4 ~ + (-a) + b

= -6x + b

(d ) -Tb + (-12b) = -19b (1) $x + (- $x) = $&x

(e ) -3x + 4x = x ( J ) 7.b + (-12) + 3.5m = lo .% + (-12)

4 . (a) x + (-5) = -4 x + (-5) + 5 = -4 + 5

(1 1

( f ) a l l numbers less than 9

pages 372-373: 10-1

(g) a l l negative numbers

5. (a) 1 5 + 8 (b) -25 + 4

(4 -9 + (-6) (d ) 22 + 30 ( e ) -12 + 17 ( f ) 8 + 5

(43) 5 + (-10) (h) 7 + 8 This deserves

emphasis s ince i t r e l a t e s d i r e c t l y t o t h e d e f i n i t i o n of sub t rac t ion .

6 . I f the b u l l e t takes t seconds t o reach t h e t a r g e t , then the sound takes 2 - t seconds t o r e tu rn . Since the d i s - tances a r e equal, t he open sentence Is

sec . The time i t took the b u l l e t t o reach t h e t a r g e t w a s 7 1 The dis tance then Is - x 3300 o r 1650 f e e t .

7. If r i s the number of gal lons of regula r gas, then 500 - r Is the number of gal lons of e t h y l . The value of t he regula r Is (30r) cen ts . The value of t h e e thy l i s 35(500 - r) cen t s . The value of the mixture i s 32(500) cen t s . Our sentence then i s

The value of t he regula r plus t h e value of t he e thy l i s the value of t he mixture.

10-1 and 10-2

The number of gal lons of regula r was 300. There were 200 ga l lons of e t h y l .

8. If t is t h e number of hours walked by the second man, then t + 1 is the number of hours walked by the f i r s t . The distance t rave led by t h e first is 2 ( t + l ) , b y t he second is

3 t Since the dis tances walked were the same, our sentence is:

The second man w i l l have walked 2 hours when he catches up

t o t he first.

10-2. Proper t ies - of Subtract ion.

The t i t l e of t h i s s ec t ion might seem t o be a misnomer, because we f i n d t h a t sub t rac t ion does not have many of the pro- p e r t i e s enjoyed by addi t ion, such as t h e assoc ia t ive and commu- t a t i v e p rope r t i e s . The point Is t h a t we always change from in- d ica ted sub t rac t ion t o addi t ion and then apply the known proper- t ies of add i t i on . Thus, mul t ip l ica t ion appears t o be d i s t r ibu- t i v e over subtract ion:

a ( b - c ) = ab - ac

only because mu l t i p l i ca t ion i s d i s t r i b u t i v e over addi t ion:

a(b + (-c)) = ab + (-ac)

I n t h i s sense, sub t rac t ion can be thought of as having the p rope r t i e s of addi t ion, but only because sub t rac t ion i s defined as addi t ion of t h e opposi te .

pages 376-381: 10-2

Answers -- to Oral Exercises 10-2a; page 376:

1. (a) True (d ) False (b) True ( e ) True (c) True ( f ) False

Answers 3 Problem Set 10-2a; page 377:

1. (a) -58 ( g ) -6x2 - gx - 6xy

(b) -9a 2 (h) - 6 ~ - 9x - 6xy

(1) 7 m - 2 x + 4

(j) 7 r n - 2 x + 4

2. The sentences i n (a ) , (c), ( e ) , and (I) are true for a l l values of the variables.

Answers t o Oral Exercises 10-2b; page 381: -- -b [also ( - l )b] c

3c -1lx -3x - 2

-2x + 5 -a - 2b + c

- 2 Y x u

pages 381-382: 10-2

2. ( a ) 4 (b) X - y - z

(c) a - x - 7

(dl 3x (e) -2x + 4 (f) - a + 2

(g) -3Y + 5

Answers - to Problem Set 10-2b; pages 381-384: -- 1. (a) -7 + 2x (e) -1 + .Olx

(b) -a - b 5 1 (f) - $x + 2y + ~x - 2y or ~x

(c) 4 - 2 c (g) -3x(2x + 5 ) (d) (h) -7m(3m - 2)

2. (a) 3 - x - 2

(c) 5a - 10 (d) -5m + n (distributive property)

( e ) - ( 5 m - n) = -5m + (-(-no (opposite of a sum) = -5m + n (opposite of the opposite)

(f) 7x + 3y - 4

(g) -6x + 4b (h) -10- + x

(j) lit + 1 2

(k) -X(X - 7) = (-X)X + (-X) (-y)

= -(x)(x) + xy = -X2 + xy

pages 382-386: 10-2

(1) (9a + 2b - 7) -(3a - 7b + 5 ) =

9a + 2b - 7 + ( - 3 4 + ( - ( - 4 + (-5)

= 6a + 9b - 1 2

(m) 3x - x2 - x ( l - x) - 3x - x2 - x + x 2

= 2x 2

(n) (x - l ) ( x + 1 ) - (x2 - 1 ) - x 2 - 1 - (x - 1 ) = 0

(0) 2m2 - 6m(m - 1 ) - m = 2m2 - 6m2 + 6m - rn

= -4m2 + 5m

(p) (x + 2)(x + 1 ) - (x + 2)x - x2 + 3x + 2 - x2 - 2x

X - t - 2

Another way:

(x + 2 ) ( x + 1) - (x + 2)x - (x + 2)(x + 1 + (-x)) ( d i s t r i b u t i v e property)

3. The sentences i n ( a ) , ( b ) , ( d ) , and ( f ) a r e true f o r a l l values of t he va r i ab l e s .

(b) (-51 (g) (51 fc ) set of real numbers . ,

equal t o o r g rea t e r than 4 (h) (2)

(d) s e t of real numbers l e s s than 2 (1) ft

( e ) s e t of real numbers equal t o o r g r e a t e r than -3

2 5. (a ) -2a + 2b 2 (d) k - 9k2 - 29

(b) -3X + 57 ( e ) n 2 + 2 3 n - 3

( c ) ga + + - 11

pages 383-384: 10-2

6 . (a) n - 8 If John I s now n years old

(b ) m = 6b i f t h e boy Is b years old and the man m years o ld

( c ) 5d = 36 i f d i s the d i s tance i n miles

(d ) j! = 2w + 2 i f w is the number of f e e t i n t he width and 1 Is the number of f e e t I n t h e length

( f ) ( 1 . l ) x i f x i s the number of pounds of candy

(g) 30x + 35(x + 40) i f x is the number of gallons of 30$ gasol ine

(h ) lOO(2d)

(1 ) 15 + 2k i f k i s t h e number of d o l l a r s I have

7 n + 1 2 - 4 + n 7. (a('-) - 96 Is the form of t he exercise .

Simplifying,

Thus n - 1 i s t h e simplest general form. S t a r t i n g with 2, 2 - 1 = 1 Is t h e f i n a l number. S t a r t i n g with 11, 11 - 1 = 10 i s the f i n a l number. S t a r t i n g with -3, -3 - 1 = -4 i s t h e f i n a l number.

8. d i s the number of dimes d + 1 i s t h e number of qua r t e r s 2d + 1 is t h e number of n i cke l s

pages 384-387: 10-2 and 10-3

The number of qua r t e r s is 4.

9 . Let n be t he number of ha l f -p in t b o t t l e s . Then 6n is the number of p i n t b o t t l e s . 39 quar t s is the same a s 2(39) p i n t s .

There a r e 12 ha l f -p in t b o t t l e s .

10. l l a + 13b - 7c - (8a - 5b - 4c) = l l a + 13b - 7c - 8a + 5b + 4c = 3a + 18b - 3c

12. (1) If a = b + c then a + (-b) = b + c + (-b) and a - b = c

(11) If a - b = c then a + (-b) = c and a + (-b) + b = b + c and a = b + c

10-3. Subtraction i n Terms of Distance. --- The r e l a t i o n between t h e d i f fe rence of two numbers and the

dis tance between t h e i r po in t s on the number l i n e i s introduced here t o make good use again of the number l i n e t o he lp i l l u s t r a t e our ideas .

You a r e no doubt aware, however, of t he f a c t that (a - b )

a s a d i rec ted dis tance and la - b 1 as a d is tance a r e very help- f u l concepts I n dea l ing with slope and d i s tance i n ana ly t i c geometry.

pages 388-389: 10-3

4 . (a) The information given can be translated:

x - 51 < 4 and x > 5 .

Since x > 5, lx - 51 = x - 5. x must be such that x - 5 < 4 and x > 5.

Hence x must be greater than 5 but l e s s than 9.

x - 5 < 4 t e l l s u s that x is between 1 and 9. But x sha l l be less than 5.

pages 389-390: 10-3

Hence x Is between 1 and 5.

( c ) x 1s between 1 and 9 .

( d ) The set of a l l numbers which are g r e a t e r than 1 and less than 9

The sentence lx - 4 1 = 1 te l ls us t h a t t h e d i s t a n c e between

x and 4 on t h e number l i n e i s 1. The sentence i s t r u e x Is 3, a l s o when x Is 5. s e t : ( 3 , 55

a l l real numbers which a r e g r e a t e r o , n a

9 than -3 and less -3 0 1 3 than 3 t h e set of a l l real numbers

pages 390-391: 10-3 and 10-4

7. Truth s e t : the Se t of a l l numbers which are g rea t e r than 3 and less than 5

The graph c o n s i s t s of a l l po in t s whose d i s tance from 4 is l e s s than 1.

8. The set of numbers which are e i t h e r g r e a t e r than 5 o r less than 3.

The graph of lx - 4 I > 1 would cons i s t of a l l po in t s whose d i s tance f r o m 4 i s g r e a t e r than 1.

9. The graph is t h e same as i n problem 7. The t r u t h s e t s a r e t h e same.

Division.

I n a manner analogous t o t h e a e f i n l t i o n of subtract ion I n terms of addi t ion , w e def ine d iv i s ion by a non-zero number i n terms of mul t ip l ica t ion by I t s mul t ip l i ca t ive inverse . The word r e c i p r o c a l " i s Introduced t o mean t h e same th ing as "mult lpl i-

1 c a t l v e inverse" . The symbol I s Introduced t o represent the rec iproca l of b, where b i s a non-zero number.

A t t h i s point It m i g h t he lp t o draw on t h e analogy between t h e rec iproca l (o r mu l t i p l i ca t ive inverse) and the opposite (o r add i t i ve i nve r se ) .

Corresponding t o each real number x there is a unique number y such t h a t x + y = O .

Corresponding t o each non- zero number x there i s a unique number y such tha t xy = 1.

page 391: 10-4

Th i s unique number y is This unique number y i s ca l l ed the opposite of c a l l e d t h e rec iproca l of the number x and i s t h e number x and is

-8

denoted by -x . J. denoted by .

The opposite of the opposite of x is x:

The rec iproca l of t h e rec iproca l of x i s x: -

For r e a l numbers a, b, c , For r e a l numbers a, b, c , a - b = c i f and only i f with b 0, a = c i f and a = b + c . only I f a = be.

The sum of t h e opposites is the opposite of t he sum :

The product of t he recipro- c a l s is the rec iproca l of t he product: 1 1 If x # O y '?I = -

xy ' and y # 0.

Again, l i k e sub t rac t ion , the operation of d iv i s ion has no proper t ies i n i ts own r i g h t , but when wr i t t en i n terms of mult i- p l i ca t ion of t he rec iproca l i t can be thought of as having a l l

the proper t ies of mul t ip l ica t ion . Thus

can be thought of as a statement that d iv i s ion is d i s t r i b u t i v e over addit ion, whereas i n reality it is a statement t h a t mult l- p l i ca t ion i s d i s t r i b u t i v e over addit ion:

-= 1 a + b c ( a + b ) - by d e f i n i t i o n of d iv i s ion 1 1 a.- + b.-

c d i s t r i b u t i v e proper ty c

a b + - c c by d e f i n i t i o n of d iv i s ion

page 391-395: 10-4

Answers t o Oral Exercises 10-4a; pages 591-392: -- 1. (a ) What number do we mult ip ly by 4 t o ge t 12?

(b) What number do we mult ip ly by 4 t o ge t -127 ( c ) What number do w e mul t ip ly by -4 t o get 12?

(d ) What number do we mult ip ly by -4 t o ge t -12? ( e ) What number do we mult ip ly by 4 t o ge t 4a? ( f ) What number do we mult ip ly by 3m t o ge t 12m?

2 (g) What number do we mult ip ly by 9 t o ge t 27x ?

(h ) What number do we mult ip ly by -13 t o ge t 26a? (1) What number do we mult ip ly by t o ge t 4?

(j) What number do w e mul t ip ly by -2 t o ge t -6a?

For ( a ) and ( c ) : "What number mul t ip l ied by - a gives us t h e product b?"

For t h e others : "What number mul t ip l ied by - b gives us t h e product a?"

Answers - t o Problem Set 10-â; pages 392-393: - 1. -4 11. 1

2 . 2a 12. -3a 3 - -5 13. x 4. -7 14. -4a 5 - -5m 15. 12 6. -21 16. -1

7. -7a 17. 3 8. x 18. -2a 9. 5ax 19. 0

10. a + b 20. -16

Answers t o Oral Exercises 10-4b; page 595: -- 1 1

1. (a) 5 (b) - 7 (4 2

pages 395-400: 10-4

2 . ( a ) True (b ) False ( c ) False

( j ) d , d i s a s s u m e d t o b e d i f f e r e n t from zero

(d ) True ( e ) True ( f ) False

mul t ip l ied by 3 * 13 -2- y ie lds 1 as product . 3

4 . I f n i s a rec iproca l of 0, then 0.n = 1, because the product of a number and Its rec iproca l s h a l l be 1. The sentence Own - 1 has the empty s e t , 0 , as Its t r u t h s e t . There is no number which when mult ip l ied by 0 y i e l d s 1.

Answers -- t o Oral Exercises 10-4c; pages 399-400:

1. ( a ) True f o r a l l values of b except zero; $ - 7 - 1

(b) Not t r u e f o r any value of a ( c ) True f o r a l l values of b and c and a l l values of a

except zero (d ) True f o r a l l values of x and a

a When we say "1f - = c " we imply t h a t b 0. b

( f ) True f o r a l l values of x except -3 (g) True f o r a l l values of x

pages 400-401: 10-4

10 w h i c h i s - - (f) -*g , 9

(g) no multiplication required

Answers - to Problem Set 10-4c; Pages 401-402: --

be 2bc (also Ã‘Ã

pages 401-402: 10-4

16 T r u t h set:

There I s no so lu t ion i n i n t e g e r s .

7. L e t x be the width i n inches . Then 7x i s the length i n inches .

8. Let x be Dickls age i n years . Then 3x I s John's age i n years.

page 402: 10-4

1 1 10. 5 x a ^ . x + 3

11. L e t t he speed of t h e wind be x miles pe r hour. The speed of the plane i s 200 - x miles per hour.

a 12. (1 ) Prove: If a - be and b 0, then g = c .

1 Proof: If b + 0 then 5 i s a real number.

Then If a = be,

1 1 a - (be) -^ mul t ip l ica t ive property of equa l i t y

1 = (b r ) ~ assoc ia t ive and commu-

t a t i v e proper t ies of mul t ip l ica t ion , and d e f i n i t i o n of d iv i s ion

a (11) Prove: If c- - c and b / 0, then a = be. a Proof: If 5 = c , then

d e f i n i t i o n of d iv i s ion

mul t ip l ica t ive property of equa l i t y

assoc ia t ive and commu- t a t i v e proper t ies of mu l t l p l i ca t ion

pages 403-405: 10-4

Answers -- to Oral Exercises 10-4d; pages 404-405:

6 - - ( e l g - 7a

Answers - to Problem Set 10-4d; pages 405-406: --

pages 405-406: 10-4

1 (a) 5 (d) 0

2 (b) 5 8 (4 - 5

10 (4 - TJ ( f ) 1

5 (-1) 5 (-2) 2 2 - a (a) g+-, - f f + (dl F + ($1 --

= i3 2a (4 - 1 + (3) -7 -2a -5 - 2a (b) 12 1 2 (f) = S - + - = 3m 3 m 3m

7

pages 407-409: 10-5

10-5. Common Names.

In t h i s and the following sec t ion we a r e i n t e r e s t e d i n t h ree commonly accepted conventions about the s imples t numeral f o r a number.

(1) There should be no lnd ica ted operat ions remaining which can be performed.

(2 ) I f t he re i s an Indicated d iv i s ion , t h e numbers whose d lv i s ion i s indicated should have no common f a c t o r .

We p re fe r

Thus, t o I l l u s t r a t e t he f irst convention we would say t h a t

"20" is not a s simple as "5"; " vy i s not a s simple as T "5"- "m" i s not as simple as 5' 7 "7" ; but 'tM" cannot be

Y "1 4l' s impl i f ied. Similar ly , f o r t h e second convention i s not as

simple a8 'Ig' and ' I 2 t 1 . sim- 3 ' l 2 ~ ; + 4" i s not a8 simple a s .

ax + 2a p l i f i c a t l o n s of t h i s kind depend on the theorem which s t a t e s t h a t a c ac a 3 = , t he f a c t t h a t = 1 f o r a # 0, and the proper ty

of 1.

The student has f o r years been "multiplying f rac t ions" according t o t he theorem proved here . It i s not a new r e s u l t t o him, but i t is now a consequence of and i s d i r e c t l y t i e d t o our definition of d iv i s ion and t h e proper t ies of multiplication. I n the pas t he knew - how t o dlvide; now he l e a r n s he divides i n t h i s manner.

I n the process of proving the theorem, It must be es tab l i shed 1 1 t h a t (-)(-) = - b d that is , t h a t t h e product of t h e r ec ip roca l s bd '

of two non-zero numbers Is the rec iproca l of t h e product of t he numbers. To do t h i s , use t he commutative and a s soc i a t i ve proper- t i e s of mul t lp l ica t lon :

= ( l ) ( l ) = 1.

1 1 Hence, (5) (z) Is the rec iproca l of bd;, 1 .e . ,

pages 410-411: 10-5

Notice t h a t we have become relaxed i n our use of t he words "numerator" and 'tdenominator". Although these words r e f e r t o numepals, we s h a l l begln t o use them interchangeably f o r numerals and numbers, whenever the context 1s c l e a r .

Answers t o Oral Exercises 10-5; page 410: --

x 2 ( c ) -( 1 = y ? Y # O Y 3 (h ) !Phis i s i n simplest form.

-1 x 1 ( d ) -(-) = - - x x x , x # o ( i ) This i s i n simplest form.

Answers - t o Problem Se t 10-5; pages 411-412: --

(xm # 0 says t h e same (k) 2 - a, a # 0 th ing as x # 0 and

# 0 ) (0 x + 1, Y # 0

3 (m) i s i n simplest form? y #

( f ) i s i n simplest form, (0 ) -l? b - a # O

page 411: 10-5

Z ( 3 x ) = 6 3x

(Zero I s excluded from the domain)

The value 1 i s excluded from the domain s ince the l e f t s i d e of the 8entence Is meaningless f o r x = 1.

x - 1 If X + l , x = l .

?x(x - 1 ) = 3 Thus the sentence - becomes 3x = 3, which i s equivalent t o x = 1.

The t r u t h ae t Is empty s lnce 1 Is not i n t he domain.

Excluding the value -1 from the domain of x, t h e given sentence 18 equivalent t o

whlch i s equivalent t o x = 1.

The t r u t h s e t is (11.

( 0 ) Exclude t h e value -1 from the domain of x.

Exclude the value 1 from the domain of x.

This means t h a t t h e t r u t h s e t cons i s t s of a l l real numbers except 1.

Fxclude 0 and 7 from t h e domain of x.

1 1 Truth set [7 , - -1. 1 and -1 are the only numbers 3

whlch are t h e i r own reciprocals .

pages 411-412: 10-5 and 10-6

* k ) 7 x - 2 + 3 x w 4 = - - - 2 No numbers need be excluded

from the domain of x .

3. Let t be t h e required number of years . After t years Brownts s a l a r y w i l l be 3600 + 3OOt. After t years Jones1 s a l a r y w i l l be 4500 + 200t.

3600 + 30ot = 4500 + 200t

(91 After 9 years t h e i r s a l a r i e s w i l l be t he same.

4. Let x be B i l l ' s age i n years now. Then 2x i s Bobts age i n years now.

B i l l i s 8 years o ld . B O ~ i s 16 years o ld .

10-6. m a c t i o n s . The main point of t h i s sec t ion i s t o develop s k i l l i n s i m -

p l i f y i n g an expression t o one i n which there i s a t most one lndl- cated d iv i s ion . !Ibis e s s e n t i a l l y means t h a t we are learning t o mult lply,dlvide, and add f r a c t i o n s .

Page - 412. We a r e again re lax ing our r i g o r I n t he use of words. We s h a l l allow ourselves t o use " f rac t ion" f o r e i t h e r the symbol o r t h e number, even though c o r r e c t l y speaking it means the symbol. Thus I n t h e preceding paragraph a prec i se statement would have said " t o mul t ip ly9 divide , and add numbers which a r e represented by f r a c t i o n s ."

Now t h a t we have begun t o r e l a x our precis ion of language, we s h a l l he rea f t e r , without f u r t h e r comment? f e e l f r e e t o use convenience of language even when I t v i o l a t e s p rec i s ion of language about numbers and numerals, a s long as we a r e sure t h a t the p rec i se meaning w i l l be understood.

296

pages 412-415: 10-6

We define the word " ra t io t ' (problem Set 10-6b) a s par t of

the language i n cer ta in appl icat ions. We a l so c a l l an equation such as , which equates two r a t i o s , a "p~opor t ion" . It &=n seems undesirable a t present t o digress i n t o a lengthy treatment of r a t i o and propertion since it i s j u s t a matter of special names f o r famil iar concepts.

Answers t o Oral Ekercises 10-6a; pages 414-415: -- 1. - + 8 . x - 3 , x # l

7. X - l , X # O 1 4 . is i n simplest form, y # -1

Answers - t o Problem Set 10-6a; pages 415-417; -- 3 7 3 - 7 - 2 1 1. ( a ) gwF = n-i.5

(1) ;, x # o

already i n simplest form

pages 415-417: 10-6

and c # 0

1 (4 x > - Truth s e t : a l l r e a l num-

1 bers greater than - 2

If t he f r e i g h t t r a i n averages p miles per hour, then t h e passenger t r a i n averages p + 20 miles per hour.

The d i s tance t rave led by the passenger t r a i n Is 5(p + 20) miles; by the f r e i g h t t r a i n i s 5p miles .

The sentence i s t r u e f o r every value of p . The f r e i g h t t r a i n could have t rave led a t any speed whatsoever.

If $c w a s t h e p r i c e of t h e c h a i r before t he s a l e , then the discount was ( .2)c. The s a l e p r i c e was t h e o r i g i n a l p r i ce l e s s the discount, g iving t h e sentence

C - .2c = 30

= 37-50 The o r i g i n a l cos t of the c h a i r was $37.50.

pages 417-420: 10-6

a number n i s the same as &I + 3 . 7 - 2

The number i s 9 .

Page - 418. The reasons f o r t h e s t e p s a r e :

Def in i t ion of s u b t r a c t i o n -a - Â ¥ g

Mul t ip l i ca t ion p roper ty of one a 1 , i f a # o a Theorem on m u l t i p l i c a t i o n of f r a c t i o n s

a Def in i t ion of d i v i s i o n : 5 = a 1 F

D i s t r i b u t i v e p roper ty -a Def in i t ion of d i v i s i o n , and 5 = - z.

The number of s t e p s needed t o do such a s i m p l i f i c a t i o n w i l l vary

from student t o s t u d e n t . After he understands reasons f o r every s t e p , he w i l l soon be a b l e t o w r i t e

Answers t o Oral Exerc ises 10-6b; pages 420-421: --

pages 420-422: 10-6

2. ( a ) x

Answers - t o Problem Set 10-6b; --

2 . ( a ) 1

(b) ,-

2 3 . ( a ) 1 5 ( - + 3 ) = W T )

3 x + 45 = 10

35 (-

(b) @I

pages 421-424:

( e ) a l l numbers greater than 1

pages 422-423: 10-6

4. Let s be the number of 4j-d stamps Mary bought. If she was charged the cor rec t amount, then s must be a non- negative i n t ege r and

15( .03) + ,049 = 1.80

I f t he re i s a non-negative i n t ege r s f o r which t h e above sentence Is t r u e , then each of t he following sentences a r e t r u e f o r t h i s value of s .

3 Since 33v i s not an in t ege r , she was charged the wrong amount.

5. He had 12 pennies, 16 dimes, 22 n i cke l s . He has $2.82.

6. John has $25.

7. If one number i s y, t he o ther number i s 240 - y .

5y = 720 - 3~ Y = 90

The numbers are 90 and 150.

It should be pointed out t h a t another open sentence f o r t h i s 3 problem Is 240 - y = yy . Here you g e t y = 150 and

240 - y = 90. Again you ge t 90 and 150 as t h e two numbers .

pages '!23-424: 10-6

The numerator was increased by 5 .

10. If h i s f a the r i s x years old, Joe i s years old. ^

The fa the r ' s age i s 36; the son's age is 12 .

11. Let x be the smaller of the two numbers. Then the larger number i s 7 - x .

The numbers: 2, 5

The sum of the reciprocals: 1 1 7 ?+?=IS 1 1 - 3 The difference of t h e reciprocals: - s" -

1 2 . ( a ) If there were g g i r l s , there were (2600 - g) boys.

and

Hence, there were 1200 g i r l s i n the school.

page 242: 10-6

.800 o r 40 rad ios were defec t ive . (b ) 55 800 - 40 o r 760 rad ios were not de fec t ive ,

40 i s the r a t i o of defec t ive t o non-defective m=ig rad ios .

The alert s tudent w i l l no t ice t h a t t he number 800 i s unnecessary information. If we suppose t h a t there were r rad ios i n t h e shipment, then

1 mr radios were defec t ive , and

l9 rad ios were not defec t ive . ^

Therefore the required r a t i o is 19 ( c ) Le t f be t he number of f a c u l t y members.

Hence, t he re a r e 126 f a c u l t y members.

5x (d) Since = 3*$ - 5.1 9 (when x / 0) , 5x and = 9

9x a r e i n

i f 5x i s

r a t i o

Then

the

the

and

r a t i o of 5 t o 9 . More p rec i se ly , '

first of t he numbers t h a t a r e i n t h e

y I s t h e o the r number, then

and 9 = Y.

page 424: 10-6

Hence, t h e numbers can be represen ted by 5x and gx, x # 0.

I f x = 7, the numbers a r e 35 and 63. I f x = 100, the numbers a r e 500 and 900.

a c ( d b d = (?)bd M u l t i p l i c a t i o n proper ty of e q u a l i t y

1 (a * k ) b d = ( c .z)bd D e f i n i t i o n of d i v i s i o n

1 1 ( a d ) ( b 9,-) = ( b c ) ( d Associa t ive and commutative

p r o p e r t i e s ( a d ) -1 = (be ) -1 D e f i n i t i o n of m u l t i p l i c a t i v e

inverse ad = be M u l t i p l i c a t i o n proper ty

of one

( f 1 If ad = bc and b # 0 and d # 0 , then

1 1 M u l t i p l i c a t i o n proper ty of e q u a l i t y

1 Associa t ive and commutative p r o p e r t i e s of m u l t i p l i c a t i o n

D e f i n i t i o n of m u l t i p l i c a t i v e i n v e r s e

M u l t l p l i c a t i o n proper ty of one

D e f i n i t i o n of d i v i s i o n

pages 425-428: 10-6

Answers -- to Oral Exercises 10-6c; page 427:

also 4- 1. (a) Z J

d - c bd also - (b' J d - c

t^

; also + (0 g

7b - 28b also -2a

(f) m a - 7b -2a

m 6 (h) 7 ; also

% x

15x ; also - Ez (i) ; also 7rT

9 x 2

Answers - to Problem -- Set 10-6c; pages 427-431:

pages 428-429: 10-6

(b) q=+-) 2 x +

pages 429-430: 10-6

(g ) the s e t of a l l real numbers

( 1 ) a l l r e a l numbers except - 4

8 . If the success ive p o s i t i v e in tegers are n, n + 1, and n + 2, then the sentence i s

The integers are 359, 360, and 361.

9. L e t the success ive p o s i t i v e Integers be n and n + 1 . Then

n + (n + 1 ) < 25 2n + 1 < 25

2n < 24 n < 12

The numbers could be any o f the p a i r s (11, 12 ) , (10, 11 ) ,

. . ., (1 , 2 ) .

pages 430-431: 10-6

10. If t h e consecut ive even i n t e g e r s a r e n and n + 2, then

The numbers a r e 22 and 24.

11. I f t h e whole number and Its successor a r e n and n + 1,

then

The numbers a r e 22 and 23.

1 2 . If t h e two consecut ive odd numbers are n and n + 2, then

But n must be an i n t e g e r , s o t h e r e are no consecutive odd numbers whose sum i s 75.

13. If t h e f i r s t number is n , then t h e second i s 5n and

The two numbers a r e 3 and a.

1 4 . If t l a t h e number of hours each t r a i n t r a v e l e d , then 40t i s t h e number of miles t r a v e l e d by t h e t r a i n going south, and 60t i s t h e number of miles t r a v e l e d by t h e t r a i n going north

pages 431-433

The time required was 1 hour and 15 minutes.

Answers - t o Review Problem - Se t ; pages 432-447:

It Is not an t ic ipa ted t h a t a l l of t he exerc i ses i n t he following review l is t w i l l be used by any one teacher . Many teachers may choose t o use some of them a s supplementary o r a s "extra c r e d i t " exerc i ses a t t he time the top ic is s tudied e a r l i e r In the course.

I n some cases it may be des i r ab l e t o use por t ions of t he l i s t as a review l i s t because the c l a s s i s completing P a r t 2.

I n a few ins tances t he completion of Chapter 10 may conclude t h e year 's course. On the o the r hand, a teacher who plans t o use Part 3 may not f e e l t h a t review is necessary at t h i s point and may omit the e n t i r e l i s t .

In no instance i s it recommended t h a t an assignment include more than 3 o r 4 d i f f i c u l t verbal problems.

(m) 100

( e ) -1, 0 h he expression i s not a number i f a = -1.)

(f) has a rec iproca l f o r every r e a l number a

( g ) has a rec iproca l f o r every r e a l number a

(h) -1

pages 433-434

(1) -3, 3

1 *3. ((a - 3 ) ( a + l ) ) ( & ) = ( a -3)(;Ã‘7

mul t ip l ica t ion property of equa l i t y

a s soc i a t i ve and commutative proper t ies of mul t ip l icat ion d e f i n i t i o n of reciprocal

a + 1 = 1 mul t ip l ica t ion property of 1

If a = 3 , then 3 + 1 = 1, and t h i s i s f a l s e .

We should - not expect t he sentence a + 1 = 1 t o have t h e same t r u t h s e t as the o r ig ina l sentence s ince our m u l t i p l i e r '

i s not a number when a = 3, and we used the mult lpl ica- a - 3 t i o n proper ty of equa l i t y i n t h e very f irst s t e p . In manipulating a lgebra ic expressions, as i n t h i s example, we have t o be constant- l y on guard t h a t we do not become s o engrossed i n "pushing symbolst t h a t t h a t

only

we fo rge t our a lgebra ic s t r u c t u r e . So long a s we remember here i s supposed t o represent a number, we a r e s a fe

a - 3 i n g a lgebra ic p rope r t i e s . When we view a s a symbol a and apply our a lgebraic p roper t ies , any r e s u l t s we ge t can

be only symbolic; t o be i n t e rp re t ed as r e s u l t s about numbers, we have t o check t o see t h a t we were a c t u a l l y using (symbolic) numbers a t each s t e p along the way.

4 . ( a ) - x 2 + 1 5 x - 1 4 ( f ) -2b 2 + 2ab

(b) a - 25 ( g ) 3a + 18b - 3c

( c ) 8a2 - 5a + 10 (h ) 3x2 - 8x + 1 9

(d ) lOn + 13p - 13a (1 ) -12s + 4 t - 10u

5. Yes, i n a l l cases

pages 454-435 1 1 6. (a ) T < T i s t r u e .

-1

1 1 (b) 7 < T; i s t r u e .

1 1 ( c ) - < - i s f a l s e .

Yes, it Is t r u e . Let a = 5, b = 2 . ( b < a) Then

Yes, it i s t r u e . An example:

1 1 If a i s pos i t i ve and b is negative, then - > 5 , f o r a the reciprocal of a pos i t i ve number i s a pos i t i ve number and the reciprocal of a negative number i s a negative number.

I f b < a, then a - b i s pos i t i ve . The proof of t h i s

follows . If b < a, then

b + (-b) < a + (-b) add i t ion proper ty of order

b + (-b) < a - b d e f i n i t i o n of sub t rac t ion

0 < a 0 b addi t ion proper ty of opposites

311

pages 435-436

If a i s t o t h e r i g h t of b on t h e number l i n e , then t h e d i f f e r e n c e a - b I s p o s i t i v e .

11. If ( a - b) i s a p o s i t i v e number, then a > b .

If ( a - b ) i s a negat ive number, then a < b .

If ( a - b) i s ze ro then a = b .

12 . I f a , b , and c a r e r e a l numbers, and b < a , then b - c < a - c . The proof of t h i s fol lows:

b + ( -c ) < a + ( - c ) a d d i t i o n p roper ty of o rde r b - c < a - c d e f i n i t i o n of s u b t r a c t i o n

1 4 . 4 - 15 = -11

The r e s u l t i n g temperature i s 11' below ze ro .

15. (-50) - 30 = -80 The new p o s i t i o n i s 80 feet below t h e s u r f a c e .

1 6 . If t h e number i s n, then

Hence, t h e number i s 9 .

pages 436-437

17. A t 11 o'clock P.M., t = - 1

A t 2 o 'clock A . M . , t = 2 .

The in t e rva l i s 3 hours.

A t 6 o'clock A.M., t = 6. A t 4 o'clock A.M., t he next day, t = 28.

The In t e rva l i s 22 hours.

18. Let the dis tance i n miles t o the e a s t of the 0 mark

correspond t o pos i t i ve numbers:

John 1 s Rudy' s pos i t ion on pos i t i on on the number the number

l i n e l i n e

1 The Distance d i f fe rence be tween

them I n miles

3 0 - (-36)l = 66 66

If a i s l a r g e r than 1, 0 < b < 1.

If 0 < a < 1 , then 1 < b . If a = 1 , then b = 1 .

If a = -1, then b = -1.

If a < -1, then - 1 < b < 0.

If -1 < a < 0, then b < -1.

If a > 0, then b > O .

I f a < 0 , then b < 0 .

Zero has no mul t i p l i ca t ive inverse . If b i s the rec iproca l of a, then a i s the reciproc a1 of b .

pages 437-438

0

No n = O p can be any number including 0.

E i the r p = 0 o r q = 0, o r both a r e zero. q must be 0 . x - 5 must be zero s ince 7 i s not zero. The given sentence i s equivalent t o

( 9 X 17 X 3)y = 0

If y > 0, then the product ( 9 x 17 x 3)y > 0. If y < 0, then the product ( 9 x 17 x 3 ) y < 0.

Therefore t he only t r u t h number of the given sentence 1s 0.

(1) x - 8 i s zero when x = 8. It follows that 8 i s a t r u t h number of t h e sentence (x - 8 ) ( x - 3 ) = 0.

(8 - 8 ) ( 8 - 3 ) = O

x - 3 i s zero when x = 3 . 3 i s a t r u t h number of (x - 8 ) ( x - 3 ) = 0.

The t r u t h s e t of ( x - 8 ) ( x - 3 ) = 0 i s (8, 3 ) .

21. ( a ) If x = 20, x - 20 i s zero, and hence

(x - 20) (x - 100) is zero.

If x = 100, x - 100 i s zero, and hence,

(x - 20) (x - 100) i s zero.

The t r u t h s e t is (20, 100) .

(j) a l l r e a l numbers g rea t e r 7 than -

(k) a l l r e a l numbers l e s s 4 than 7

(1) {Î

pages 438-439

22. ( a ) 1

3 5 - - 2ab b + .m=p=- a + 5a lla b +ô

*(I) + 8a + l2 TMS may a l s o properly be l e f t i n i t s

a - 8a + 12 o r ig ina l factored form.

6x x (d) ,q 1 . = x

The truth set Is [O).

= 25 + ((-10) + (-25)) by the associative property of addition

by the commutative property of addition

by the associative property of addition

pages 439-440

15 < 18 , . thus "1 < 55 9 " i s t r u e .

7 4 - 2 8 A . z = g and ~ ~ q - m 20 3

" 9 < 7 I' i s t r u e . 27 2 8 ; thus ^ 5 < ^

7 Then, by the t r a n s i t i v e property, fy < vp i s t r u e .

27. Let e be t he number of u n i t s i n t h e length of each edge. Then he i s t h e number of u n i t s i n t h e perimeter and e 2

i s the number of u n i t s i n t h e a r ea .

Now make the length o f t h e edge 2e.

New perimeter 8e The perimeter i s mult ip l ied by 2.

New area 4e2 The a r e a Is mul t ip l ied by 4.

28. ( a ) Set A i s closed under mu l t i p l i ca t ion . Set B i s closed under mul t ip l ica t ion .

(b) C = (0, 1). Se t C Is a subset of both s e t A and s e t B, but i s a proper subset of B only.

x + 3x + 5 i s not a 29. >dt . The only value of x f o r which ,-(

7 7 r e a l number i s x = . 2x - 7 = 0 i f and only i f x = 5 . The set of r e a l numbers o ther than 0 i s closed under d iv i s ion .

30. Since 2 a a = & . - 9 3

(a) If a < 24, then 9 < 24

and b < &24)

Hence b satisfies t h e i nequa l i t y b < 36.

page 440

2 31. If a i s one of t h e numbers, then a # 0 and Ã i s t h e o t h e r number.

1 1 (a) a < 3 . If 0 < a < 3, then ->, 2 and > ? , by t h e m u l t i p l i c a t i o n p r o p e r t y of o r d e r . If a < 0, then 2 - < 0. Thus, t h e o t h e r number i s greater than a or 5 - less than 0.

1 1 ( b ) a < - 3 . Here - > - 2

a ~ l s o , s i n c e 2 and - > - 7 a < 0, < 0. Thus, t h e o t h e r number i s g r e a t e r than

A {' - - and l e s s than 0. 3 -

and F a r e no t equal f o r a l l va lues of a, b, and c . ^ (For example, l e t a = b = c = 2)

33 No 2 3 A coun te r example: 2 + 3 # 3 + 2, s i n c e - s r # ~

34 1 If ~ = a + ~ and a = - then 2 '

pages 440-441

35* If a i s between p and q, then 1 i s between - P

1 and . Suppose p > q, then q < a < p. Since a < p, -a

1 1 . Since a > q, - < . Hence, g is between 3 > p q

1 1 - and - . P q

l8y f e e t , i f 6y Is the number of yards 24f inches, i f 2f i s t h e number of f e e t 8k p i n t s , i f 4k is t h e number of quar t s (n - 10) years , I f she i s now n years o l d (16k + t ) ounces, I f k i s the number of pounds and t i s the number of ounces l 4 4 f square inches, i f f i s the number of square f e e t (100d + 25k) cen ts , i f d i s the number of d o l l a r s and k i s the number of quar te rs (100d + 25k + l o t + 5n) cen ts , i f t he re a r e d d o l l a r s , k quar te rs , t dimes, and n n icke ls n + 1, i f n i s t h e whole number - i f t he number Is n n p 5280k f e e t , If k i s t h e number of miles 2(5280k) f e e t , i f k is the number of miles

In these open sentences, the phrases and numbers o f ten give a c lue t o t he poss ible t r a n s l a t i o n s . I n each p a r t , Ju s t one in t e rp re t a t i on Is given, f o r suggestive purposes only, and t he re is no implication t h a t t h i s In t e rp re t a t i on i s t h e "best1' one; pupi l s should be encouraged t o look f o r more than one meaningful t r a n s l a t i o n . Note t h a t with c e r t a i n t r ans l a t i ons the var iab le is r e s t r i c t e d t o t h e set of whole numbers, whereas w i t h o ther t r a n s l a t i o n s t he re Is no such r e s t r i c t i o n .

(a) My grandfather i s l e s s than 80 years o ld .

(b) His annual salary is 3600 d o l l a r s .

(c ) The assets of a c e r t a i n bank a r e more than one hundred mil l ion d o l l a r s .

pages 441-442

The sum of t h e ang les of a t r i a n g l e i s 180Â°

The l e n g t h of a r e c t a n g l e i s 18 inches more than t h e width. The a r e a Is 360 square inches .

The l e n g t h of a r e c t a n g l e i s t h r e e t imes t h e wid th

and t h e area does not exceed 300 square inches.

The number of u n i t s i n t h e l e n g t h of a r ec tang le is two more than t h e number of u n i t s i n t h e width . A s i d e of a square i s one u n i t longer than the width of t h e r e c t a n g l e . The a r e a of t h e square i s g r e a t e r than t h e area of t h e r e c t a n g l e .

Farmer Jones had 30 sheep which he expected t o s e l l f o r $20.00 a head; some of t h e sheep d ied , but he s o l d t h e remainder f o r $24 a head, r e c e i v i n g as much as o r more than he had o r i g i n a l l y expected.

The s i d e s of an e q u i l a t e r a l t r i a n g l e and a square a r e such that t h e pe r imete r of t h e t r i a n g l e i s equal t o t h e pe r imete r of t h e square .

The sum of f i v e consecut ive numbers Is l e s s than 90, and t h e l e a s t of t h e numbers Is g r e a t e r than 13.

I n each c a s e above t h e response could have been given i n t h e form of one sentence by use of connect ives . Sometimes, f o r t h e sake of c l a r i t y , i t is better t o use s e v e r a l s h o r t e r sentences i n making a t r a n s l a t i o n .

38. ( a ) If n i s the number, then t h e number diminished by 3 Is n - 3 .

( b ) If t i s the first temperature, the temperature a f t e r it rises 20 degrees Is t + 20 degrees .

( c ) If n i s t h e number of p e n c i l s purchased a t 5 c e n t s each, t h e c o s t Is 5n c e n t s .

( d ) If t h e number of n i c k e l s i n my pocket Is y and t h e number of dimes i s x, t h e amount of money I have i s

( l o x + 5y + 6) c e n t s .

I f the number i s n, then the r e s u l t of increasing i t by twice the number i s n + 2n.

If the first number i s x and the o ther is y, t he f i r s t increased by twice the second i s x + 2y.

I f the number of weeks is w, the number of days i s

7w

I f x i s the number of melons and y i s the number of pounds of hamburger, t he t o t a l cos t i s 29x + 59y cen t s .

If n i s the number of inches i n t h e sho r t e r s i d e of a rectangle , n + 5 i s t h e number of inches i n t h e longer s ide , and the a r ea i s n(n + 3 ) square inches .

I f x Is the population of t he c i t y i n Kansas, then one mil l ion more than twice t h e population i s 2x + 1,000,000.

If x i s t h e number of d o l l a r s s a l a r y pe r month, t he annual s a l a r y i s 12x d o l l a r s .

I f b i s t he number of d o l l a r s i n Bet ty 's allowance, the number of d o l l a r s i n Arthur's allowance i s 2b + 1.

If h i s the number of hours, t he d i s tance t rave led a t 40 m.p .h. i s 4Oh miles .

If t h e number of d o l l a r s i n the value of t he proper ty i s y, t he r e a l e s t a t e tax Is h ( 2 5 ) d o l l a r s .

If the number of pounds E a r l weighs i s e , t h e number of pounds Donald weighs is e + 40.

r - 1 i s the number of miles t he first c a r t r a v e l s i n an hour, if r is the number of miles the following c a r t r a v e l s i n an hour.

If x i s t h e number of pounds of s teak , t he cos t i n d o l l a r s i s 1 . 5 9 ~ .

If the number of hours Catherine works i s z, t he number of d o l l a r s she earns is .75z.

If the number of gal lons i s g, t he cos t i n cen t s i s

39. ( a ) y i s Mary's s i s t e r l s age. 1 6 = y + 4

( b ) b i s t h e number of bananas. gb = 54

( c ) n i s the number. 2n + n < 39

( d ) b i s the number of d o l l a r s i n Be t ty t s allowance. 2b + 1 i s t h e number of d o l l a r s i n Arthur's allowance. 2 b + 1 = 3 b - 2

( e ) t i s the number of hours. 40t = 260

( f ) t i s the number of hours t h e t r i p took. 501 > 300, i f we assume t h a t t h e maximum speed Is not maintained f o r t h e e n t i r e t r i p , o r 5 O t = 300, i f we assume that t he maximum speed is maintained. The sentence 50t > 300 gives a cor rec t t r a n s l a t i o n .

(g ) h i s the number of f e e t of e leva t ion of Pike's Peak. h > 14,000

(h) n i s the number of pages i n t he book. 1.4 = 0.003n + 2( . l )

( 1 ) Let p be t he number of people i n any c i t y i n Colorado. 3,000,000 > 2p + 1,000,000 2

(j) x < (x - l)(x + 1 ) . This i s a co r r ec t t r ans l a t i on . However, i t i s not poss ib le t o f i nd any value of x f o r which it i s true. Using t h e d i s t r i b u t i v e proper ty we get:

Y? < x2 - 1, and t h i s is f a l s e f o r every x .

(k) y i s the number of d o l l a r s I n t h e valuat ion of the proper ty .

pages 443-444

w Is the number of pounds E a r l weighs. 152 - > w + 40 n i s the counting number. n + 1 i s i t s successor. n + (n + 1) = 575

n is the counting number. n + 1 i s Its successor. n + (n + 1) = 576. Thia sentence i s f a l s e f o r a l l counting numbers. I f a number i s odd, i t s successor i s even; i f t he number i s even, Its successor i s odd; i n e i t h e r case, t h e i r sum cannot be even.

n i s the first number. n + 1 i s the second number. n + (n + 1 ) = 576. Here the so lu t ion s e t i s not t he empty s e t s ince t h e domain of n is not r e s t r i c t e d t o the counting numbers.

f i s the number of f e e t i n t he leng th of one piece of board. 2f + 1 I s the number of f e e t i n t h e length of t h e other piece . f + (2f + 1 ) = 16

y i s the number of years o ld Mary i s now.

y - 6 i s t h e number of years o ld Mary was s i x years ago.

y + 4 is t h e number of years o ld Mary w i l l be i n four years . y + 4 = 2(y - 6 )

t i s the ten ' s d i g i t . u i s the u n i t 1s d i g i t . l o t + u i s the number. u + t is the sum of t he d i g i t s . l o t + u = 3 ( u + t ) + 7

pages 444-445

(u) n is t h e number. 3 ( n + 17) = 192

40. m i s the number of months t h a t have elapsed s ince h i s weight was 100 I b s .

175 + 5m = 200

41. ( a ) n is t h e number. n < 7 and n > 1 - -

(b) b i s the number Bet ty chooses, and b - < 7. n i s t h e number Paul chooses, and n - < 5. Both a r e counting numbers, s o b > 0 and n > 0.

If b = 1 and n = 1 , b + 3 n = 4 ; i f b = 7 and n = 5 , b + 3 n = 2 2 ; hence: b + 3 n > 4 - and b + 3 n < 2 2 . -

( c ) b i s t h e number Bet ty chooses, and b < 7. n i s t h e number Paul chooses, and n - < 5. Bet ty choose6 a counting number, so b > 0.

Paul chooses a whole number, so n - > 0.

Hence: b + 3 n > l - and b + 3 n < 2 2 . -

(b) t i s t h e number of one-hour periods a f t e r the i n i t i a l hour. 35 + 20 t i s the parking fee.

( c ) h i s t h e t o t a l number of one-hour periods parked. h - 1 is the number of one-hour periods a f t e r the i n i t i a l hour. 35 + 20(h - 1) is t h e parking f e e .

43. (a) l O O x + 40y i s the t o t a l number of gal lons .

(b ) 120(100) is t h e number of gal lons from the first pipe i n 2 hours.

40y i s t h e number of gal lons from the second pipe i n y minutes, where y > 120.

pages 445-446

120(100) + 40y i s the t o t a l number of gal lons In y minutes, y > 120.

* ( c ) loox + 4oy = 20,000 ~f x i s 0, 60, 120, 160, 180, 200 and y i s 500, 350, 200, 100, 50, 0 the sentence i s t r u e .

44. c Is .the number of degrees Centigrade. 1 . 8 ~ + 32 i s the number of degrees Fahrenheit . 1 . 8 ~ + 32 < 50

45. d i s the number of d o l l a r s Harry rece ives . d + 15 i s the number of d o l l a r s Dick rece ives .

2(d + 15) i s the number of d o l l a r s Tom rece ives .

d + (d + 15) + 2(d + 15) = 205 4d + 45 = 205

4d = 160 d = 40

Harry must receive $40. Dick must receive $55. Tom must receive $110.

46. Last year ' s cos t was lOOd cen t s pe r dozen. This year 's cos t is lOOd + c cen t s pe r dozen.

H a l f a dozen b a l l s w i l l c o s t load + cen t s . 2

47. Since the amounts a r e proport ional t o t h e ages 7 and 3, they may be represented a s 7x d o l l a r s and 3x d o l l a r s .

Then 7x = 16.80 and 3x = 7.20. The older ch i ld receives $16.80 and the younger, $7.20.

pages 446-447

48. Let x be the new average. Then 8x i s t h e t o t a l number of po in t s received by the 8 pupi l s who remained i n t he c l a s s . The t o t a l number of po in t s received by the 10 pupi ls is 720. Hence,

8x + 192 = 720, 8x = 528,

and x = 66.

Hence the new average i s 66.

Addition, sub t rac t ion o r mul t ip l ica t ion of any two numbers of the set gives a number of t he s e t .

Division may not give a number of the s e t . For example, 2 5 i s not an even i n t e g e r .

Finding the average of p a i r s of numbers from the s e t 2 + 4 o r may not give a number of t h e s e t . For example, - 2

3 i s not an even in t ege r . Thus, t he s e t of even in tegers i s closed under addi t ion , sub t rac t ion and mul t ip l icat ion, but i s not closed under d iv i s ion o r pairwise averaging.

50. If t h e f i r s t s h i r t cos t x do l l a r s , then

The f irst s h i r t cos t $5, s o he l o s t $1.25 on i t .

If the second s h i r t cos t y d o l l a r s , then

Y + -2537 = 3.75, 1 - 2 5 ~ = 3 075,

Y = 3 .

The second s h i r t cos t $3 , s o he gained $0.75 on it.

Thus, he l o s t $0.50 on t h e s a l e of the two s h i r t s .

page 447

51. If n i s the number of n ickels , then 12 - n i s the number of dimes, 5n i s the number of cen ts i n n n icke ls , and 10(12 - n ) is the number of cen ts i n (12 - n) dimes. Since the t o t a l number of cen ts Is 95, we have

There were 5 n icke ls and 7 dimes.

52. I f t i s the number of hours he r i d e s i n t o t he woods, then 5 - t Is the number of hours t o r i d e ou t . 4t i s t h e number of miles he went one way and SO i s 15(5 - t ) . Hence,

18 He can r i d e I n f o r 3rn hours s o he can go a d i s tance of 18

(- x 4) miles i n t o t h e woods.

53. I f s i s the speed of the wind i n miles pe r hour, then the speed of the plane i s 200 - s miles per hour,

1 and the dis tance t rave led Is %(So0 - a ) miles.

So , $(200 - S ) = 630

The speed of t h e wind i s 20 miles p e r hour.

Suggested Test Items --

1. Simplify each of the following: (a) -4 - 6 (h) , where y # 0

^ (b) 12 - 4 - 3 (1) (2a - 1 ) - (a + 2)

+ where s # - 1 (k) Yi7T-r

where x # 2 and x # -2

2. If m = - 4 and n = 3 f ind t h e v a l u e o f (a) m - n (4 l m - nl rn - n

i i iTii

3. Simplify each of t h e following:

4. For what values of the var iab les in Problem 3 i s each of these expressions no t a real number?

5. Find the t r u t h s e t of each of the following:

(a) $ + $ = 7 1 (d) - = Y a

(c) g - +x = 2x

6 . For what values of the var iab le i s each of the following t rue?

(a) 0 - x = 0

(b) 5x = 0 (c) x -0 = 3

7. Find the t r u t h s e t of each of the following:

(a) y - 3 = 3 - y 2 ( 4 m > 1

8. I f a < b, which of the following numbers a r e pos i t i ve?

( a ) a - b (d) a, if ab < 0

(e ) ( b - a ) a 2

9. What number must be added t o -2x + 3y - 4 t o g e t x - 2y + 2?

2 10. By what number must be mul t ip l ied t o g e t 3ab?

3 11. I f the numerator and the denominator of the f r a c t i o n 7?

are each increased by x, where x i s pos i t ive , the value 1 of the f r a c t i o n i s increased by . Find x.

A student l i ved a t a boarding house, where he paid r e n t a t the r a t e of $1.50 per day, except on those days when he was ab le t o work f o r the boarding house owner. Whenever he worked f o r the owner f o r a day, the owner charged him no r e n t f o r t h a t day, and gave him $8 c r e d i t toward h i s r e n t f o r the month. The s tudent paid $8.50 r e n t f o r the month of January. Write and solve an open sentence t o f i n d out how many days he worked f o r the owner t h a t month. ( ~ i n t : I f the student worked n days, f o r how many days d id he pay r e n t ? )

13. Horatio i s making a scale model of a building. I f the scale 1 i s -- that i s , if a length of 60 f e e t on the building i s

represented by a length of one foot on the model~how long should he make the wall of h i s model which i s t o correspond t o a 25-foot wall of the building? Write and solve an open sentence f o r t h i s problem.

Answers - t o Suggested Test Items --

4. (a) If a t l e a s t one of a, b, and c i s 0 (b) If a = b

(c) None (d) If b i s -2, 0, o r 2

6. (a) a l l r e a l values

(b) 0 ( c ) f o r no value of x

(dl 2 (e) a l l r e a l values except 2

( c ) a l l values of x which a re greater than -2 and less

than 2, except 0.

( b ) a l l posi t ive r ea l ( d l ft numbers

The numbers i n ( c ) and ( e ) are posit ive; the others negative .

12(3 + X) - 3-3(4 + x) = 4 + x x = 2

If the student worked n days, then he paid rent f o r (31 - n) days. Then

1.50(31 - n) - 8(n ) = 8.50 46-50 - 1.50n - 8n = 8.50

46.50 - 9.5n = 8.50 38 = 9.5n 4 = n

w The student worked four days during the month of January.

If the w a l l of the model i s y feet long, then

(21 The wall must be ft . (5 inches) long.

pages 447- 448

Challenge Problems

Pa r t 2

1. The r e l a t i o n " >Â¥I does - not have t h e comparison property. For example, 2 and -2 a r e d i f f e r e n t r e a l numbers, but n e i t h e r i s f u r t h e r from 0 than the other; i n other words, n e i t h e r of the statements - 2 >- 2" and "2 >Â -2" is t rue.

The t r a n s i t i v e property f o r " ~ " would read: If a, b, and c are r e a l numbers f o r which a > b and b y e , then a + c . This i s c e r t a i n l y a t r u e statement as can be seen by s u b s t i t u t i n g the phrase "is f u r t h e r from 0 than" f o r >- wherever i t occurs.

The r e l a t i o n s " >Â¥I and I' > " have the same meaning f o r t he numbers of ar i thmet ic : "is f u r t h e r from 0 than" and "is t o the r i g h t o f" mean the same thing on the ar i thmet ic number l i ne .

2. By the d e f i n i t i o n of t he product of two r e a l numbers, we have

ab = la1 . lbl o r ab = - ( la1 -1b I ) .

(1 ) If ab = l a l o lbl , then

lab1 = l a 1 l b l l

= la1 l b l , s ince lal-lbl 2 0.

page 448

3 . Prove t h a t t he number 0 has no r ec ip roca l . Proof: Assume t h a t t he sentence of t he theorem i s f a l s e . Then 0 has a rec iproca l , say a . This would mean t h a t

Since the product of zero and any r e a l number i s zero, it follows t h a t

0 = 1.

This sentence is f a l s e . Thus our assumption t h a t zero - has a reciprocal i s a f a l s e assumption, and it follows t h a t zero has - no rec iproca l .

4 . Prove t h a t the rec iproca l of a pos i t i ve number i s pos i t i ve , and the reciprocal of a negative number is negative. Proof: The statement follows immediately from t h e de f in i -

1 t ion , a x - = 1, s ince t h e product of two numbers is a pos i t ive i f and only i f both numbers a r e pos i t i ve o r both numbers a r e negative. (proof by cont rad ic t ion would a l s o be poss ible . )

5. Prove t h a t t he rec iproca l of t he rec iproca l of a non-zero r e a l number a is a .

1 1 Proof: Since i s the rec iproca l of = by the de f ln i - - a 1 1

t i o n of a reciprocal , It follows t h a t (Ã‘ (Ã‘IÃ = 1 . - a

1 Similarly, s ince i s t h e rec iproca l of a , it follows

1 t h a t (a)($) = 1, or , by t h e commutative property, (y) ( a ) = 1.

1 1 1 Compare (K)(T) = 1 with (Ã‘) (a = 1. We see t h a t t h e - a

1 1 number g has rec iproca ls Ã‘< and a. Since any non-zero -

a r e a l number has only one rec iproca l , i t follows t h a t

1 = a, which i s what we wanted t o prove.

7. From the preceding exercise the student will, we hope, Infer that for all real numbers a and b,

In case some of the more capable students are interested in seeing a proof of these statements, we give the following.

The statement that lx + yl - < 1x1 + lyl for all real numbers x and y can be used to prove the three statements above: With x - a - b and y = b, we have

a1 = l(a - b) +bl < la - bl + lbl. By the addition property of order.

la1 + (-lbl) < la - bl

la1 - lbl < la - bl la - bl > la1 - lbl.

Similarly, x = b - a and y = a leads to the sentence

lb - a1 > lbl - lala Since lb - a1 = I-(b - a)I = la - bl, this gives.

la - bl > bl - al*

pages 448-449

But lbl - la1 = -( la1 - l b l ) , so t h a t we now have

Therefore, la - bl 2 llal - Ib l I .

8. The dis tance between a and b is found t o be a t l e a s t as great as t h e dis tance between 1 a 1 and 1 b 1 , because a and b can be on opposite s i d e s of 0, while la1 and lbl

must be on the same s ide .

9 . The two numbers a r e 3 and 5.

m Though the above is the suggested approach t o t h i s problem, some s tudents may t r y t o do i t by using the d e f i n i t i o n of absolute value.

If l x - 4 1 i s 1, t h a t is , l x - 4 1 I s a n o t h e r name f o r 1, then (x - 4 ) must, by d e f i n i t i o n of absolute value, be e i t h e r 1 o r -1. Thus,

10. The t r u t h s e t of t he sentence lx - 41 < 1 is the s e t

3 < x < 5 .

Rather than using formal methods f o r so lu t ion of t he inequal i ty , the student w i l l be guided by the question: What i s the set of numbers x such t h a t the d i s tance between x and 4 is l e s s than I? A s i n t he case of t he preceding exercise, the s tudent may work d i r e c t l y from the d e f i n i t i o n of absolute value ins tead of by the suggested approach.

For example : If x - 4 1 0 , then lx - 41 = x - 4

But lx - 41 < 1

So x < 5

If X - 4 < 0 , then lx - 41 = - (x - 4

~ u t lx - 41 < 1 So - x + 4 < 1

-x < -3 x > 3

Thu s , x > 4 - and x < 5 o r x < 4 and x > 3 .

F i n a l 1 y , 3 < x < 5 .

11. The graph of t h e t r u t h set of x > 3 and x < 5 i s

It I s t h e same a s t h e t r u t h s e t of lx - 41 < 1.

1 2 . I n some of t h e fo l lowing e x e r c i s e s t h e methods described i n connect ion with t h e s o l u t i o n of Problems 6 and 7 above may be used by t h e s t u d e n t s . The method of t h e d i s t ance on t h e number l i n e Is our main o b j e c t i v e h e r e .

(a) Truth s e t : (-2, 1 4 ) ! I $ ! !

Graph: -2 -1 0 1 2 3 4 5 6 7 8 9 10 I t 12 13 14

( b ) Truth set: ( 4 )

Graph :

( c ) Truth set: (8, 1 2 )

( d ) Truth set: Real numbers x such t h a t x < -3 o r x > 3 .

Graph : 3

( e ) Truth set: A l l r e a l numbers

Graph :

pages 449-450

I Y I = 1 Truth s e t : {-I, 1)

Graph : - 0 I

Truth s e t : Real numbers y such t h a t 4 < y < 12. Graph :

l l l l l l l l l l l l

The empty s e t 0

Truth s e t : [-22, -16) - l - w - M - 4 ' ! 2 ! ! ! ' ! ! ! ! ! ! ! ! ! ! ! ! !

Graph : -22 -16 0

Truth s e t : (-14, 4 )

13 . Prove: 1 1 (Z) = - (;'

Proof:

1 = (-1) .- a Defini t ion of mu l t i p l i ca t ive

inverse

14 . Prove: If a < b, a and b both p o s i t i v e r e a l numbers, 1 1 then < .

Proof : a < b Given

1 1 1 1 a ( z * < b(z Mult ipl icat ion proper ty of

order; i s pos i t i ve , a E since a and b are p o s i t i v e .

1 1 1 1 ( a < (b d7 Associative and commutative

proper t ies of mul t ip l ica t ion

Defini t ion of mul t ip l ica t ive inverse

Mult ip l icat ion property of 1

15. Prove: If a < b, where a and b a r e both negative 1 1 r e a l numbers, then 5 < g .

Proof: a < b Given

1 1 1 1 a ( < = p) < b(g el?} Mult ipl icat ion property of

order

(Since g and 1 a r e both negative numbers, ( k - t) is a - ~

p o s i t i v e number. ) The remainder of the proof i s iden t i ca l t o t h a t i n Problem 14 . Al ternat ively , s ince a < b,

-a > -b. Because -a and -b are both pos i t i ve numbers, 1 Problem 14 allows u s t o assert t h a t -= < -E , and

1 1 - - < - . Taking opposites, again we have, a

1 1 1 - i s negative 16 . If a < 0 and b > 0, then g < 5 because a 1 and p i s p o s i t i v e .

17 a b - + - = 1 1 c c a(-) + b (-1 Defini t ion of d iv i s ion

1 = (a + 1')- Dis t r ibu t ive property

a + b = - c Defini t ion of d iv i s ion

18. a b a d b c - c + 3 - -(-I c d + 7(7) Mult ipl icat ion property of 1

Commutative property of multi- p l i c a t i o n and the theorem:

Proved i n Problem 17

pages 450-451

19. (a ) Yes, because the product of any two numbers of the s e t i s a member of the s e t .

(-1) x ( - J ) = J ; ( - J ) x (-1) = J Hence, (-1) x (- j ) = ( - j ) x (-1)

Hence, (-1) x j x ( - j ) = (-1) x J x (-j)

Hence, 1 X (-1) x j = 1 x (-1) x j

(e 1 1 x 1 ~ 1 . Hence, 1 i s the reciprocal of 1.

(-1) x (-1) = 1. Hence,-1 i s the reciprocal of -1.

j x ( - j ) = 1. Hence,-j i s the reciprocal of j.

( - j ) x j = 1 . Hence, j Is the reciprocal of -j.

( f ) If x i s a number such t h a t j x x = 1, then

If x = -j, then J x x = j x (-1) = 1. Hence, the t r u t h s e t i s (- j ) .

( g ) Similar ly , t h e t r u t h s e t i s (-1). Multiply by j,

s ince j i s the rec iproca l of - j .

(h ) The t r u t h s e t Is (1 ) . Multiply by ( - l ) , s ince (-1) i s t h e rec iproca l of l2 o r (-1) .

2 (1) The t r u t h s e t is ( 1 ) . j3 = ( J ) x J = (-1) x j - - j . Hence, mul t ip ly by j, s ince j i s the reciprocal of

-J

20. A r a t e of 3 minutes p e r m i l e i s 20 m .p .h. Thus, the 360 time going i s -r,- , o r 18 houre. 3 milea per minute

360 i s 180 miles pe r hour. Thus the time re turning i s , o r 2 hours. The t o t a l time Is 18 + 2, o r 20, hours, t he t o t a l d i s tance 2.360, o r 720, miles and the average r a t e is ws- 720 , o r 36 m.p.h.

21. S t a r t with t h e sum t.

Then, f o r t h e t en numbers, the new sum i s

3 ( t + 10-4) - 10-4 = 3 t + 80.

For 20 numbers,

The new sum, then, Is 160 more than th ree times the o r i g i n a l sum.

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