STUDENTS TEXT

FIRST COURSE IN ALGEBRA PART I

SCHOOL MATHEMATICS STUDY GROUP

i 1

I YALE UNIVERSITY PRESS

School Marhenlatics Study Group

First Course in Algebra

First Course in Aloebra a

I)rcpai-t.d undcr thc supcrvi.iion oi

th r Ilnnol o i i Sanlplc 'I'cxthooks

nf rlic School Mnthcniatic~ Stud!- Group:

f rnnk B. Allcn L > . c ) t ~ k I ' o \ ~ - l l ~ I l l ~ ~ I 1 1 ~ 1 1 S i l ~ i ~ n l

F.dn,i~i C:. L l c ~ ~ ~ ~ i a s Tafi hchool

13on~ld E. R~chmond W illianls Collc gc

C h ~ r l c r E. Kickarr Y RIC LTr~iversit~.

1 1~111-y Stv;~in New 7'ricr To\i I ~ S I I I ~ I li4cl~ 5 ~ 1 ~ ~ ~ ~ ~ 1

llobcrr J. W2lkc.r Corncll Univci-iit\.

Neiv I Invcn and London, Yalc Lnlvc.rsiti. llrc\\

All rig], tr ;c~scr\-ccI. This book i-IIny no t

bc r ~ p r i d u ~ ~ r d , i t1 tvh<)Ic or in part. ill

an)- &I-m, ~ v ~ r h o i l : ~vri t tcn pcrmis~ion hcml t11c l~ibllshcr<.

Flila~icial support fc>r the Sciltlrli M.itliinl~tics S t ~ ~ l j ? - Group Ilas been p r o v ~ d r - d !>r rlic National

S C ~ C ~ L C L ' Foi i~ lda t ion .

FOREWORD

The increasing contribution of mathematics to the culture of the modern world, as well as its importance as a vital part of scientific and humanistic education, has made it essential that the mathematlcs in our schools be both well selected and well taught.

With t h i s in mind, the various mathematical osganizations 5n the Unlted States cooperated in the formation of the School Mathematics Study Group (SMSG) . S E G includes college and univer- sity mathematiclans, teachers of mathematics at a11 levels , experts i n education, and representatives of science and technology. T h e general ob j ec t i ve of SMSG 1s the Arnprovement of the teaching of mathematics in the schools of this country. The National Science Foundation has provided substantlal f'unds for the suppart of thls endeavor.

One of the prerequisites f o r the improvement of the teaching of mathematlcs in our schools is an improved curriculum--one whlch takes account of the increasing use of rnathematics In science and technology and in other areas of knowledge and at t he same t l m e one which r e f l e c t s recent advances In mathematics i t s e l f . One of the first projects undertaken by SMSG was to enllst a group of outstanding mathernatlclans and rnathematics teachers to prepare a serles of textbooks which would illustrate such an Improved curr iculwn .

The professional mathematicians In SMSG believe that t he mathematlcs presented in this text is valuable for a l l well-educated citizens in our society to know and that it Is important for the precollege student to learn in preparation f o r advanced work in the f i e l d . A t the same time, teachers In SMSG believe that it is presented In such a form that it can be r ead i ly grasped by students.

In most instances the material w i l l have a familiar note, but the presentation and the p o l n t of view w i l l be different. Some material will be e n t i r e l y new to the traditional curriculum. This is as it should be, for mathematics Is a living and an ever-growing subject, and not a dead and frozen product of antiquity. T h i s healthy fusion of the old and t he new should lead students to a better understanding of the basic concepts and structure of rnathematics 2nd provide a flrmer foundation f o r understanding snd use of mathematlcs in a scientific society.

It is no t intended that t h i s book be regarded 3s t h e only d e f i n l t k v e nay of presenting good mathematics t o students at t h h s level. Instead, it should be thought of as a sample of t h e kind of Improved curriculum that we need and as a source of suggestions for the authors of commercial textbooks. It I s s i n c e r e l y hoped t h a t these t e x t s w Z l l lead t h e way toward inspiring a more meaningful teaching of Mathematics, t h e Queen and Servant of the Sciences.

Below are listed t h e m e a of a l l t h o s e who p a r t i c i p a t e d i n any of the writ ing Se88 ions

a t which t h e foliowing SMSG t e x t s were prepared: F i r s t Course 111 A l ~ e b r a , Geometry,

In te rmedia te Mathematics, Elementary Func t ions , and I n t r o d u c t i o n to Matrix fl~ebra.

H.W. Alexander, Earlham College F.B. Allen, Lyons Township High School, La Grange, I l l i n o i s

Alexander Reck, Olney Hi@ School, Phila- de lphia , Pennsylvania

E.F. 2eckenbach, Univers i ty of C a l i f o r n i a at Los Angeles

E.G. Fegle, School Mathematics S t u d y Group, Yale Univers i ty

Paul Berg, Stanford U n i v e r s i t y Emil Berger, Monroe Hiph School, S t . Paul,

Mlnnesot a Arthur Bernhart, University of Oklahoma R.H. Bing, University of Wisconsin A.L. Blakers, Universkty of Western

A u s t r a l i a A . A . Blank, New York U n i v e r s l t y S h i r l e y Eoselky, Frankl in High School,

S e a t t l e , Washington K.E. Frown, Department of Health, Educa-

t l o n , and Welfare, Washington, D.C. J .M. Calloway, Car le ton College Hope Chipman, U n i v e r s i t y Hi@ School, Ann

Arbor, Yichigan R.R. C h r i s t i a n , Universi ty of B r i t i s h

Columbia R.J. Clark, St. Paul b Schaol, Concord,

New Hampshire P.H. Daus, University of C a l i f o r n i a a t Loa

Angeles R . 3 , Davis, Syracuse University Charles DePrlma, C a l i f o r n i a Institute of

Technology Mary Dolc i a n i , Hunter College Edwin C . Douglas, The Taft School, Uater- town, Connecticut

Floyd Downs, East H i @ School, Denver, Colorado

E . A . Dudley, North Haven H i g h School, North Haven, Connect icut

Lincoln h r s t , The Rice I n s t i t u t e Florence Elder , west Hempstead High School,

N e s t Hempetead, New York k1.E. Permson, Newton Hipb School, Newton-

v i l l e , Yassachusetts N. J . F i n e , Univers i ty of Pennsylvania Joyce D. Fonta ine , North Haven High School,

North Haven, C o n n ~ c t i c u t F.L. Friedman, Massachuset ts Institute of

Technology Es ther Gaase t t , Claremore High School,

Claremore, Oklahoma R.K. Getaor, U n i v e r s i t y of Washington V.H. Haag, F r a n k l i n and Marshall College R .R . Hartman, Edlna-Morningside Senior KZgh

School, Edina, Minnesota M.H. Heins, Unlversity or I l l i n a i a Edwin Xewitt, University of Washington Martha Hildebrandt , Prov18o Tawnshlp !-if&

School, Magwood, I l l l n o i a

R.C. JurRensen, Culver M i l l t s r y Academy, Culver, Indiana

Joseph Lehner, Michigan S t a t e U n i v e r s i t y Marguerite Lehr, Bryn M a w r College Kenneth L e i s e n r i n ~ , Untvera l ty of Michlean Howard Levi , Columbia U n i v e r s i t y Eunice L e u l s , Laboratory High School,

Un1vers:ty of Okl zhona M.A. Llnton, William Penn Char te r School,

P h i l a d e l p h i a , Fennsylvania A.E. L lv ings ton , U n i v e r s l t y o f Waahington L.B. Loomis, liarvard Univers i ty R.Y. Lynch, P h i l l i p s meter Academy,

Fxeter, New Hampshire W.K. F:cNabh, Hockaday School, Dallas,

Texas K . G . Hlchaels , Nor th Haven Hi~h School,

North Haven, Connect icut E.E. M o i s e , University o f Michigan E .P. h'ortnrop, U n i v e r s i t y of Chicago 0.J. Peterson, Kansas State Teachers

College, Emporia, Kansqs 5 . J . P e t t i s , Univers i ty of North Caro l ina R .S. P i e t e r s , F h i l l l p s Acadeny, Andover,

Massachusetts H.O. Poll&, Bell Telephone L a b o r a t o r i e s k 'a l ter Prenowitz, Brooklyn College G.9 . P r i c e , U n i v e r s i t y o f Kansas A.L. h t n m , UniuePsi ty of Chicagb Persis 0. Redgrave, Norwlch Free Academy,

Norwlch, Cor.nect1cut Mina Rees, Hunter College D.E. Richmond, Williams C o l l e ~ e C .E. R i c k a r t , Yale U n i v e r s i t y H a r v Rudeman, Hunter College High School,

New York Ckty J .T. Schwartz, New York Univers i ty O.E. S'anai t is , S t . O l a C College Robert Starkey, Cubberley H i g h Schouls,

i 310 Alto, C a l i r o r n l a Phllllp Stucky, Roosevelt High School,

Seattle, i lzshington Henry Swain, Hew T r i e r Township H i g h

School, Winnetk?. I l l i n o i s Henry Syer, Kent School, Kent, Connecticut G.B. Thomaa, ?Inssschusetts institute of Technolorn

A.W. Tucker, Pr ince ton U n i v e r s i t y 3.E. Vaughan, U n i v e r s i t y of I l L i n o i s John Wzgner, U n l v e r s i t y o f Texas A .J. Xalker , Cornell Unlversi t y A . D . Wallace, %lane Universlty E.L. W-llters, William Penn Senior High

School, York, P e m s y l v a n i n Warren White, North High School, Sheboygan,

Wlsconzln D.V. Widder, Harvard University W i l l i a m Wooton, P l e r c e J u n i o r College,

Woodland Hills, C a l i f o r n i a J.B. Zant, Oklahoma S t a t e Universi ty

CONTENTS

Chapter

1 . SETS AND THE W E R LINE . . . . . . . . . . . . . . 1 1- P . S e t s and Subsets . . . . . . . . . . . . . 1 1- 2 . T h e N u m b e r L l n e . . . . . . . . . . . . . . 7 1- 3 . Addition and Multiplication on the

Number Slsle . . . . . . . . . . . . . . . 14

. . . . . . . . . . . . . . . 2 . NUMERALS ANDVARIABLES 19 2- 1 . Numerals and Numerical Phrases . . . . . . 19 2- 2 . Some Properties of Addltion and

MultipLication . . . . . . . . . . . . . 23 . . . . . . . . . 2- 3 . The Distributive Property 29 2- 4 . Variables . . . . . . . . . . . . . . . . . 34

3 . SENTENCES AND PROPERTIES OF OPERATIONS . . . . . . . 41 3- 1 . Sentences. True and False . . . . . . . . . 41 3- 2 . Open Sentences . . . . . . . . . . . . . . 42 3- 3 . T r u t h s e t s of OpenSentences . . . . . . . 45 3- 4 . Oraphs of T m t h Sets . . . . . . . : . . . 47 3- 5 . Sentences Involving Inequalities . . . . . 49 3- 6 . Open Sentences Involving Inequalities . . . 50 3- 7 . Sentences WithMore ThanOna Clause . . . . 52 3- 8 . Graphs of T r u t h Sets of Compound

Opensentences . . . . . . . . . . . . . 54 3- 9 . ~ummarg of Open Sentences . . . . . . . . . 56 3-10 . Identity Elements . . . . . . . . . . . . . 57 3.11 . Closure . . . . . . . . . . . . . . . . . . 60 3-12 . Associative and Commutative Proper t ies . . . . . of Addition and ~ u l t i ~ l i c a t i o i 61 . . . . . . . . . 3.13 . The Dis t r ibu t ive Property 66 3.14 . Summary: Properties of Operations on

Numbers of Arithmetic . . . . . . . . . . 72 . . . . . . . . . . . . . . . . . Review Problems 73

. . . . . . . . 4 . OFEN SENTENCES AND ENGLISH SENTENCES 77 . . . . . 4- 1 . Open Phrases and English Phrases 77 . . . 4- 2 . Open Sentences and English Sentences 82 . . . 4- 3. Open Sentences Involving I n e q u a l i t i e s 86 Review Problems . . . . . . . . . . . . . . . . . 90

PREFACE

To The Student:

This textbook I s w r k t t e n f a r you t o read. It Is no t just n l i s t of problems.

Your mathematical growth and, as a conseqxence,

t h e satisfaction and enjoyment which you der ive from

the study of algebra will depend l a r g e l y on careful reading of t h e book. F o r this reason, you will f i n d It

Important t o develop effective habits of reading mathematics.

Reading mathematics is n o t the same as reading a

novel. YVJ w i l l f i n d t h a t there are t h e s when you will

unders tand only p a r t of a paragraph t he fkrst tlme you read I t , a l i t t l e more the second time, and will f e e l sure of yourse l f only after the t h i r d reading. Some-

times working out details or examples w i t h paper and

penc i l w i l l be necessary.

You have a new and enriching experience ahead of

you. Make the most of it. Go t o it.

Chapter 1,

SETS AND THE NUFJTBEZ LINE

1 Sets and Subsets -- Can you give a description of' the following:

Alabama, Arkansas, Alaska, Arizona?

How would you de sc r ibe these?

Monday, Tuesday, Wednesday, Thursday, Friday?

I n c l u d e :

Saturday, Sunday

in the preceding group and then describe all seven. GPve a

description of the collection of numbers:

1, 2, 3, 4, 5; of t h e collection o f numbers:

2, 3, 5, 7, 8. You may wonder if you drifted into the wrong class. \ h a t do

these q u e s t i o n s have to do w k t h mathematics? Each of t h e above

collections is an example of a s e t . Your answers to the - ques t ions shou ld have suggested t h a t each set was a part icu lar c o l l e c t i o n o r members or elements with some common characteristic.

This characteristic may be only t h e characteristic of being listed together.

The concept of a set wLll be one of t he simplest of those you

will learn in mathematics. A set is merely a c o l l e c t i o n of

elements (usually numbers in our work). Now we need some symbols t o Indicate t h a t we are forming o r

d e s c r i b i n g s e t s . If t h e members o f t h e set can be l i s t e d , we may

I n c l u d e t h e members w i t h i n braces , as far t h e s e t of t h e f i r s t

five odd numbers:

C1, 3, 5, 7, 91; o r the s e t of a l l even numbers between 1 and 9;

12, 4, 6, 83; o r t h e s e t of states whose names begin with C :

[California, Colorado, Connecticut].

Can you list all the elements of the set of all odd numbers?

Or the names of all c i t i z e n s of the United Sta tes? You see t h a t in these cases w e may p r e f e r o r even be forced t o g i v e a descr lp-

t i o n o f t h e s e t wlthout a t t empt ing t o l i s t all I t s elements.

It i s convenient t o u se a c a p i t a l l e t t e r t o name a set, such

as

A = r1, 3, 5, 73 o r

A i s the s e t of a41 odd numbers between 0 and 8. R child l e a rn ing t o count reci tes the first few elements of *

a s e t N ,;:hich we call t h e s e t of counting numbers:

N = (1, 2, 3, 4 , " . * I . We write t h l s set with enough elements t o show t h e pattern and

then u s e t h r e e do ts t o mean "and so f o r t h . '' When we take the

s e t N and inc lude t h e number 0, we call the new s e t the s e t W of

71hole numbers :

W = { O , l , 2, 3 , 4 , . . . I . An i n t e r e s t i n g ques t ion now arises. How shal l w e descr ibe a

s e t s u c h as the s e t of all even whole numbers which are greater

than 8 and a t t h e same time less than lo? Does t h i s set c o n t a l n

any elements? You may no t be inclkned t o c a l l t h i s a s e t , because t h e r e is no way to list its elements. In mathematics we

say t h a t the set which c o n t a i n s no elements is described as t h e

empty, o r -J n u l l set . Me s h a l l use t he symbol $ t o denote the empty set.

Warning! The set; ( 0 ) is - not; empty; it contains the element

0 . On t h e o ther hand, we never write t h e symbol f o r t h e empty

s e t wfth braces .

Perhaps you can think of more examples of t h e null set, such

as the s e t of a l l whole numbers between $ and 3. Notice t h a t when w e t a l k i n terms o f s e t s , !ge are concerned

more with collections of elements than with the individual

e l e m e n t s themselves. Certain s e t s may contain elements which a l s o

* kle sometimes refer t o c o u n t i n g numbers as " ' n a t u r a l numbers."

Csec. 1-11

belong to o t h e r s e t s . For example, let us c o n s i d e r the sets

R = ( 0 , 1, 2, 3, 43 a n d S = 10, 2, 4, 61. Form the s e t T of a l l numbers xhich belong to bo th R and S . Thus,

T = [O, 2, 43. !-vie see that every element of T is also an element of R . We say

t h a t T Is a subse t of R .

If every element of a s e t A belongs to

a s e t B, then A is a s u b s e t o f B.

Is T a s u b s e t of S?

One result is that every set is a subset of I tself! Check 1 f o r yourse l f t h a t ( 0 , ?, 3 , 43 is a subset of itself. We shall

a l s o agree t h a t the null s e t @ is a subset of' eve ry set.

Problem Set 1 - l a

1. L i s t t h e elements of the set of

( a ) All odd 1:1hole numbers from 1 to 12 inclusive.

(b) All numbers from 0 to 50, inclusive, which are squares of

whole numbers.

( c ) A 1 1 two-digkt whale numbers, each of !lrhose units d i g i t is

twice i t s t e n s d i g i t .

( d ) All ! h o l e numbers from 0 to 10, I n c l u s L v e , whfch are the square roots of whole numbers .

( e ) A l l cities in t h e U. S. with popu la tkon exceeding f i v e

million.

( f ] A l l numbers less than 10 which are squares o f whole

numb,ers . ( g ) Squares of all those whole numbers which are less than 10,

(h) All whole numbers less than 5 and a t the same time

greater than 10,

4 2. Given the following sets:

P, the set of whole numbers greater than 0 and less than 7; 1 3 Q, t h e s e t of count ing numbers less than -T;

R, the s e t of numbers represented by t h e symbols on the faces

of a di-e;

S, t h e set; 11, 2, 3, 4, 5, 61,

(a) List the elements of each of the s e t s P, Q, R .

(b) Give a d e s c s i p t i o n of s e t S.

( c ) From the answers to (a) and (b) d e c l d e how many possible

descriptions a s e t may have.

3, Find U, the s e t of all whole numbers from 1 to 4, i n c l u s i v e ,

Then f i n d T, t h e set of squares of all members of U. Now f i n d V, the set of all numbers belonging t o both U and T.

(Did you i n c l u d e 2 in V? B u t 2 is not a member of T, so t h a t

it cannot belong t o bo th U and T . ) Does every member of V

belong t o U? Is V a s u b s e t o f U? Is V a s u b s e t of T? Is U

a subset of T?

4. Returning to problem 3, l e t K be the s e t of a l l numbers each

of which belongs e i t h e r t o U o r t o T o r t o both. i id you i n c l u d e 2 in K? You are r i g h t , because 2 belongs t o U and hence belongs to either U o r t o T. The numbers 1 and

belong t o both U and T but we include them only once I n K.) Is K a subse t of U? Is U a s u b s e t of K? Is T a s u b s e t o f KT

Is U a subse t of U?

*5. Consider the four sets @ A = ( 0 )

II

B = 10, 11

c = ( 0 , 1, 21

How many d l f f e r e n t s u b s e t s can be formed from the elements of

each of these f o u r sets? Can you tell, without writing o u t

the s u b s e t s , the number of s u b s e t s in t h e s e t

D = [o, 1, 2, 33?

t h a t I s the r u l e you discovered f o r doing this?

A s e t may have any number of elements. Ha::? many elements a r e

in t h e se t of a l l odd numbers between 0 and LOO? Could you count the number of elements? Do you need t o c o u n t them t o determine how many elements there are?

How many elements a r e i n the set of whole numbers which are

mult iples of 5? ( A mult iple of 5 i s a whole number times 5.) Can you count t h e elements of this s e t (that is, w i t h t h e c o u n t -

i n g corning t o an end) T

Consider a s e t ~ f h o s e elements can be counted, even though the

job of c o u n t l n g would e n t a i l an enormous amount of t ime and e f f o r t .

Such a s e t is t he s e t o f a l l living human beings a t a given

instant. On t h e other hand, there a r e sets whose elements cannot

possibly be counted because the re I s no end to t h e number of elements.

We s h a l l say that a set I s f i n i t e if the elements of the s e t

can be counted, w i t h the counting coming to an end, ex if the se t i s the n u l l set. Otherwise, w e c a l l It an infinite set. We say

t h a t an L n f i n i t e s e t has infinitely many elements.

Sometimes a f i n i t e s e t may have so many members t ha t we

prefer t o abbreviate its I l s t i n g . For example, we m i g h t wslte t h e

s e t 2 of a l l even numbers between 2 and 50 as

E = 1 4 , 6 , 8 , . . ., 481.

Problem S e t 1-lb -- 1. How many elements has each of t h e following se t s?

( a ) The s e t of a l l whole numbers f r om 10 t o 27, inclusive.

(b) The set of a l l odd numbers be t l~een 0 and 50.

( c ) The set o f a l l mult iples of 3.

( d ) The set of a l l mul t ip les o f 3 from O t o 99, i nc lu s ive .

( e ) The set of all mult iples o f 10 from 10 t o 1,000 inclu-

sive,

2. C l a s s i f y t h e following sets (finite or infinite):

( a ) A l l natural numbers.

(b) All squares of all c o u n t i n g numbers.

(c) A l l citizens of the United States.

( d ) All natural numbers l e s s than one b i l l i o n .

( e ) A l l n a t u r a l numbers grea te r than one b i l l i o n .

3. Given the s e t s S = 10, 5, 7, g ] and T = [O, 2, 4, 6, 8, 103. ( a ) Find K, the s e t of a l l numbers belonging t o both S and T.

I s K a subse t of S ? O f T? Are S, T, K finite?

(b) Find M, the s e t of all numbers each of which belongs t o

S o r to T o r t o both. (We never include the same number

more than once in a s e t . ) Is M a s u b s e t of S? Is T a

subset of M? Is M finite? ( c ) Pind R , t h e subset of M, which contains a l l the odd

numbers i n M. Of which others of our sets i s t h i s a subset ?

( d ) Find A, the s u b s e t of R, which con ta in s a l l the members of M which are multiples of 11. Dfd you find that A has no members? \hat do we c a l l t h i s s e t ?

( e ) Are s e t s A and K the same? If not, how do they d i f f e r ?

( f ) From your experience with the last few problems, could

you draw the conclusion that subsets of finite sets are also f l n i t e ?

4. Referr ing t o the definition of a mult ip le of 5 given above i n the text, de f ine an even whole number i n terms of mu l t i p l e s .

I s 0 an even number?

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lines, s:lch A; on a r u l e r ;r 3 therncneter. Suppose -!? r~o:!

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Using the? intervsl bet:;eer, ' ~ k ~ d s e pc?i:lt.: =IS 2 l i ~ . ' L t ~ , f lb3?. . ; l lP2, 2:1d

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T T P , , , 1.1kcl %i?e;.e goints ,"-=rn left to righ:, lzbeling cdr!! p o i n t :;ith

c - 3 1 b e . r'l~::' l i n e r;;~'.; 1 7 3 k ~ LLkc t h i :: :

!.;e sic.? that 123211 :!hole numbel, i:; f o l l o n e d c!l the rli.gP.l: b:! it? 3~1~~.;;,;3r. I.:;,at is trle succcssgl, r ~ f 133'; 3f 1C.@c, jSC32' : Cnc,7;,;e ~2

1~ r3g? 3 cc!,lr:: j.ng n < ~ n b ? ~ > s,: you <an i 1 n a ~ i r . e . Doc; it I!?v.c. n

s ~ ~ i c e ; s o ~ S Gi:;c 2 rule for f:!lLilng the z d c c e s s z ? of a c?nn:ing

i?urr.::~3r.. :I:3~t Cicelj : i ~ l s ?mpl;yT/ d , ~ n c t h i n g , ~ i n c e ? ~ c ! - I coua;ir,g

z u m b e ~ h::: a s;;cces;or, :,>~2r:? c a n n ~ i , I.)? r! l la rgcs t c-,~.Arltr.~ng :?::mi-,er,

and .he set b f :minting n u l ~ , b e ~ s i s - i . ?~Tin ; te .

::tt-c:?zd tc 2 7:ir.t J? t h e e;;f,ended l i n e sr;d P.jFl?;. r 3 i ~ t S O far

pFrrcl;F u ,, 3n~:0- -,IUL x 3 5et::eec s e t s cf !1~15ers 2nd sets of pintz o n a

Ilr;e is s n i:i_!o3 ::e s k e l l u s e c:?.::y :ir.es i:1 tllic. c;Jr;?.

3:31.i;l!:g .!i% t h e l i n e .:k.ich p ~ i n t c 3 ~ e 1ebel i .d iYb!

::hslc .A~L;:): er:, ::? can l a b e l a t h e r s>int,z b y c i i ; . l c i l l : ~ i ~ - ~ t z r _ : z l s

1 I I I I e tc.

0 1 2 3

+ 1 1 I I I 1 I I I e tc.

0 1 2 3

I I I 1 I 1 I 1 I I L I I I I 4 I I e t c .

0 1 2 3

e t c .

, :, :? -:- <, . l e -:isu3li zu 2:) . ~ n e n d l n g zc'i sf 1 i r . e ~ ?;:ith thc i n t e r T i s l s I ' > . ~ ~ i d c d ict; i . ~ c c e s z i ~ c l y g-eater nunbers of p a r k s .

:J;.: p:;t 211 these tdge ther 2 3 c.re line. :ie :~:'.~:e 3 laSelir . ,

:f 521-I.- c;- . resp~r:dir?g tc x : : I ~ of Lhe e l e ~ e f i t ~ l : uf' 3 se t of

n l - ~ ' ~ ? - ~ \ : .A*. L" -. * L- - . ; l ~ i z i : . 31-e called r i t i = , ~ ~ l nurcbers .

etc.

d i d a?8?ve, s ~ ~ u r n h e r ~ 1':ne. Th1? nur~ber s s q c l a t c i i : i ? h :i ; ; i n t i .;

celled L?I~? c .~3rdin; !Le of the p o i n t . . ,

1i.1; t l ~ l r ; tii.15 let L I Z re./ic::! what is mea:it b;r :L '!i 'rcr,tic.r:. I t

I 1 & 11 f'3r t h e same nxjnt8:r 2 . Thc ~:ur;lbcr nancjl '3? ~:+~i- l .-. l , ; ~ ' 7 : ~ 2

10 '5 y2 l j rcsen ted i ) ; ~ f frs ?r,ian;l; : , F, -6, c t i . . ~ ' i i ~ j ~ ; , ':I:*? 2 1 1 ~ : : ; 1 ' ? - ~ '

I 1 :!e sh*ll also rcz l:i t?r briat, i ~ u t , sll. i'-,: ::?,;:lA;

:.e;rt'%cn? r n l ; l ~ ~ - : a l n;lmbcr.r;, bat i-;: def in1 tL;n e-;-17,: v,.:l;ln:~.r. 1

) T'.l?;.e 7 x 2 rr;.:~y po.;sl2:lc narr:::s f o r the 5ame :-umber.

( I \ ) ::c , ~ i : ; s v ca pra3r e::s ~ ' o ; , l c l c ~ t i n g t h e aoirlt o : : ~ the :lur:lbe~

1::-!ct r;~.?raspor:J-lr;g to a n y ~ i . , ~ c r l r o t i u n a l nur~be , l ;>~ t h a t

::,, 1'0:- CLZI - . el~-r::;.:lt o-T the s e t cf ~ a t i o r i a l i-iurhuru

t k 1 2 r c - 1,; a cc;.r,esp:~:~d:i:~g p o i n t on the tlumber line.

Y L ; ' ~ C:,33.: t h i n k -:e ; . i S . l l u s e u p t!ze s u p p l y of points in t h i s

procc::; of ?~s!;gt~ing n:~xbcr ' s t:, ~ 3 ; r . z ~ . Are t!c s u r e -',- ,.lat betc!sen

3,:: k:;o dl:;c;n:'~ ! j~ . in t . ; , n> r::at",r i-~o:i clc:se, t he re is 23c:';;le t,

p o l r , t '! !;E: zar: 3nss:f.r '~211:: ques t i z r l f c r p;;i~-it.; car>res?c! lding L:.

1 - e l : : CF,..cJse t : ;o z;lcPL ~3ll:ts, for

1 L - l x z L n i a I - ::e k1;c.; ~:1-3t - .;

5 2 3

cm 7 j c: , 1. n - 1- , Il'.-", - 'L,

< . I : -, 7, ,?LC., ar,d : ha:: t h e r,ar,:es :

'd d T? T*

I , J ? l , l ~ ~ : < : ~ : ~ -L;,: ~ 1 , ~ ; : ~ ; ~ TL-: , , .-., 7 ;ervc-: to ah:,,,, t h a ;r. !;he;.s s r ; ~ r lc t

[ s e c . 1-21

o n l y a great many p o i n t s b e t w e n any t w o giver! points, but

i n Y i n i t 2 l y many p o i n t s . Nolw we are q u i t e s u r e t h 2 t e v e r y r a t i o n a l number c o r r e s p o n d s

Go a p ~ i n t on t h e number l i n e . Do you t h i n k t h a t every poinr on t h e number line (to :.he r i g h t of 0) corresponds t o a ratiilnal nurnber? In o t h e r wards, cia you t h i n k we can label every poi.nt

to t h e right o i 0 w i t h a r a t i o n a l number? The answer t o t h i s q u e s t i o n , amazingly, is "No. I ' La te r t h i s

year ;.la sha l l pFove t h i s Pact k c you. And we s h a l l soon

sssdcia tc :-lumbers wirh points to the left o f G . Meanwhile, we

assun.e t h a t e v e r y p o i n t t o the r imt of c? has a n u m b e r c , > o r d i n a t e ,

a l t h o u g h s ~ m r of t h e s e numbex? are - no's rational.

To s:~mmariz-: the above statcmepts : There arc i n f i n i t e l 7

many points on the n ? : ~ i o e r line. There are alsc I n f i n i t e l y

rnany points w i t h ~ a t ; i o n a l nuinbers as coordinates. Indeed, t h e r e

a rc i n f i n i t e l y many p o i n t s bet::ieen each pair of point:; or, t h e

number l i !?e. A1 though xe have seen t h i s on ly f o r p o i n t s x i t h

r a t i c t ~ a l cool*dLnates, Zt is z l s o t r ~ e for al l . p ~ i n t : ; .

In Chapters 1 t h rough 1; we shall be concerned with the s e t af

numbel-s c3nsLzting of C! and a l l numbers corresp~nding to p c l l n t s on

t h e r i g h t ~f C. I n thpse chzpters whell xe speak of "n~: :~tbers of

arithnetic" we shall mean numbers of this s e t .

Problen: S e t 1-2a -- 1. Ho;l mar!;; numbers ar? t h e r e between 2 and 3 ? 3 6 t ; ? ~ e e r ~ - and 500

3 2 3 rn'? List t ~ o numbers betxeen 2 a n d 3; bot:ueen and m.

2. O n t h e namber line i n d i c a t e w i t h heavy d c t a the pcints r.:hose

coordinates a r e

(b) R a t i o n a l numbers r e p r e s e n t e d by fractions -r:.Lth iieno::ii- 1 20 naco rs 5, beginnLrig u i t h and endirig v ? i t h -.

<

3 ' 3 ( c ) 0, c.5, 0.7, 1.1, 1.5, 1.8, 2, 2 .7 , ..,, , l l . l .

[ sec . 1-21

( 2 ?;hii,h 2f t h e sy : : i h l s s~L~;.:!I in parts ( A ) and ( c ) a re n o t

frac ' , ic ,zs? Vhich ?f "rl~e e:?mb'~ls i n ( a ) an3 ( c ) ?epr.esa?nt

rEt ion51 n ~ ~ ~ b e r s ? -1 . In thc p r u b l c 3 7(b), t h o coo;>dir.at; of the p o i n t a s s o c i a t e d v;.:ith

- '9 c o u l d hevc had a n o t h ~ r nane , t h r c a m o n name sf a n a t u r a l : >

nunlb?r. i h d t ? : ~ u l c ! t h l s be? Ca.r you write f o u r 3;her names

irf the : ~ u r n b ~ i ~ that c i ~ u l d serT;e as t h e c v o r d i r r a t s or' t h i s

r:)i ?k?

11 7 - . m.,,rit: S ~ X ranes f:>r thc coordlnnte of the p o i n t a z s o c i s t e d w i t h

-. r. ,,.I t ? ~ a : ~ i ~ ~ b + r l i r . 2 !!e s e c t h a t sorne p o i n t s lie to t h e r i g h t of

: t hws , sonc t o t ? e l e f t ~f 3tPt~1~3, S M ~ be tueen o t h e r s . Iforr

i s tr ,e p o l n t ; l i th cooz-d lna te 5,5 l o c a t e d nlt,h resp?c , t to tl-,e

p j i l n t ; r:i-,h csor3;nate 2? Which is the greater of 3 . 5 and 2':

'J ,>'., .is ., V . . the 1,alr.t : i i t h c=z:*dinst? 1 . 5 loca ts< l:it,h r e s p ~ c t to

Lhe p ~ i l ? t ?::.%?I caordina", F? Which is the g r e a t e l - (;f 1.5 ; j i i c i 27

22 1s 1 . hct i :?en i:l?Lck success i7 . re s:b.c,Le numbers i , : l l l ;O:J find T ?

?cl - m - 22 - - 0 - <''-kat.er lhan 5.1':' 3ves :,he p o i n t ?l'ith c o o r d i n e ~ e 7- lie i

*7. ::hat can ::cl s a y a u o u t a s e t S af !.iilole nucbers if it hzs t i ~ e

t : ,n chayscteris tics :

(1) 2 is a r ~ e i r . ~ : ~ v t r.f S;

2 ) :!l?encvel- a nufiber -2elongs to S it: s u c c e z s o y a l s5 t . e l c~?gs Z''

Let us r,:t~~;*;-i 2 2 t h e idea si' a s e t of numbers and visuzlize 3

sek ul: 5:-,c n\ll-,kcr line. Bor e~;nplr_-, ~ a c h elcrce:?: 'af t i le s e t

is a r iun3er a s s c c i a t e d ,;:ixh a _ n o i ~ l t , on he n.dm%er l i n e , ?;F;. c a l l I. L A ~ S se? sf s s s o r i a t e d points the graph of ?.he set A . Let as

i n d : c a t c t h e p o i n t s of t h e gra3l-r by markirjg then spe - i a l l ; . i,:itl:

h02vy d z t s :

Thus, t h o graph nf 2 set of n:~ribcrs Is ti15 c o r ~ e s p o n d i n g c c t nf

p o l r : t ~ on the r.u::her line ;:hose c g c r d l n a t c s sre the ncn;l.ers oi'

the set, a n d 3nly t : i o s e p~ints.

Ii? gassir .g, ::ie no", t t h n t the g r a p t ~ s of the se t Y 11f c o u n t i n g

r . u ~ ~ L - ~ r s a.ld che s e t o? ?;hole nunb,.rs a r e :

A A * 7 r eetc. w - 0 1 2 5 6 7 3 4

F x r n t h e s e dlagrarr.5 I!e see I r r m e d i a t e l y t h a t N 1:; a subset a ? ';!.

T h e graph5 iif t h e s.?ts u f e x f e n and odd ~ . h o L e n u ~ l l z r s are :

EVEN &b--t+ e t c. 0 1 2 3 4 5 6 ?

Prcblem Sst 1-2b -- 1. G i v e n the s e t s S = { r J , 3 , l l , 7) and T =- [C ) , 2, !:, 6, 8, '13).

(a) Find the s e x , t h o s e t cf all numbers b e l z n g i r r g ~o hot!!

S and 7 , a n d tile set M, t h e s e t g f a l l n u m h r s each sf'

:!l:ich belongs t o S G r T or ta both.

tb) D r a : four i d e n t i c a l numbcr lines, one bel~?; the ~ t h e r .

On 3~lccezs i . ; e lines sho!! t h e g r a p h s of t h e sets S, T, K, and M.

( c ) schemes do you see for ob ta in i r lg the g:?apilz of %

14

3rd I-:! fror:; ',h? graph,; ( : . f S and T,'?

5, lC]. 2 . ro r l s ide r the sets A = {o, 5, '(, 9) and F! - (1, 3,

(a) D I ~ : tr!?, '_dentricsl . : lumber l i r : ~ ; , gne bclc:; t he other.,

31td on t h e z c lines shcs! the grapi~.; o r k aqd. B.

(b) If set C is t h e set of r ~ > ~ r r ~ b e r s xhich sr r ncmbers of b o t l ~

A a1;d E, ?;hat do y o u i n f e r a k o u t s e t C f m n t h e graphs

qf A and b? :hat is the name of +;hi; set?

7 - Y. 3:;ery f i l j i k e set except the null s e t has the p roper ty t ha t it can be pa i red off, one-to-one, wlth a f i n i t e s e t of n a t u r a l

numbers. F o r example, t h e s e t of a l l l e t t e r s of the English

alphabet can be paired o f f , one-to-one, wlth the s e t of t he

f i r s t 26 natural numbers .

An I n f i n i t e s e t , ho;.:c:rer, c a n n o t be pa i red qff, o n e - t c - m e ,

:ri.th a f i ~ i t e set. Furthermore, it has tI-12 S ~ l r ' p r i ~ i n g grg-

p e r t y that all i t s members can be palred off', ol:c-to-one,

~ l t h The elements of sorne prijpc? subset or' i t s e l l ' . (A p r o p e r S E ~ ~ ~ ~ ~ of a ;St is ~ n e which dces g o t conLalt-, :?11 the ~5;e-

meats of t!le s e t . ) Hex can the set ot' 1:~hole n u ~ l b e r s ( x h i c h

j-s ~ n f i r ~ i t e ) be pired off, sne-to-one, !r i t ;h t h e s e t of a l l

r n ~ l t i p l e s of 3 (;.:hi& iz a prope r subset o? tkic set of

1-3. A d d i t i o n and T ~ u l t i p l i c a t i o n on t h c Nunber L i r a ? , -- Ye ! ~ e v e seen hbw to portray sets of nurr :bcrs on a n u m b ? ? line

I*i-,,n l e t us u s e the ntlrnber line t~ .;isual:Ze zc?Cl4io:: 2nd : n u l t Z -

plicatidn of numlters,

At the beginning we recall that

diagrans, al t .13ugh ";~ejr te r rx inztc 31; t h c sari!? 3,:iit. T!-LE: 15, - !! "5 + 3" 2nd "', + : are : i i f f e r e n s sy,t.>3l>; , " .~ r - :;.lc ::z.me :lu::lbi.r ;<.

I..c , J - mag $ : r ~ l l i . l e e l 5:hcthcr s d a i t ? : : n c,:] t:le r.l.arr.he~ ll2c ~ : i l L a l : . ; ~ ) ; ~

be pn:;s?t.>, ;;1:11 r.he sum zf an;. t v c i r ra1;iorinl n.~rn3erz on t h e

nunker line I z si!r1ilal3 t~:, mat; 3f' a c d i t l ~ ~ l if ~ r c r l ecs l l t h 3 f , fcr

As a dii'l'erc-?l e:iample, c o n s i d e r

. > ' . .li.;n L a s:;nbol f ~ r thi: ~:un:'ue:> 3 b t a i :?cd 'cy a d d i n g t v s c 3 I s . 3n C' :h? ncrrSur l i ~ e ;:e u 3 e d e segment f r~n O to 3 and move t o t h e

1,ii:J;t r ~ m O a d i s t s ~ c c of ~1;!3 s u c h segments. T h e terminal p o i n t < - 7 ; .,.* 2o:clrdi.12tc 2 ti 3 .

F?o;n $he rl;?,gra,;ls r:le obserir,? t h a t thcsc t r r o m d l i l ~ l l c 3'i.ions cn , . i.:Ir: :1l:-r.kc.r 1 t a e aye d i f f - . r e n t , b u t again t hey ierminafe a t the

s ? x e 7 3- in t .

Problem j e t 1 -3 -- I . P ,e r f zrm t i ~ ~ f i.llcis;:l:lg ~ p t - i ~ a t i ~ a S OR t he n u W e r 1ir.e:

( a ) j; + 6 ( d l 5 X 2

( f ) 4 x 2

2 3 e s c r i b c 9 ?rr,cedl;i-c f o r s u b t r x t i n g numbers on th? number l i e A p ~ j l ; : ;rdLrra procedure to

* <, ;, '- m--1 : ~ i z . ~ a L i z e th? ~ i l l t i 1 ; L i r ~ t i m zf rat:oml nurr:l?cr? on the 2 , f . - <AJ

- - - 1 0 ; : co:;sl.de :I 1 3 t t

' l X b , 1,:rhicJl i: a symbol < d

*z,.* 7 - s-->- 2 3

L l L J - L . ~ g t : i z - t h l r d ~ ijf T. 02 t h e nur.ben l i n e i:e divide - 3 ';hc ,:t?,~;?r.t r'rz?n O to L L L , ~ t h r ee eqi:zl payt,s snd l z c a t e t h e

3:-7-??r-? - - - - th? r:811z::ing s p e r a t . ' c , n . ; or, th2 n~:: .be~ l i n e :

' I 'i 2 (a) + x -, (2) t x ?. -

:1,~13b:r '.? Give zn exxr~:~le . 0,) I f ::e r m l t i p l ~ : 22.y t .~i : odd nun3ers, .:111 c h c i r p r o d u c t

r l ? :Lb a r odd r~rr;~.?:~? $1-:e aE exa~ple.

Co.:s:de.- t h e sefu T =; 11, 2 , 3, h]. If :;:? select 2-97 elsrr:c?.t

-- ' -- +: . , i 3 set arid add to It m y olcrr.cxn c,t' the scl; ( i i - i ~ l - . : Y ? r - ~ ~

:!,yi<]i:lg 2:. ?le~pr,(- ':; i t s e l ? ) , :,.:-,zt :z t!-le se': 5 "f 211

- ~ ~ ~ : ~ ~ i c ~ ~ ~ 2 = L3, 1 j . c;I'J:J~"~ an;; clenent cf T aand L, *

r;Lilki:21y :t b y a:,;! c l ~ r . ? : nf &, i n c l u d i n g T.he 2,?71*? elenen:;.

- 3: '7 - . This set, i:< P = {l, 3 , 5, 7, 9, . . . I . I, is a sujs?t

.:I: c, 2nd :.:: dclr;:ci;!e t n : ~ s ?rope*lt;: cf ;I-,? ~ 2 3 n:;mbers '23'

? . ,L2-nz ?-.: tilat t:;e s e t 12 is " c l a s e d i l n d e r : 7 i l i t i p l i c a t i z n . " 2~

, ::e c ~ y t ,h?t . j.f 3 sci; Ei is gi-,?en, arcl E r,err,:iil;~ L,

o3,2e~c t lon i:; a > p l l e d t ~ , ;I1 p y i r s sf clcn?nts ( ~ f K, s!:d i? t h e

( Is i;hc sa+; 2 aSo7:e c l ssed under addition? ( t : ) Ic: ti-1:3- ~ e t ; of u-..en nu.nuers c l o s e d under addition? Under

t F - 1 ~ ~ p e r h t i r ~ r ! ~f takir ig t h e 3 ,~erage:

( c ) lo t.1-.c ??t: cf : - . ' r .31~ n.lrr'nel-z c l c s e d ur-der e d c i i t l m ? Under : - q u l t i ? l i ca t igr : ' ?

(d) 3.; ::~t qf ?~' : i?nal nuybel-2 c l o s e d u g d e r a d d i t i o n ? 1 : . - ~ * ,,- A ,, r : . ~ l L i ~ l i c 3 t i ~ > ~ ?

* S? . ( 2 ) D.::c;.iimZ i? st? at iz c l o s e d under t : ? ~ o p e r a t i o n "t::'lce

tke ? r o ~ u c c . "

(L) 3 e s c r i b i : a zck tka"si cl lcspci ur?de? ;he c;t.ratigfi ' ' t x i c e

;;-LC !;r.z<,~ct a:13 ad(< cne. It

?hap ter 2

NUFER.%S AND VIIRIABLFS

P - 1 . Nwnerna l u and Numerical Phrases.

McsZ of y o u r l i f e you have been readinfi , : ~ r l t i n g , . tali:ing

about , and worliiw with numbers, -- You ha9re a l s r , used nany d i f f e r e n t

names fo? the same number. Some numbers have one or more common

names which arE the ones most, o f t en u s ~ d in referring t o t h c num- b e r s . Thus thc cornmon name f o ~ tlle number r l v ~ is "5" and f o r one

gross is "144" . Comm names f o r the rational number 0 1 - l e - h ~ l f a r e "1" and "0 .5 " . A problem ir. a r i t i m e t l c c a n b e regarded as a i lrob- 2

lenl of finding a common name f o r a number k~h ich is q i v c n I n some

o t h e r wag; f o r example, 17 times 23 is f o u n d b:~ arithnetic r o bc

the number 391.

The names of numbers, as distinguished from t h e numbers them-

selves, a r e c a l l e d numerals. Two numerals. f o r example, which re-

presen t t.he same nunber are t he indicated 31x1 "!: * * ' ' -- - - snri t h c

i n d i c a t e c p roduc t "2 x 3". The number represerlted in each case i.s

6 and we say t h a t " t h e number 4 + 2 i s h", "th1? number 2 ': 3 ir, G " ,

and " t h e number 4 + 2 is the same a s t h e number 2 x 3 ' . These

statements can b e w r i t t e n more b r i e f l y as " 3 t- 2 = 6", "2 x 3 = 6'': and "il - 2 = 2 x 3 " . This use o f the equal s i s n illustrates 1:s

general use wlth numerals : .4n equal sign standlng be tween t xo num-

e r a l s i n d i c a t e s tha t the numerals represent --- the same number.

We s h a l l need sometimes 50 e n c l o s e a ~ u m c r a l in parentheses in

orcer t o nake clear that it r ea l l y is a numeral. Jence it is con- v ~ r ? < e n t : t~ 1-egard the symbol obta i r -ed by - r ~ c l u u i n g a numeral For a

given number in parentheses as another numeral Tor the same nunber.

Thcs "(4 A 2 ) " Is another nuneral f o r i; and we m i ~ b t wri te " (4 + 2 ) = 6". h order to save w r i t i n g , t h e symbol f o r mul t i p l ica-

ticn "v" is o f t e n replaced by a d o t " . " .Hence " 2 x 3 " can be w r i t t e n

as " 2 . 3 " . A l s o t o sav? ~mit'ng, we agree that two numerals p l a c e d

side-by-slde is an indicated product, FOT example ";l '( ' i - J L ) " is taken t o mean t h e same as ''2 x (7 - 4 ) " . N o t i c e , howe-J?F, that "?? i s already cstabllslled a s the common name SOP the number twefity-

three and so cannot be i n t e r p r e t e d as the indiczted p r o d u c t "2 3 " .

A similar exception is "2;" which means "2 - a" rather than I t 2 ., 1,I A T .

1 : ~ xa:r. h o r ; e v ~ r , ?rrite ' ' 2 r 3 ) ' ' or "(2)(5)" 'in p l a c e of "2 x 3" . 1 71 1 It cr t r ? 1 !I i r l ; , "2 x F" m i g h t be replaced by 2 .F 4 -

C@n;ict?r t h ~ e x p r 2 s z i o n "2 Y 3 + 7". Is t h i s a nimcral? F e r -

ha?:. ;vu l;~lll acre2 that it. ia a ince 2 x 3 = 6 and hence ? :Y 3 -+ 7 = 6 - ;

On tllc other bar-d, someon€ t?7 r;e night d z c i d e that;, since 3 1- 7 = IC,

2 x 3 + 7 = 2 ; 4 3 3 = 2 0 .

Let us ~ x a n l r ~ o r,he expression more carer 'u l ly . 3ol.1 d o p!e read It;? !,;hat n u m c ~ a l s are Im-olved in it? Obviously "3 " , " 3 '!, and "7 " a r c

nur:erals, b u t ~ b ~ h a t ahcl~1.t "2 x 3" and "3 '7":' It is tru-. + ,ha t

"2 Y 3 " , as an i nd i ca t ed product.. and "3 + ? " , 3s an i n d i c a t e d sum, a r e bo;h numerals. Rov!eve-, ~ i t h i n :he expression "2 x 2 c ? " , -- -- ii' " 2 - 3" 5s an f n d i c a t e d prcduct, ther. " 3 + 7 1 1 cannot be ~n i cd4-

I I cacz:l sun,, r.,r; if "2 i 7 " I s an Indicat.=lcl s-m. Chen "2 x , cannot

h e arL 5ndlc:ted product.. 'I"nerefon~, without addi t . io r~al . l n f f i m a s l o n

to dec:de between these al ternati71es, the expression "2 Y, 3 ! 7'' is r e a l l y not a r~wneral s i n c e it does n c t represent a d e f i n i t e number .

A n n t h e r cxprcssicn 1 1 1 wlifch th; same proSlen a~ises i s "10 - 5 ). 2". In order b~ a v o i d the con ? u s i o n i n cxpre~sluns of :h is k i n d . xe

s h i l aqrec t o give multiplica5ion .- preference o v e r a d d i t f o n and sub - ----A

+ n ~ ,,,c%iorl ~x i l ezs ot i lerwise i n d i c a t e d . In otI-.er words, " ? .: 3 -c 7"

t r i l l be r e a d v.l_Lh " 2 Y, 2" as an Indicated pr-ocluzt, so that: ? Y 3 4 7 = 13, 3Lm:larly, "10 - 5 x 2'' w i l l b e mad w i t h "5 Y 2'' as e n i n d i -

Parcnt,heses can also be used t o Fndicate how we in t e n d f o r qn Exp:,,. ,2,-on <-. <. a c o 5e read. We have only to enclose w i t h i n parentheses

t hose p a r t s oY thc espresslon whlcn are to be taken as a xmeral .

PLUS, in the case o r "2 v 3 t 7 " , we c a n wi7:e " ( 2 >: 3 ) ; 7" if we I I !:r?nt, 2 Y 3'' t o be a numeral an3 "2 x: ( 3 + 7)" If ;v? want " 3 ; 7''

... t 3 be a nun;ral. sn a t h e r ~0rd3, t h e operations indicated w i t h i n

pa ren thcoc r , 31-2 talcsn f i r n t . . You should alw2ys f ~ e l f ree t -o i n s e r t

v!';~;~'ie.:rer papent.hew:', are necdcd t o remuljc a11 doubt a s to how an

e x p r e s s i m is to be r e a d .

Another casc i n ts'hlch v!e need to agree cn how an €xpr?ssinn ~ h o u i d be read :s i l l u s t r a t e d by the f o l l o w i n g example:

It is understood t h a t the exprezsfons above and below the fraction

bar a re t o be t a k e n as numerals. Therefore the express ion is an

ind ica ted quotient of the numbers 5(6 - 2 ) and 13 - 3 .

Problem Se t 2-la -- 1. Write sFx o the r numerals f o r the number 8. How many numerals

are there f o r 8? 2 . In each o f the following, check whether the numerals name the

same number.

(a) 2 + 4 x 5 and 22 ( e ) 4 + 3 x 2 and ( 4 + 3) x 2

( b ) ( 2 4 4 ) x 5 2nd 30 (f) ( 3 + 2) c 5 and 3 -t ( 2 -+ 5 )

( c ) 3 x 3 - 1 and 6 ( g ) 14 - 3 x 3 and 2

2 1 2 1 ( d ) 2 r 5 i l arid ( 2 x 5) + 1 (h) ( 4 + 5) + g and 4 i (5 + 3) 3. Write a common name f o r each numeral.

( a ) 2 x 5 + 7 ( 9 ) 3 5 + 7 x 3 )

(b) a 5 -t 7) ( h ) 4(5) + 8

( e ) (8 - 316 + 4

(f) 4 + 15(2) + 5

Expressions such as -+ 2 " , "2 x 3", "213 + 71, 1 "(1 - y)(16 + 4 ) - 5" are examples of numerlcal phrases. Z ~ c h of

t h e s e is a numeral formed from s impler numerals. A numerical

phrase is any numeral g i v e n by an expression which involves other

numerals a long wi$h the s i g n s for operations. It needs to be

e m p h a s i ~ ~ d that a numerical phrase must actually name a number,

tha t is, it must be a numeral. Therefore, a meaningless expression

such as " ( 3 +) x ( 2 +)-' ' is not a numerical phrase. h e n t h e ex-

pression "2 x 3 + 7" is not a numerical phrase without the agree- ment g iv ing preference t o multiplication.

Numerical phrases may be combined to form n~zac r i ca l - sentences;

i . q. , s e r r t e ~ ~ c ~ s which makc statements about r.mbers. For example. 2(3 - 7 ) = ('2 + , 3 ) ( 4 t 0 )

13 a sentence v;hic;? s ta tes t ha t the number r e p r e s e n t e d by " 2 ( 3 -t 7)" is t h e same as the number represented by " ( 2 + 3 ) ( 4 + 0)". It is read " 2 ( 3 + 7) i3 equal to ( 2 + 3 ) ( 4 + O ) " , and you can ea s i l y

v e r i f y that it i s a true sentence.

Zonsider the sentence,

( 3 4- 1)(5 - 2 ) = lo.

This s e n t e n c e asserts tha t the number ( 3 + 1)(5 - 2 ) is 10. mes

t h i s bo ther you? Perhaps you are wondering how t h e author could

have made such a r i d i cu lous mistake In arithmetic, because anyone

can see t h a t ( 3 -t I ) ( ? - 2 ) is 12 and no t 10 ! However,

" ( 3 . t 1)(3 - 2 ) = 10" i s s t i l l a p e r f e c t l y good sentence in s p i t e

cF the fact t h a t it :s fa l se .

The important Tac t abou t a sentence Involvinr : numerals is t h a t

1k :s either t r u e or false, b u t not bcth. Much of the wol-2 In

algebra is concerned w i t h the problem of dec id lng whether or not

c c r t a l n sentences i n l ~ o l v i n e ; numerals arc true.

Problem Set 2 - lb

I . T e l l wn ich of the felio~ing sentences are true and which are f a l s e .

16 , (a) ( 3 -- 7)'; = 2 + 7 ( b ) 16 (f) - - I 4 - 3 = - A . ( a - 3 )

2

(b) '~(5) + ?4(8) = 4(13} 20 ( g ) 5 ( 7 + 3 ) = lo(, 4, 1)

3

1 1 12 12 (c) 2(5 -i F ) = 2(5) .+ 2 ( $ ) ((h ) = T = 5 ( p )

(d) 23 - 5(2) = 25 (i) 3 ( 8 + 2) = 6 x 5 7 + 9 ( e ) - = 2 7 + ( 5 ) 1 2 t (2 x 3 ) = 12(9)

( 1 3 + 7 ( 9 + 2 ) = ( 3 +- 7)(9 -1- 2 )

2. ':;rite a common name for each numeral.

( a ) 3 -+. 3(5 - 2 ) - ( 3 - 5) (b) Z.2(5) + 7.G

" 3 ( c ) (b)(5) + 2(" 33)

Isec. 2-11

3 . You are explaining the use of parentheses C G a f r i e n d i.rhz does

n o t know about them. I n s e r t parentheses in each o f the follow-

in5 expressions in such a !my t h a t th? expression will s t i l l be

a n ~ ~ e r a l for t h e same ntlmber. 1 ( a ) q r G c 3 ( c ) > x 3 - ! 4 x 3

( ) 2 . 5 - 1 6 . 2 ( a ) 3 x 5 - 4 4. I n s e r t parentheses in each of the fcllowing express ions so tha t

t h e r e s u l t i n g scn tence is truc.

(a) 7.0 - 7 - 3 = 6 ( 3 x 5 - 2 x , ! ! = 7

( b ) 3 5:7=36 ( k ) 3 7 5 - 2 ~ 4 = 5 ?

( c ) 3 . 5 + '7 = 22 1 1 (1) 1 2 x p - ~ x 9 = 5 1

( d ) 3 . 5 - h z 2 (m) 1 2 ~ ~ - ~ ~ 9 = 3 1 1 - ( e ) 3 - 5 - 4 = 1 1 1 1 ( n ) 1 2 ~ ~ - - ~ 9 = 1 8 3

( f ) 3 ,<- 5 1 2 y 3 = 23 ( 0 ) 3 + 4 * 6 + 1 = 4 9 (g ) 3 x 5 4 2 ~ 4 = S 4 ( p ) 3 4 - 9 ' 6 + 1 = 3 1

( h ) 3 ~ 5 + : ? r k = 6 8 ( q ) 3 " ; a . 6 + 1 = 4 3

(i) 3 s 5 - 2 ~ 3 = 3 6 (r) 3 . i 4 . 6 + I = 28

2-2. - Somc P r o p e r t i e s - of Addltio~ and Multiplication, >."tie end of Chapter 1. you r e c a l l e d additlon and Its

r e p r e s e n t a t i o n on the number line, 'We a re now going t o c o r l s i d ~ r

some of the n r o p e r t i e s q of addition. First of a l l , a d d i t i o n is a binary operation, in the sense t h a t it is always performed on two - numbers. This may not seem very reasonable at f i r s t s i g h t , since

you have o f t e n added l o n g strings of figures. Try an ~xper iment ;

on yourself. Try to add t h e numbers 7 , 8, 3 slmultaneously. No

*Property, :n the most familiar sense of the word, Is something you have. A p r o p e r t y of addition is something additi~n has; i . e . , a characteristic o f addition. A similar common usage of the word would be "sweetnees is a proper ty of sugar".

[ s e c . 2-21

matter hcw you attempr It, you are f o r c e d t o choose t w o of t h e nwn-

bers, add thern, ar.d thcr. add the thi .rd to th j . s sum. T;?us. t&en w e

writ- 7 + 8 3 , we really mean (7 -+ 8) 1- 3 o r 7 t (8 t 3 ) . NC 1 ; ~ s

par3ntheses here, as we have I n the p a s t , tc single out c e r t a i n

groups ohnunlbers to b e operated on f i r s t . Thus, (7 -t 8) + 3 i r n -

p l l e s that :qe add 7 and 8 and then add 3 t o that sun, so tha t we

think "15 3". Simila?ly, 7 + ( 8 - 3 j 1npl:e~ thzt t he sum of 8 and 3 is added to '(, g r l v i r g 7 t 11. Let us now g o one s L ~ p further and observe thaL 15 i 3 - 18, and 7 + 11 = 18. We have thus found

t ka t

(7 + 8) t 3 = 7 -I. ( 8 + 3 )

3s a true sentence. Check w h e t h e r o r not

13 a true sen5ence.

Check s i r n l l a r l y

(1.2 t 1.8) + 2.6 = 1.2 + (1.3 + 2.6), and

It 1s apparent t h a t these senten~es haye a common pat tern,

and they ~ l l t u rned o u t bo be t rue. He conzlude t h a z cl?or;r sen-

tence having t h i s p a t t e r n is t rue. This l o a s roge r ty of' add i t lon

of nwnbess; t.Je hope Lhat y o c w 3 1 1 t r y t n formulate it f a r yourself.

!Compare yuurn e f f o r t w i th a statement such as t h i s : If you ad3 a

zecand number to a flrst number, end then a t h i r d n m b e r t o thelr sum, t h e outcome is t h e same If you add t h e second numker and the

th i rd number an? then add their sum t o the f i r s t nmber .

This p r o p e r t y o f addition is called the a s s o c i e t i v e propert .y

of addition. T t Is one 3f the basic f a c t & about t h e number system-- - one of t h e f a c t s on wtllcl~ all o f nrathematf c s depends. I t l c i d e n t a l l ~

it Is o f t e n handy for cuttine Aown t h e w o r X in adding. In the "2 + 1'' is a n o t h e r name f o r 2, second cxarn~le abcve, f o r instance, - 1 so that "5 i ($ + prodcces a simpler addition t h a n

3 1 I, " ( 5 + 7) I . 3ml.larly, in t h e t h i r d example, 1.2 + 1.8 = 3;

which of t h e two expressions "(1.2 + 1.8) i- 2.6''

25

and " 1 . 2 + (1.8 + 2 . 6 ) ' leads to a simpleer second addition? 1. Now l e t us l o o k a t the f o u r t h exarnpl a. Neither "(i I ?) 2 1 1

,I 1 1 2 + 7 nor Y + [ d

7 + -) " g ive s a part:cularly s inp le f l r s t sum to help us 3 r-,

c I w i t h the second sum. If we could on ly add 7 t o 7 first, t h i s 1 would give I, and addlng - t o 1 i s easy! What tie would l i k e is to 2 1 1 2 I1 take t h e first i n d i c a t e d sum in t i 7 , a d write it i n s t e a d

,l 1 1 as (H t 3 - ) 1 1 , 1" order t o p u t 11$" next t o "3'. To do t h i s , we need

t o know t h a l

1 1 1 1 S + T = F + S

1s a t r ue sentence.

Although we can perfectly well do t he a r i t h e t l c t o check this,

let us first cons ide r some s i m p l e r examples. Is the sentence

3 + 5 = 5 + 3 t r ue? Perhaps you think: ''If I earc $3 today and $5 tornorrnw, I shz l l earn the same amount as If L earn $5 today and $3 tomorrow."

Perhaps Cohn t h i n k s : "Walking ; b e e mi les before l u n c h and f i v e

mlles a f t e r lunch covers the same distance as walking 5 miles be-

fore lunch and 3 miles af ter l u n c h . "

Now r e c a l i t.he nwnber 21ne. What Bid we find o u t , in Chapter

1, about moving from 0 to 5 and t h e n moving three units t o the

right.. and how d i d t h i s comparc with moving f r u m 0 to 3 and then

moving 5 u n i t s t o the r i g h t ? What does t h i s say about 5 + 3 and

3 -t 5? Similar ly, on t h e number l l n e , dec.l.3e w h ~ t h e r ;he f o l l o w -

ing are true sentences: 0 + E = 6 + 0 , 1 1 2F + 5 = 5 : 2-* 2

T h l s p roper ty of addikion 1 s probably yery faml l ic r to you.

It is called the - commutati7e p roper ty of addition. Try t o formu- - l a t e it for yourself , and compare your s t a t e m e n t with the follow-

: If two numbers are added in different orders , t h e results

are the same.

The a s s o c i a t i v e property o f addition t e l l s us that an indi-

cated sum may be wr i t t en with o r without parentheses as group in^

symbols, as we wish, The commutative property, in t u r n , tells us

t ha t t h e additions, which m e always o f two numbers a t a time, may

be performed in any order . For instance, i f we consider

32 + 16 -t 18 -+ 4, the associative property tells us that we do n o t have to use par-

entheses to group t h i s i n d i c a t e d sum, because any way we group I t

g i v e s the same r e su l t . We may, if we wish, just add 36 to 32 , then

18 t o t h e i r sum, and then 4 t o that sum. The commutative p r o p e r t y

t e l l s us that we may choose any o t h e r order . For purposes of mental arithmetic, it is eas ie r t o choose pairs whose sms are

multiples of 10 and consider them f i rs t , We may think of

" 3 2 + 16 -k 18 + 4" as " ( 3 2 + 18) t (16 + 4) ", where t h e indicated

sums mean that we first add 32 and 18 (because t h i s g i v e s t he "easy"

sum SO) , t h e n 16 and 4, and finaily t h e t w o partial sums 50 and 20.

I n our thinking, we first used the commutative proper ty to inter- change t h e 16 and the 18 in the or ig ina l indicated sxm.

Problem Set 2-2a -- 1. Consider various ways t o do the following computations mentally,

and find the one t h a t seems easlest (if there is o n e ) . Then

perform t h e indicated additions in the eas ies t way.

(a ) 6 + ( 8 + 4) 2 2 1 8 ( b ) 5 + 5 + 1 + 1 + 5

1 2 !I ( e ) 2- + 35 + 6 + 73-

5 1 (f) ( P 5 + 1) + 6

5 8

(g) (1 .8 + 2.1) + ( 1 . G + -9) + 1 . 2

3 1 ( d ) 7i + 5 (h) ( 8 t 7 ) + 4 + ( 3 + 6 )

2. From the t i p o f a mouse's nose t o the back of his head Fs 32 W

millimeters; from the back of h i s head t o the base o f hls t a i l t 71 miilimeters; from t h e base of h i s tail to t h e t i p o f his a tail 7 6 millimeters. What is the length of t h e mouse from the

t i p of h i s nose t o t h e t i p of his tail? Is he the same l e n g t h t a' from the t i p of h i s tall t o the tlp o f his nose? Why do you ! 8

t h ink we Tnclude t h i s exercise here? 1

We shall now l ook at the corresponding properties of multipli- [ nl ca t i on . Cons ider this sentence,

1 1 (7 x 6 ) r 5 = 7 x (6 x 5) .

Check whether o r n o t t h i s is a t rue sentence; be sure to carry out the multiplications as i nd i ca t ed . Similar ly check t h e truth of the

sentences

and 3 3

( T x 7 ) x 4 = = x (7 X ' i) .

Once again, we f i n d t h a t these sentences are t r u e , and t h a t t h e y

f i t a common pattern. We conclude that a l l sentences of t h i s

pa t te rn are true, and call this property o f multiplication the

associative proper ty - of multiplication, Reca l l y?ur effort towards

stating in words the associative proper ty of a d d i t i o n , and make a

similar statement f o r the associa t ive proper ty o f multiplication.

In the examples above, the indicated multiplications were n o t

always equally d l f f l c u l t . In the first sentence, '17 u 6 ) x;",

1 is more work t o c a r r y o u t than "7 x ( 6 x gY' whlch becomes "42 x 7 , which becomes jus t ''7 x 2". Which phrase in t he second sentence

is easier to handle? Thus the associative proper ty of multiplica-

tion, j u s t as the associative proper ty of addition, can be used

t o simplify mental arithmetic.

In the third sentence, neither form l e d t o a p a r t i c u l a r l y

simple second product. The eas ie s t th ing t o do would be to take 3 3 x 4 first, even thougn and 4 are no t adjacent in e i the r phrase,

and then to mult iply by 7 . Is this l ega l ? We could be s u r e that

it is if we knew t h a t

were a t r ue sentence. This is a poss ib le interchange we might l i k e

t o make before applying the assoc ia t ive proper ty . (What would be another?)

As in the previous section, make up some simple problems

about walking or earning money which v e r i f y the t r u t h o f a sentence

such as

2 X 5 = 5 X 2 .

You have also had the experience in Chapter 1 of seeing on the

number line tha t 2 X 3 = 3 X 2

is a true sentence. You a l s o know, from al7iL1mletic, t h a t you may

perfom long m u l t i ~ l i c a t i o n in either o r d e r , and j?ou have probably

used t h i s to check your work :

A l l these, in various 3ituations. are instances o f t h ~

commutative propcr2y of rnu l t i p l i ca t j on : -- If two nlmbers are to be

multiplied, t h e y niay b e multiplied in ei;her o r d e r with the same

result . A s in t h e case of addition, t h ~ ass3ciativc and commutati,~e

p r o p e r t i e s of multiplPcation bell us that we may, in a n i c d i c a t e d

prodcct, k h i n k of t h e ambers grouped a s we choose, and msy a l s o

-earrange such a produc t at will. Thus in finding 9 x 2 x 3 x 50, it is con:ferilent to handlc 2 x 50 first,, and t h e 1 t o m u l t i p l y

3 x 3 , or 37, by 100.

Problen Set 2-2b

1. Consider v a r i o u s ways t o do t h e following co~putatl~ns mentally, and f i n d the one that seer.s eas ies t (if there is

one). Then p e r f o m the indicated operations in the e a s i e s t way.

(a) 'I x 7 x 25 1 1 5 (f) 2 Q X g

1 (b) 5 x (86 r 5 ) ( E ) 6 x 8 x 125

( e ) 73+62-27 ( h ) (1.25) x 5.5 x 8 (d) ( 3 x 4 ) x (7 x 2 5 ) (1) (2 x 5 ) x 1.97

2. Is it easier t o compute

95'( 222 x 222 o r ? - x z

H:vr ~ lb3ut 3.E9 or 1 i l l

:< 141 x 3.89 ?

The D i s t r i b u t i v e P roper ty . 2-3 - John c o l l e c t e d money in h i s homeroom. On Tuesday, 7 people

gave him 15$ each, and on Wednesday, 3 people gave him 15# each.

How much money d i d he c o l l e c t ? He f igured ,

15(7) + 15(3) =

105 + 45 =

150.

So he c o l l e c t e d $1.50.

But now we shall ask him to keep d i f f e r e n t records . S ince

everyone gave him the same amount, it is a l s o p o s s i b l e to keep an

account only of t h e number of people who have pa id him, and then

t o multiply the t o t a l number by 15. Then his f igur ing l o o k s l i k e

this :

The final result is the same in both methods of keeping accounts;

theref o r e

15(7) + 1 5 ( 3 ) = 15(7 t 3) is a t rue sentence. Since t h e above t rue s e n t e n c e means tha%

15(7) + 15(3) and 15(7 + 3 ) are names f o r the same number, we might

also have written

15(7 + 3 ) = 15(7 ) -t 1 5 ( 3 ) . h%lf the money John collected is to be used f o r one g i f t , and

one t h i r d of it f o r another. How much i s s p e n t ? Again, t h e com-

pu ta t ion can be performed i n two ways:

As before, we have found a true sentence,

Another way o f writing the same t rue sentence dould be

Let -2s t-y another exm-ple. M r . Junes 2med a c - t t y l o t , 150

feet deep, :vifh a f r o n t of 162.5 f e e t . Adjacent LJ h i 3 lot, and

separated f r o 3 it by a Tcnce, i s another lot with the same depth,

but wi th a mnt of 3nly 3'1.5 :eet. What a r e the areas , In square f e e t , o f ~ c k cT these two l o t s , and :(fiat; is th~ir s7m? M r . Jmss SWJS the seculrd 2 0 t and r e m ~ v a ~ t h e fence . Now w l i a t I s the area

3f' the lot? Tqe nl.mber of squ21-e f e e t In t h e n s w lat is

150(162.5 - 3 7 , 5 ) =

150(200) = - 162.5' - -. - - - - - 37s' 3,3000.

The t o t a l area of tfie Ewo W

seyaratc l o t s i s 150' L: 2

15~(>62.5) + 150(37.5) = W L 2 4 3 - { 5 + 5.625 =

3 OOUO . ~ I U S ,

I T , E ~ ua l o c k more closely at ~ W O of our t r u e sent?nces. iq-2

i : ~ o te

15(7) + 15(3) = 15(7 t 3 ) .

15(7) represcn5s one r r m k c r , which ire hs -~e zhoson t o vrl-lte 23 ar:

i nd ica t e3 7radx: ; so does 15(2) . Thus 2 5 ( 1 ) s 15(3) Is an i nd i -

ca ted sum or t t m n-mbe~s , On t h e c u ~ e r hand, 7 -c 3 regresents a

number ~ ~ : h i c , l ~ we have chosen 5 0 wrlte a s an I n d i c a t e d zum, and so

1.3(7 t 5 ) is an Fndlcated groduc?,. %us the sentence 15(7 ) .i- 15(3 ) - 15(7 +- 3)

asser23 that ti12 ind ica ted sum and Ehe i r l d i c a t c d p r n t u c t are names ~ Z F t h e same nunbcr. P e true sentenc

1 1 I i50(& . + -1 3 = SO(,) i- 150(-) L 3

mn!ics a s i m i l a r sta:ement.

Problem 3e t 2 - 3 3

1. TesS the t r u t h of t h e following sen tences :

( a ) 2.4 + 5.4 = 7.9

( b ) 15.2 = 7 . 7 + 8-2 ( 25(4U + 3 ) = 25(Ji0) + 25(3) I d ) 3(2) + 6C3) = 9(6) ( 13(19 + 1) = 13(19 ) -I 13 (1 )

1 1 1 1 (f) ? ( F + a) = pH A 2~

( h ) 3 ( 2 .?) Z(1.5) - S ( 2 . 5 + 1 .TI

It appears that we fsund a p a t t e r n by vrhinh t r u e s e r t e n ~ e s

may be formed. Try to say t o yourse l f in varfous ways what; this

pattern Is . After you have made an effort, wmpare your result

with t h e following: The p r o d . ~ c t of a n m b e r times the s w n of two

I otYLers i s the same as the product of t h e first and secohd p l u s t h e

prcduc t of t h e first and t h i r d . . This p r o p e r t y is c a l l e d - t h e

d is t r ibu: ive p r o p e r 5 for mu1 tipllcation - o v e r - addition, o r j u s t , as we s h a l l f r e q u e n t l y say, the distributive property.

As in the case of t h e o t h e r p r o p e r t i e s we have s tud i ed , the d i s -

I I

t r l b u t i w p r o ~ e r t y has rn:lch t o do with arithmetic, both w i t h d e v i c e s

1 for mental facility and with the very foundations cf the s u b j e c t .

/ In our f i r s t exampl?, t h e comparison in arithmetic labor between

: 1115(7) -1- 1 5 ( 2 ) " and "15(7 -t 3 ) " favors t h e i nd i ca t ed product, be- i cause '; t 3, or 10, 'leads t o an easy mu1:iplication. In t h e next

example, hovrever, t h e comparison between 11150($) i- 150($) " and - I I T I "150(: -t- =;) Ssvors the Ind ica ted sYm, because it is more work to c 3 1 1 add t h e f ract ims - and 7 than it i s t o add 75 and T O . Ifnich form 2 V

was eas ler in the t h i r d example? In the sentences of W l e m Set 2-3a?

More imporbant t han these niceties of mental manipulation is t h e

rno le . of the distributive p r o p e r t y in much o f our aritihme t i c tedmique

suck! as, f o r example, I n lone; multiplication. How d o we perform

62

This r e a l l y neans tha t we take 62(20 t 3 ) as 62(29) + 6 2 ( 3 ) , o r 12'10 136. (The "0" at t h e end of "1240" is understood in our 1

l o n g r n u l t i p l l c a t i o n form.) Thus the distributive proper ty is t h e

foundat ion o:' t h i s standard technique . Suppose we wish ta cons lder several ways of c ~ m p u t i n ~ t h e i n d i -

c a t e d product

Thls phrase does n o t q u i t e fit t h e pa t t e rn of t h e distributive

p r o p e r t y as we have discussed it so fa r . You c a n probably guess

on t h e bacis of your p rev icus experience , that 2

i s a t r u e sentence. L e t us, however, see how the properties as

we have discovered ;hem thus far p e r m i t us to conclude t n e t r u t h

of t h i s sentence. First we ]mow that

is a true sentence (by what property of rn .x l t ip l i ca t ion?) . ?Tien 3. we may apply the d k t r i b u t i v e property as we have dissevered I t

thus f a r t o wfte

and apply the commutative proper ty twice nore t o wri te

The last s t e p , which would be unnecessary if we were just trylrg 1 t o compute "(I + Ti)1211 in a simple fashion, finally leads to the

A d e s i r e d sentence, n a m e l y

This sentence, once agaln, seems t o have a slmple form, and in fact suggests an a1 ternete p a t t e r n for the d l s t r l b u t i v e property which

Isec. 2-31

33

I s obtained frcm u!lr p r e y i ~ u s pattern ty ae>bcral a p p l i c a t i o n s of tk

commutative p r o p e r t y of multiplication. I n your own words, s t a t e

t h i s alternate p a t t c ~ ? n . What pattorn is suggeszed by t h o sentence 1 1 1 1 (g)12 i ($12 = (; J + h)lp 7

Problem S e t 2-3b pp - -

1, Follow t h e paY.tt?rn of any c o n v e n i e n t form o f the distributive

property in cornplct ing each of t h e following as a true sentence:

( a ) lF(3 + a ) = 12( ) - 12( )

(b) 3 ( 4 (7) = 3 ( 5 + 7) ( c ) (2.5 - '..!I) - - (2.5) , + ( k? ) !~

I d ) 1 1 YE + F )

( 1 7( 1 " 6( = 13( (f) (3 .i 1 1 ) ~ =

1 2 Consider in two d i f f e r e n t ways and in each case decide which,

if any, is t h e easier computation. Ther. perf c r n Lhc. i n d i c a t e d

o p e r a t i o n s in the eas ie r way.

( a ) ?7(&) + 27(6) ( c ) ( 2 . 3 . I 11.6) -- 7.7

(b) (+)12 " + ($132 3 (f) 6 ( F i $1 I

( c ) ( . 3 6 - .2.4).5 ( g ) :71(.8) + .?{.:l)

1 ( d ) 12(5 - 5 ) (h) 3 ( 2 -. 7 -I 6 - - 5)

/ 3. The f i g u r e belo:: sho7vs a number or' triansles

I D I H

I : h a t re1af; ian hoicla b e t v ~ e e n t h e area o i triangle ACD and t h e

I areas cf Lhe L w c f;riangles EFT and PC97 Use the formula f o r

( the area nf a trian~le in terns al le:-gths of base and u l t i -

I tude

[ s e c . 2-21

. Write a common name f o r

1 1 (~int) Thin!< of i 5 as one numera l , and don't start working wti l you have thought of a way of doing this exercise which

i s n ' t much work.

5 l:J~ite the common names f o r

6. Write cormnon names f o r

(a) 3 s (17 + 9) x 3 ( g > 7 ( 4 ) t 42

(b) = L p (h) 3(?) 1 + 5 2 1

( c ) 5 ( 3 + 8)10 (I> 3 ( 1 7 ) .I 12

1 3 ( 7 ) + 3 ( 1 1 ) ( J ) 6 (19) +

(4 3 0 ) + 3 ( 2 ) 1 ( Ic ) (10 -+ 2 ) x !: x - 2

(f) 3 ( 7 ) r- 6

2- 4. Variables.

C o n s l d ~ r , f o r a moment, a simple exerclse in menta l arithmetic:

"Take 6 , add 2, multiply by 7 , and d i v i d e by 4". Fol lowing these Instructions, you w i l l no doubt think of the

succession of numbers

6 , 8 , 56, 14 and obse rve t h a t 1'1 is t h e answer t o the exercise. Pretend now

tha t your best f r i e n d is a b s e n t from c lass and t h a t you have

promised to g i v e him a de t a i l ed r e p o r t on the dayt s work. lt1l.th

your f r i e n d in mind, you write down the instructions for each s t e p

of t h e exercise as fol lows:

Does t h i s method o f vr r i t ing the exercise g i v e more information o r

less Information: It c l e a r l y has the advantage of showing exac t ly

what o p e r a t i ons a r e involved in each s t e p , but; it does have the

disadvantage of !lot endim up w i t h an answer to the e x e r c i s e . On

t he o t h e r hand, t h e phrase "w" is a numeral for the answer

14, a f a c t which is r e a d i l y shown by doing t he i n d i c a t e d arithmetic.

This is an imaginary situation in 1'111ich you were led t o record

f o r your friend t!ze form of the exercise, r a t h e r t h a n just the

answer. It illustrates a p o i n t of' view which is basic t o mathe-

mat ics . Tnere w i l l be many places in this course where it is t h e

~ t t e r n or forn o f a problem which I s of primary importance rather .-

than the answer. As a matter of f a c t , we are rarely Interested only in the answer to a problem.

P y the exe rc i s e with t h e following instructions:

" ~ a k e 7 , multiply by 3 , add 3 ,

m u l t i p l y by 2 , and d i v i d e by 12. I'

What is the phrase which shows a l l of the opera t ions? Is it a

numeral for the same number you ob ta ined mentally?

L c t us now d o one of these exerc ises w l t h the added fea tu re

t h a t you are permitted t o choose a t t h e start any one of t h e num-

bers from the s e t

s = 11, 2 , 3 , . . . , 10ooj.

The instructions this t i m e are : " ~ a , k e any n m b e r f rom S, multiply by 3 , add 12, d iv ide by 3: and subtract 4."

This exerc lse might be t h o w h t of as actually cons i s t i ng of

1000 d i f f e r e n t exercises in arithmetic, one for each choice of a

starting n m b e r f rom the s e t S. Consider the exercise obtained

by starting with tile n m b e r 17. The a r i t h e t . i c method and the

pattern method lead to the f o l l o w i n g s t e p s :

Eat tern I . .('

2(1;)

.3(1() 2- 12

11; 9rher. ;';21'13~, :ilc f ' k ~ a l pl.~??rr- n b t a i n 4 i : ~ t h ~ "3a t t~ r1 : " 13 a. 1 . o 7 , T2.y xone norc c i -~oiccs f'rorn t:?e s e t 3 . ;;!ill you

2iy;:;ays e:~c? up ,i: t : ~ ? :,,ut:fLcr you c k ~ o z ~ 3t t ! ; ~ S ~ . Z P ~ ; ,: Onc nc',hcd

9: at-,s;.:e:-;fi;' ti::: 33e.;';iL>r. :.:3121a be to 8,~2r!.: cv.t; er.ch 0: t h e 105C8

' : T e f i ,zc, - , .?c;.l j:a~:e zlr ;a<:? cu.^:p,s.?<, j7j-r,p !,:r~rk-

i n 2 t:?c p3t tc - rn for. :3c-..rerr2 c z s c z , S!~at ft r,:y not, b c x c e s s a r y

- c:-~osen :r.om \:he zed; ,?. !-r, Tac t , :'.? i.:e r ~ f ' c ? t;c; t i l ~ number chosen

by t h e xord " n m b e r " , Lhe step; in t h e cxel-cise b?corne:

1:: o r d e r tc savc r ! ? l t l r l i ~ ; , 3er:ctc t h c chczcn n u - ~ S c r by "n". Rle1-1 ,

t h c s:,eps bccomc : . .

I J 1 7 Note tha t n I s used her9 as a numeral f o r the chosen number and

t h a t t h e ph-aze in each of t h e other steps is a l s o a r,umeral.

(Thus, 2f n happens t o be 17, t h e n t h e Lndicated product "2n1' is "3n t- I? - 'I a numeral f 3r 51. ) Ln parclcular , :he phrase l o a 3

numeral rclr the "answer" to the exe-clse. Morecver, by t h e dis- t r l b u t i v c p r o p e ~ t y for numbers,

3n + 12 = 3n 4. 3 ( 4 ) = 3 (n - F 4). Henc f

Thel-ef o r e ,

Since 'In" can represent % particular Element ul" Lhc set S , wc

conclude t k a t t h e end r e s u l t In t h i s exercise is indeed alnayc ths

sane as t h e number chosen at the start.

The above d i s c u s s i o n Illustrates the grezk p c w r of methods

based on pattern o r form rather t h ~ n on simple a r i t h r n c t i c . The

m e t h ~ d ? in a sense, enables us to replace 1000 d i f f e r e n t arithmetic

problems lyr a s lng le problem! Would =he discussion of the exercise be chenged in any e s s e n t i a l

way if we had decided to denote t h e chosen rmmber f r o m S by some l e t t e r o t h e r than n, say rn or x'?

A l e t t e r used, as "n" was used i n t 5 e above exercise, :o d e n ~ t e

one of a given s e t of numbers, is c a l l e d a var lable . In a gi7jen

computation involving a v a r i a b l e , ;he va~iable Fs a numeral which

represents a d e f i n l t e though unspe~ifled nmbe- from a g i v e n s e t

of a&vlssible r:u;nbers. The admissible n - m b c r s r o r the variable "r?"

in the above exercloe a r e th? who15 nmbers from I to 1000. A num-

ber hkich a gl-/en variable can r c p ~ c o e n t is c a l l e d a value o f the

variable. The se t of values of a variable is sometimes c a l l e d i;s

domain. The donmin of t h e v a r i a b l e "n" in t he a b w e cxercise is -- -- the set S = 1 1 , 2 , 3 , . . ., 1000). Unless t h e domain of a

variable is spec l f i . c a l l y skated, we s h a l l assume :t Lo be t h e set

of a l l numbers. For t h e time being, we a r e considering only the

numbers of arithnetic.

Problem ,Set P - ) i . -- I. If t h e sum of a c c r t a i n number t and 3 is doubled , which '2f

the following would b e a c o r r e c t form:

2 t + 3 or 2 ( t + 3 ) ?

2 . If 5 is a d d e d to t w i c e a certain number n and the sum is

divided by 3 , which is t h a correct form:

3. If one-fnurth < ~ r a c e r t a i n :-lumber x ie ~ d d e d Co m e t h i r d ? f

f o u r times the same number, which is the correct form:

4. If the number o t gallons of m l l k purchased i s y , whlch is the

co r rec t form f o r t h e n~mtber o f quart bottl~s t h a t wlll eontair ,

it:

4, 0, f ?

5. If a is t h e number of f e e t i n the leng th of a c e r t a i n r c c t -

angle a n d b is the number or" f e e t in t he w i d t h gf the same

r ec ta rz le , i s e i t h e r form the co r rec t form for t h e perimeter:

a + b o r at., 7 3 6. If 2 is 2, b Is 3 , c Is I - , m 1s 1 and n is O . then what '4

I s the v a l u e of (that iz, what number is r e p r e s e n t e d b y ) :

(a) 6b a c ( f ) n ( c + ac)

( c ) 6 ( b + ac) (h) m(b - )LC)

{e) nr: + ac ( J ) a + 2(b - rn) 7. Find common names for

( a ) ZC 32, when C is 85

[ s e c . 2-41

(b 1 h(a+ 2 ' I , when h 1 3 4. a is 3, and b 1s 5

( c ) P ( 1 + rt), when P is 500, I) is 0.0k. t is 2

( e ) jwh, vhen 1 is 18, w i s 10, h is 6 8. If a is 3 , b is 2 , and c Is li, t1Icl1 what is t h r value of'

9 . Summarize tile new Ideas, including d e X n i t i o n s , which ha.:t?

been introduced in t h i s chapter.

3-1. Sentences, True and False --- Lhon we make assertions about numbers we write sentences, such

as

( 3 + 1)(5 + 2 ) = lo

Remember t h a t a sentence is either t r u e o r false, but n o t b o t h .

This particular sentence i.s fa l se . Some sentences, such as the one abo-le, involve the ve rb "=",

meaning "is" or "is equal to". There are o t h e r v e r b forms that we sha l l use in mathematical sen tences . For example, the symbol

"#" >dill mean "is not1 ' or "is n o t equal to". Then

8 + 4 # 28.2 1s a true sentence, and

24 3 + 4 # T

i s a f a l s e sentence,

'

Problem S e t 3-1 -- Ifi~ LC:? .?A? "Y?e f 71 1 ?vLng ~en5en~es aye ?,rue ', Which are f a 1 ce ?

1. 4 + 8 = 1 0 + 5 11. 65 x 1 = 65 2 e + 3 = 1 0 i - 1 12. 1 3 ~ 0 = 1 : !

3 . Open Sentences -- We have no t r oub l e deciding whethe? o r not a numerical sen-

tence Is t r u ~ , h c a u s e such a sentence l ~ ~ v o l v e s s p e c l f l c numbers. Ccns l d e r the sentence

x + 3 = 7 . Is this scntcnce t l h u e ? You wlll protest t h a t you d o n ' t know what

nu-nber "x 'I represents ; w i t h o u t this inf oma t ion you cannot dec ide .

In t he sane way you cannot d e c i d e whetker the s e n t e n c e , "He is a

d u d o r , " 1s true mtil "he" is Identified. In t h i s sense, tke va~iable "x" I s used In much t h e same way as a pronoun in o r d i n a r y

languaze . Consider the sentence

Sn + 1 2 = 3 ( n + 41, with which t a l e rvorlked in Section 2 - 4 whcn the i d e a uf a variable

! a s f lrst i n t r o d u c e d , Again we cannot d e c t d e whether thi .3 sentence

Is t r u e E n t h e bas i s a f the s~nt3nce a lone , but here we have a

different sitvat ion. A= before, we could decide if we knew what

number "n" represents . Eut i n t h i s case we can d e c i d e without

knowing t he value of n . Ik can r e c a l l a g e n e r a l p r o p e r t y of

number; to sl-low that ;his ser.tence is true no matter number - "nu =resents . ( \ h a t d3d rJje all this general p r o y e r t g o f numbers?)

We say that sen tences such as

x i 3 = 7

and h

J n + 12 = 3 ( n + $1, ~vhicn c~ntain variabl~s, are Dpen sen tences . The word "open" is

suggested by t h e r a z t that wc do n o t knox whether tney ar2 t rue wi thou t more i n f o r m t i o n . A n opcn o e n t e n ~ e 1 s a s e n t e n c e I n v o l v i r g

one or nore variables, and t h s question o f whether it is true is

le l ' t open urtil we have e c o ~ h additional information to d e c i d e . In t h e same way, a phrase Invulvl .ng one o r more var'ables Is ca l l ed

ax open phrase ,

Problem Set 3-2a -- In each of the fol lowing open s e n t e n c e s . de t e rn ine whether

t h e s e n t e n c e is true If the v a r i a b l e s have t h e suggested -.:slues:

1 7 t x = 12: l e t x be 5 2 , 7 + x 1 1 2 : l e t x be 5 3 . y + 9 2' 11: l e t y be 6 4. t + 9 = 11: l e t t be 6

5 5x + 1 7 # 3 ; let x be 3; l e t x be 4

6 . 2y + 5x = 23; let x be 4 and y be 3 ; let x be 3 and y

be 4 7 . 2a - 5 # (2a + 4) - b ; l e t a be 3 and b be 9: l ek a b e

3 and b be 9 8. 5m + x = (2m + 3 ) t x; l e t x be 4

If we are g iven an open sen tence , and t h e domain of t h e v a r i a b l e

is specified, how shall we determine the va lues , if any, of t he

variable that will make it a t rue s e n t e n c e ? We could guess various

numbers u n t i l we h i t on a "truth" number, but a f t e r the f i r s t guess,

a b i t of t h i n k i n g could guide us.

Let us experiment with the open sentence "2x - 11 - 6". A s a

first guess. try a number x large enough so that 2x I s greater

than 11; l e t x be 9 , Then

Z X - 11 = 2 ( 9 ) - 11 = 7.

Thus, the numeral on the lest represents 7, which is different

from 6 . Apparently 9 was t o o large; so we try 8 for x. Then

- 11 = 2 ( 8 ) - 11

= 5 . Here the numeral on t h e left represen ts 5 , which is a l s o d i f f e r e n t

from 6 , S i n c e d was t o o small, ne try a number between 8 and 9. 1 say b. Then L 1 2x - 11 = 2(%) - 11

= 6 . "6 = 6" is a true sentence; so we find tha t the open s e n t e n c e

1 "2x - 11 = 6" Is true if x is %. Do you t h i n l r t he re are o t h e r

values oi x making it t r u e ' Do you think every cpen s e n t e n c e has

I value of' t h e varla'ble which makes I t true 1: which ma.ies it f a l se?

Frcblem S e t 3-?b -- - Determine what numbers, if any, will make each of t n e r o l l u w -

ing open scntenc~s t r u e :

1. 12 - ;,T = 3 6. ' I x - A x = 14

2 qy '- 5 = Q5 7. s + 3 = s t > ?

3 . 3x - 2 = l ( 3 2 , t + 2t # 2'{ + 3t !I. 3x - .2 = 15 9 . t + 3 = 3 + t

5 . 'IX -7 3~ y 11: 10. ( X + I)? # PX -! 2 i

If a vzriable occurs in an open sen tence i n the form " a . a f f t

meaning ''a mul:ripllied by a", it is convenient t o write "a.al' ,7 as " < " "

d , r e a d "a squared". i

Problem S e t 3-2c -- Try t o find values of the variables which make t.he following

open sentences t r u e : 2 ( a ) x = :J ( e ) r. -1 2 - 9

( b ) 5

- x' r 0 P (f) ( x - 1) = 4 3

( c ) x - x 2 ( g ) - x = 0

3 2 ( d ) x - 7 - 3 (h) x - + 7 = 7

[*hat is a ~ ' a l u e of x f o r which

is a t r u e sentence?

A number of interest t o us l a t e r 1s a value of x rr L' x = 2" i s a t r u e sentence. We c a l l t h i s number

root of 2, and mite it G. Later you w i l l flnd

is t he coordinate o f a p o i n t on t he number 1 F n e .

where on the number l i n e would it; l l e r

for whlch

t h e square

t h a t vT Approximate

3-3. Truth Set5 3f Open Sen tences ---- Let the ??ma!.? n f t':e - r r i = r b l c i r t!-;e se?tep?ce

1 1 x = 7 >,

be t h e s e t a h a l l numbars or' arithmetic. If we speclfy that x

has a p a r t i c u l a r yalue, the!> the 1-esult ing sentence i s t r u e or is

talse . Par I n s tsnce , If x is --- the sentence ---- is -

0 3 1 - 3 - 7 rqa1se

1 3 ~ 1 - 7 . f a l s e

I - 1 3 + - = 7 2 2 i'al s e

2 3 + 2 = 7 f a l s e 1 , - 3 + 4 = 7 t r u e

C 3 + 6 = 7 f a l s e In t h i s way the sentence "3 t x = 7" can b e thought of as a

sorter: it s3rts the domain of the va r i ab l e inzo two subsets. i ~ u s t as you might s o r t a deck of ca rds into two subseYs. blaci i and

1 red, t h e domain of t h e variable is s o r t e d lnto a subset 3: 311

1 those numbers which make t h e sentence t r u e and ano the r s u b s e t zf

a l l those numbers ~ h i c h make the sentence fa lse . iiere we sFte t ha t

4 belongs t o :he f i r a t s u b s e t , while F, 1 , t belong t u t h e

/ second subse t .

i The t r u t h s ~ t of an open sentence in ore v a r i a b l e is t h e set --

I of all those numbers from the domain of t he varlable whi:h make the sentence t rue . If we do n o t s p e c i f y o t h e r w i s e we shall con-

/ tinue t o assume that t h e dornaln of the v a r l a b l s is t h e set o l 3 7 1

[ numbers af' arlthmrtlc. ( R e c a l l t h a t t n e numbers of arithneti:

consist c f 0 a d all nurr~bers w h i c h a re coord ina tes of p c i r t s t,n

I the r i g h t ~ f ' O. )

Problem S e t 3-3a -- 1, T e s t w!ret;tler' t h e number be longs t o the Eruth set 3 f t ne p i .~er?

open sen tence :

(a) 7 T x = i2; 5

(b) 7 + ? f 11: 6

( c 5x 4- 1

# 3; 3 I

I ( e ) x -:- - = 2.

X 1 3

(f) 3m = rn .I 2m: 15 3

( g ) n- + Pn f n ( n - 2 ) ; 3

2. With each of t h e following open s e n t e n c e z i s given a set which

contains a l l the n m b e r s b e l o n g i r g to i5o t r u t h s e t , with

p l ~ s s l b l ) r some more. You are to f i n d Lht? truth set.

1 2 ( 3 ) S ( X - 5) = 17; 10, j. 3, 1)

(b) x' - (kx - 3 ) = c ; (1, 2, 3, 2)

2 1 7 1 ( x + - - -x = 0; [1, 2 , 6, t: t.

( a ) x -, y_ - :J; (1, 2 , 31

(e) X ( X t i = 2x; 10, 1, 2)

(0 5x + - / = 3 ; lo, 2 , L j

1 (5) x :- 1 = 5x - 1.; { I . 7, 21

) x + 7 - x - ' ( ; [O, 2 , 3 )

3 . I:'rit~ en oper, ser~zen<e whose truth s e t I s t h e null set P.

i hny i'orm;llas used in s c l c n c e n:ld business are i:i t he forms

of opei? sente:~:es in several v2r iable i ; . For example, :he f urmula I :.I = iEh ,I

I;_s used t o find ',he -i:olume uf' a cone. The var iab lc n r e p r e z e n t s I* ?.he n-umber of > m i t z in Lke height of the cone ; B represents the !

nu:incr of square uni. ts In the base; V r e g r e s ~ n 2 2 t n e number of

c u b i t ~ m i t s in ',he v j ~ l ~ i S . ihen r.:alues a r e s p e ~ i f l c d f o r a l l but

one o: :;he . ;ariables in such a formula, the resu l5 ing open z e ~ t e n n c e ' i contain:; ,Jne remainLng var ieLle . Ti le3 t h e t r u t h n e t 09 t h i s sen- 1 tencc gives i n f o m a Z i o n &out the nurr.ber representei by this

~ r a r l a b l e . I

Continuing "he cxmple , l e t us consider a partiruler ?one whose

volume is 66 cubic f e e t and the area of whose base is 33 square f 'cct . *om %is Ini 'ornatlon we de te rmine ti:at 7 is 66 an3 B is 3 3 ,

and we w y i t e the ccrrcspondlng open s e n ~ c n z e in one va r i ab l e h ,

[ s e c . 3-33 I

1 f,t = q ( 3 3 ) h .

L.

h e t n ' 3 h :;et o? t h i s scnto!?.:c is { E l . Applying This information t o t h e cone, Tvie f i n d that ' n e he igh t

of t h e cone is I ' ee t .

Problcm Se t 3-3b -- 1. The formu1.a ;awd ko change a temperature F muasured in degrees

32hrenhc i t zo ;ha c o r r e s p c n d i n ~ : temperature C in decrees

Cent,i;:,rade i s 5 C = - ( F - 3 2 ) . 9

Find t h e -:slue of' C when 7 is 86. 2. The f o r n u l a used to compute simple i n t e r e s t is

i = p r t .

where I is ?he number of m i o l l a r s of interest, p I s t h e nun-

ber of dollars of p r i n c i p a l , r is the i n t e r e s t rate, and t

is t he number of years. Find t h e v ~ l u e of t when i is

L P G , r is 0.0!:, and p is 1000.

3. A form:lla used in physics t.s ?elate pressure and volume of a

g iven a?omt o f a gas at constan5 kernpsraturc is

pv = PV, where ?J is :he number of cubic unltz of :,olume a t P units

of pressure and v is t h e number of cubic u r i t s of volume at p tulits cf pressure. Find t h e value o f V when is 600,

P is 7 5 , and p is 15.

~ 4 . The fmml-ila Tor t h e area of a t r a p e z o i d is

where A is ti;€ n m b e r of square units in the area, B is the

number of units in ohe one base, h is t h e number of units in

t h e other base , and h Is the nunber of u n i z s in the he igh t .

Find t h e -%~alue o f B when A is 20, b is ' 8 , and h I s j-!.

.3 -4 . Graphs of Truth S e t s --- The r raph ol' a z e t S of numbers , :. t v r i l I. kp c ~ : 8 ' 1 l erl , is

the s e t of a l l p o i n t s on t he number l i n e corresponding t o t h e num-

bers of S, and only those p o i n t s .

[ sec . 3-41

T h u s , the graph of the truth s e t of an open sentence c o n t a i n - ------ iw one var iab le is the s e t of a l l points on t he number

l i n e whose coordinates are the values o f the varl-able which

nlxe t h e op?n sentence t r u e . L e t us draw t h e graphs of a

few bpen sentences. Sentence Truth S e t Gr;lph

(aj I = 2 (23 I L A h I 1 1 I

0 I 2 3 4 5

A l l numbers A 1 A 1

of arithmetic I 2 4 5

except 3

( c ) 3 + x = (7 + x) - 4 A l l numbers of arithmetic O I 2 3 4

(f) 2x -t 1 = 2 ( x + 1) la 1 I I I L I

0 I 2 3 4 5 (Graph contains no p o i n t s )

You w i l l n o t i c e in (b) that we i n d i c a t e that a p o i n t is included

in the graph if it is marked w i t h a heavy dot, but n o t included

I f it is circled. The heavy lines i nd ica t e a l l t h e p o i n t s t h a t

are covered . The arrow at the r i g h t end of the number l i n e i n (b)

and ( c ) indicates t ha t a l l o f the points to the r i g h t are on t h e

graph.

Problem S e t 3-4 -- S 5 a t e t h e +rut17 s e t 9 f e ~ r _ t l apen s e n t e n c e ~ n d d r p w i t s ~ r ~ p 3 :

- 1. x 4 , = 10 6. 3 + x f 6

2. ? x = x - t 3 7 . 2 x + 3 = 8

S . x + x f 2x 8. 5 f 3 n + 1 " x + 3 = 3 + x 9. ~ . ( l ) f y

5. ( x I (0 ) = x 10. x2 = 2x

[ s e c . 3 -41

3 - 5 . Sentences I n v o l v i n g h e q u a l i t i e s - If we cons ider any two d i f f e r e n t numbers, t hen one is l ess

than t h e o t h e r . Is t h i s always t rue? This suggest,^ ano ther ve rb

form that; we shall use in numerical sen tences . We use t he symbol

'1<'1 t~1 mean "is less than" and ">" to mean "is greater than". To avoid confusing these symbols, remember that in a t rue

sentence, such as

8 < 12 or

12 > 8,. the p o i n t of t h e symbol ( t h e small end) I s di.rected toward t h e

smaller of the two ~ u m b e r s .

Find the two p o i n t s on the number l i n e which correspond to 8 and 12. Which p o i n t i s t o the left? Will the lesser of two num- bers always correspond t o the p o i n t on the l e r t of the o t h e r ?

Verify youT answer by locating on t h e number l i n e p o i n t s correspond- 5 3 8 ing to severa i p a i r s o f numbers, such as and 2 . 2 : 5 and r . L J

t 1 Just as "f" means "is not equal to", "$" means is n o t g rea te r

than". '&at does "# I' mean ?

Problem S e t 3-5 -- iShich D? t he following sentences are t rue? Which are f a l s e ?

Use the number l i n e to h e l p you dec ide .

1. 4 4 - 3 < 3 + L ; 7. 5 . 2 - 3 , s < 4.6 2. 5(2 + 3 ) r 5 ( 2 ) + 3 8 . 2 +1.3)3.3

3 - 6 . - Open Sentences - Invo lv ing Inequalities I a a t is the t r u t h s e t of the open sentence

x + 2 > 4.r We can answer tRIs ques t ion as follows: We know t h a t the truth

s e t o r

r ; + 2 = 4

is ( 2 1 . \ hen x is a number greater than 2 , then x + 2 is a number g r e a t e r t11an 4. Whcn x is a number less thx? 2, then x + 2 is a number less than 4. Thus, e v e r y number grea te r than 2 makes t i le sentencc true, and every other number makes it false.

That is, the truth s e t of t h e sentxnce "x + 2 > 4'' is t h e set of all numbcx g r e a t e r :ban 2 .

The graph cf this t ru th s e t is t h e s e t of a l l p o i n t s on the number lirie whose coordinates are greater than 2. Thi s is the s e t

o f a l l p o l n t s which l i e t o the r i g h t of the po in t wi th coordinate ;

As another example, consider t h e graph of t h e truth s e t of 1 + x < 4 :

Truth s e t Graph

A l ! numbers ?f a r l t h tne tPc

from 0 to 3, i n c l u d i n g 0 , 0 I 2 3 4 5

not including 3 .

T t is cus tmarg to c a l l a simple sentence Involving " = 1 1 an equation and a sentence involvkg "(" o r " ) I 1 ar. inequality.

Problem S e t 3-6 -- 1. Dctemine whether Lire i r id lcated a c t of points is the graph of

the $ruth s e t of t h e given open sentence. If t h e graph I s not the graph of t h e truth zct, explain why

( a ) 2 + x = k L L I C I I J 1 w - 1

0 1 2 3 4 5 6 7 -

(b) 3 x = 5 I L * 1 L I I L

I

0 1 2 3 4 5 6 7 ( c ) 2y = 7

0 1 2 3 4 5 6 7 A * - A L A ( d l x > l 7 1 w T - I -

0 1 2 3 4 5 6 7

(4 > 5 1 1 - 1 I L r L

0 I

I 2 3 4 5 6 7 2. Draw the graphs o f t h e t r u t h s e t s of t h e f o l l o w i n g open

sentences:

(a) Y = 3

(b) x # 2

t c ) x > 2 ( d ) 3 - i ~ = ~

(4 3 + Y # 4

(h) 3 t Y > h

(i) 3 .t y < ' l

( J ) m < 3 ( k ) m > 3 (1) x(x I 2 ) = 'IX

(f;) F X > 5 (n) 3a -I 2 = 3 ( a 2)

3. Below are some graphs. For each graph, find an open sentence

whose t r u t h s e t is the s e t whose graph is g i v e n .

(4 1 0 1 2 3 4 5

4. If t h e domain of the variable of each open sentense below i s

the s e t [ O , 1, 2 , 3 , h, 53, find t h e truth s e t of each , and

draw its graph.

( a ) b x = 7 ( d ) x .- 3 < 6 I I 'D) !?x + 3 = 6 (e) 2x + 6 = ?(x 3 )

( c ) 2x > 5 t ! 5. If t h e domain +of the variable of each open sentence in

problem is the s e t consisting of 0 , 5, and all numbers

[ greater than 0 and l e s s t h a n 5, find t he t r u t h s e t of each

i and draw its graph.

/ 6. Which o f the t r u t h s e t s in problems 4 and 5 aare f i n ; t e sets?

!

3 - S e n t e n c e s Nith Nore Than One Clause ----- A l l the sentEnces disr -ussed so far have been simple---that is, ,

t h ~ y conteincd ~ n l y one verb form. L e t us c o n s i d e r a sen tence i

s c ~ h as

A t I = j a n d 6 + 2 = 7 . c

Your f i r s t impression may be t h a t ? ~ e have w r i t t e n t w o sen tences - ! But if you read the sentence from left to r i g h t , ic w i l l be one t corpound s e n t e n c e with t h e c o n n e c t i v e and between I w c clauses.

-- - n So in mathernnt ics , as well as in Engl ish , we encounter sentences

(declarative scntcnces) which are compounded o u t of s implesen tence i 1 Fiecall t h a t a numerical sentence i s e i t h e r t rue or false. The

compound sentence

4 L 1 = 5 a n d 6 + 2 = 7 is c e r t a i n l y Yalse, because the word - and means "bo th" , and here !a

Problem S e t 3-7a -- -

the second of the two clauses is f a l s e . The compound sentence ! : B 3 $ 1 + 2 a n d L + 7 > 1 0 i

i:: t r u e , because both clauses are t r u e sentences . I

'a1

' thich of t h e following sentences are true?

1. : + = 5 - 1 a n d 5 = 3 i 2

In gene ra l , a compound sente!~ce with the connectLvc - ~ n d is

~ F u ? if a l l i ~ s clauses are tru? sentences; otherwise, it is lalse.

?,. 3 > 3 1- 2 and 4 .I- 7 < 11

$ 1

5. 3.F i 9.:; # 12.6 and 7 11 8<12

Consider n e x t the sentence

4 + 1 = 5 o r 6 i 2 - 7 . Tnis is a n o t h e r type of compound sentence, this t l m e with the

c o n n e c t i v e or. Here we must be v e r y c a r e f u i . Poss ibly we can

g e t a h l n t from English sentences. If.we say, h he Yankees o r [ s e c . 3-71

t he Ind i ans will wln the pennant", we mean tha t exac t ly cn? cf

the two w i l l w in ; ce r t a in ly , t h e y cannot both ain. But when v;z

say, "My package or your package will arrive within a week, " it is p o s s i b l e t ha t both packages may a r r ive ; here we mean t h a t m e

o r more of the packages w i l l a r r i v e , including the posslbili i? :ha+

both may a r r l v e . The second o f these interpretations ol' "or" iz the one which tu rns out to be t h e b e t t e r suite3 for ow ~ o r i c in

mathematics.

Thus we agree t h a t a compound sentence wi th the c o n n e c z i > ~ e - o r I ' is true if one or more of i t s clauses i s a t r u e sentence; otherwisr,

i t i s false. ' we classify

4 + 1 = 5 o r 6 + 2 = 7 I as a t r u e compound sentence because its f i r s t clauae is a t r u e I sentence; we a l s o c l a s s i f y 1 5 < h + 3 or 2 + 1 # 4

as a t rue compound sentence, because one o r more of I t s c l ause s is t r ue (here b o t h are t r u e ) . / Is t h e sentence

I 3 + ' 2 + 1 o r 2 > Q + 1

Problem Set 3-7b -- Which of the rollowing sentences are t r u e ?

1. 3 = 5 - 1 o r 5 = 3 + 2

3 - ? . S r 2 p h s of Truth Sets of Compound ope^ St?ntenc~_.s --- c'zLll ~rcijlerns in graphing have so far i n v o l h ~ c d only simple

s e n c e n z e s . Graphs o f compo-~1'1d open sentences r equ i r e spec i a l

handling. L e t us c o n s i d e r the open sentence

Tnc c l a u z ~ s cf ;his sentence and t h c corresponding graphs of t h e l r

t ruch s e t s arc :

if a number bcllongs t o t h e truth set; rll? the s e n t e n ~ e " x > 7 "

o r t o t h c truth s e t of the sentenc~ "x = 2", it is a numtcr be lon

in: t o f;he t r u t h set of the compound sentence " x ) 2 or x - 2". 'I"r.erefore, e.3er-y number grea te r than o r equal to 3 belongs t o the

t r u t h set. On the otncr hand, any number l e s s than 2 makes b o t h

clauses of %he compound sentence false and so fails t o belong t,o

its truth s e t . Tt-le g raph of t h e t r u t h s e t is then

j : > i o r ; c = 2 2 0 1 2 3 4

We abbrevia te the sentence "x > 2 o r x = 2 " to "x > ? ' I , read - " x is 5 r e a t . e ~ t h a n or equal t o 2" . Give a corresponding meaning f'oy "<'I

Let us nakr: a precis? staterr,ent of t h e p r i n c i p l e in-rolved:

The graph of t h e truth a c t of a cvrnpaund senCcnce w i t h connec;

il:c - or consictz of the s e t of a l l p o i n t s which bclong to e i t h e r

one of t h e graphs of t he two clauses of the compound sentenc.e.

Sinally, we cons ide r the problem of f i n d i n g t h e grsph of an

open s r n t e n c e such as

x > P and x < j! .

Acair! ?Te begin wlth the two c lauses and the g raphs of' t h e i r t r u t h

zets: I I A e

x > 3 0 1 2 3 4 5

x < q 0 1 2 3 4 5

[sec . 3-81

55 Then it, fo l lows (using an arguimrnt, r,.irnilar t o that above) that ; t h e

graph of the t r u t h se t of the compound sen tence i s

Sometfic-s we write "x ) 2 and x < 4" as "2 ( x < j ! " .

We see t h a t the graph of the t r u t h set of a cor.pound zentence

with connecti-,!e - and c o n s i s t s of all p o i n t s l:ihich are common t o

the graphs of the truth s e t s of t h e two c l a h ~ s e s of t;h? com?ound

senterce . It has required many trords, carefully cnosen, t o d e x ribe t h e

various connections between sen tences , truth s e t s and graphs, We

consistently r e f e r r e d to the graph, of the t ru th a e t u r zn open

sentence. In t he fu ture , let us shorten this phi-ase to t h e graph

of a sentence. It w i l l be a s i m p l e r d e s c r i P t I o n , and no confusion - - w i l l result if wu r e ca l l what is r e a l l y meant by the description.

For t h e s2me r e a s o n s I+/? shall t'Lnd it convenxent t o s p e a k of 1 the p o l n t 3 , 3r %IF! p o i n t ,., when we mean the po;ln: w i t h coorriinate - 7

3 or :he po ln t with caordinate $. Points and numbers are d i s t i n c t

' ent i t ies t o be sure, but they c.orrcspond exac t ly on t h e number llne. t lhent. :~er~ tllei-e is any pozalbility of' confusion we shall

I remember t o give t h e complete descriptions.

Prcblern S e t 3-9 -- Construct the graphs of the following open sentences:

x( 3 o r - x = 3

x $ 3 a n d x # 4

- 7 . Summa3 -- of Open Sentences.

We have examined some sentences and have seen that each cne

can bc c l a s s l f i e c as eithcr true o r f a l s e , b u t not both. We have

cs tab l i shed a s e t o f symbols t o indicate r e l a t i o n s between numbers:,

" = I f means "is" o r "is equal t o "

" # I 1 means "is n o t " or "is n o t equal t o "

"<" mear,s " is l ess than" I

C

">" means " is greater t h a n " "<I f - means "1s less t h a n or is equal to"

")" - means "is g r e a t e r than o r i s equal t o " We have discussed compound sentences which have two clauses.

If the clauses are connected by the word or, the sentence is t r u e ir at l e a s t one clause is t r u e ; otherwise it is fa l se . If t h e --- c lauses are connected by a&, the sentence is true if both c lauses are true; otherwise it is false.

An oper; sentence is a s e n t e n c e c o n t a i n i n g one or nore v a r i a b l --

The truth s e t of an open sentence containing one :rariablc is -- -.

t h e s e t of a l l t h o a e numbers which make the sentence true. The

open sentence a c t s as a sorter, to sort the domain of t h e v a r i a b l e

i n t o two subsets: a subset of numbers which make the sen tence

true, and a subset v ~ h i c h make tihe sentence f a l s e . The g r a p h o f a sen tence is the graph of the truth s e t of the ---

sentence.

Problem Set 3-9 - -- State the truth s e t of each of the following open sentences

and construct its graph. Some examples of how you m i g h t s t a t e

the t r u t h s e t s are : Open Sentence Truth S e t --

x 4 3 = 5 (21

2 x f x t 3 The set of a l l numbers of arithmetic except 3 .

X L 1 < 5 The s e t of a l l numbers of arithmetic less than 3. 2x > g 1

- Tie set of numbers consisting of $ and all num- 1 bers greater than 'lT.

[ s e c . 3-91

2. 2 + v < 1 5 12. ( 3 ~ ) ~ = 36 3. 2 x = 3 13. 9 + t < 1 2 or 5 + 1 # 6 4. G > l t 3 and 5 + t 2 4 14. 5~4-3(1;1

5 . 6 > 1 ; 3 o r 2 t t = l 15. (n - 112 =

7. x 4 2 = 2 o r x + 4 = 6 17. x 2 + 2 = 3x

X 8. 7 > 3 18. t + 6 < 7 - and t + 6 > - ; 9. t + 4 = 5 or t 4 5 j r 5 19. 3 ( x 4- 2 ) = 3x -t 6 XI 3 a # a + 5 2 0 . t + 2 # 3 and 8 + 2 < 5

3-10. I d e n t i t y Elements

Consider the truth s e t s of t h e open sen tences

5 + x = 5 3 -1. j. = 3 ,1 1 '7 + a = 2 ~ .

cI

Do you f i n d that the truth s e t of each of ehese sen tences is

(o]? For what number n is it true thzt

Here we have an interesting proper ty whtch we sha l l c a l l the addition p r o p e r t y of zero. l ie can state this proper ty in words: -- " ~ h e sum of any number and 0 is equal to t h e n u m b e r . "

We can s t a t e t h i s p r o p e r t y in the language o r a lgebra as i'ollows:

For every number a,

a + O = a .

Since adding O t o any number g i v e s us i d e n t i c a l l y the same

number, 0 is o f t e n ca l l ed the - identity element for addition. - -- Is t h e r e an ident= element f o r multiplication? Consider the

truth s e t s of the following open sentences:

3x = 3

2 2 y = 5 .7 = .7y

n ( 5 ) = 5.

You have surely found t h a t the truth set of each of these is {I]:

Thus

n (1 ) = n

seems t o be a true sentence f o r all numbers. How could you st;ai;e

i n words this property, which we shall call the m u l t i p l i c a t i o n

p r o p e r b -- of one? We c a n also s t a t e this In t he language of slgebra:

For every number a, a ( l ) = a.

We can see t h a t the i d e n t i t y element f o r multiplication i s 1. Tliere is another property of zero which will be ev3dent; if you

answer the following questions:

1. What is t he result when any number of' arithmetic is multi-

p l i e d by O?

2 . If the product of two numbers is 0 , and one of the numbers

is 0, whas can y o u tell about t he other number?

The p2operty that becomes apparent is c a l l e d the

property -- of zero, and can be stated as follows:

FoY every number a,

a(0) = 0.

These properties of ze ro and 1 are very ussful. ??r i n s t a n c e ,

we use the m u l t i p l i c a t i o n property of 1 in arithmetic in working

with r a t i o n a l numbers. Suppose we wish t o find a numeral for i;. in the form of a rraction with 18 as its denominator. Of the many

2 3 5 names for 1, such as F , 5, 5' ' ' . , we choose "3" bezause 3 is th 3

number which multiplied by 6 sives 18. We then have

8 = 20)

= g(;) =@j

15 = rn'

[ s e c . 3-10]

$ and 2, n m e

and 9, but it

of 6 and 9 . ) In order ,

. Find a cormnon

27

SUppose we now wish t o add & t o 2. To do this, w e desire o t h e r Y V

s which ere fractions w i t h t h e same denorni-

nator. mat denominator should we choose? It must be a m u l t i p l e

cannot be 9. Thus 36 , o r 18, cr 5 4 , o r my others, are poss ib le choices. For simplicity we pick t h e

amllest. which is 18. (This is called the least common m u l t i ~ l e . - 5 - now, to add $ to 6, we already know that

;+ 5 , name f o r 3-

7

+ = - ' % (Why d i d vr

- + 5(21)

- - - (Note use of the ?(21) d i s t r i b u t i v e

Problcm Set 3-10 -- I n prcblerns 1 t o 10, show how you use t h e p rope r t i e s of 0 and

1 t3 find a common name f o r each of t h e following:

( a ) If you know that tk-e p r o d ~ ~ c t of two numbers is 0 , and

t h a t one of the numbers is 3, vhat can you t e l l about

the other nunber?

(b) If the produc t of two numSers is 0 , what can you t e l l about at l e a s t one of t h e numbers?

( c ) Does the multiplication proper ty of 0 p r o v i d e answers to these queotionc? Is a n o t h e r property of -5 lmplled

here?

3-11. Closure

In our work so far we h a ~ e often combined two numbers by addi-

t lon o r multiplication t o ob t a in a number. We have n e v e r doubted

t h a t we always 2 g e t a number because ou? experience is t h a t we

always do. However, there are some primitive t r i b e s who can count

sec . 3-11]

only t o th ree . Suppose you tried t o teach such people t o add -- what would you tell them when you came t o "2 + 2" and "2 + 3"?

Obviously, you would have t o enlarge t h e i r s e t of numbers until

the sum of any two numbers would be a r-umber of the s e t .

T?e s e t of a l l numbers of arithmetic is such a s e t . If you

add any two of these numbers the sum is always a number of this se t . When a certain o p e r a t i o n is perf'ormed on elcrncnts of a glven

subse4 of the numbers of arithmetiz and the resulting number i s always a member of the s m e s u b s e t , then wc say t h e t c l ~ e subset; is -- - closed under t h e opera t ion . We say, therefore , tha t t he s e t of

-- - -- numbers o f arithmet-1.c i s c l o s e d under addition. Li l rewlse , s ince -- -.

the produc; of any two numbers is always a number, t h e s e t of nm- bers is closed lmder multiplication. W? s t a t e thesf? properties i n t h e language of algebra as f o l l o w s :

Closure Proper ty - of Addit ion: F o r every number E and every

nwnber b , E t b is a number.

Closure P r o p e r t y - of Mult ip l ica t ion : For e v e r y number a and

e v e r y number b , ab is a number.

5-12. Associative and Cornmutative Properties of Addieion and -- - -- - Multiplication. --

In Chapter 2 we d i s c u s s e d a number of patterns f o r f o r m i n s

t rue sen tences about numbers, and saw that these p a t t e ~ n s were

closel:r c o n x c t e d w i t h many of the techniques o f arithmetic. idhat

were some d these patterns? For ixxtance, xe found t r u e szntences

such as

( 7 + 8 ) + 3 = 7 + ( 9 t 3 ) ar~d

( 1 . 2 t 1 . 6 ) -:- 2.6 = 1.2 .t (1.8 t 2 . 6 ) . We concluded a p a t t e r n rnr t r u e sentences f ron these examples,

which we ver3alized as f o l l o w s : If you add a second number to

a first n m h ~ r , and tnen a third number ;o t h e i r sum, the outcome

is the same i f you add the second number and the third number, and

then add t h e i r sum to the f i rs t numker. What was the name of this

pr3pe r ty?

[ s e c . 3-12]

The algebraic language with which we have been becoming familiar permits US, as in the case of the proper t ies o f 1 and 0

which we have just s tud ied , t o g l v e a statement about the above

p r o p e r k j in this language. We have th ree(not necessarily d i r f e r e n t )

numbers t o dea l w i t 1 1 a t once , Let us call the first "a", the

second "b", and Che t h i r d "c" . "Add a second nwnber to a first number1' is t h e n interpreted as " a + b t l ; "add a third number to t h e i r sun" is interpreted as "(a + b ) + c". (Why dfd we I n s e r t

t h e parentheses?) Write the second half of our verbal statement i n the languase of algebra. The words "the outcome is the same"

now tell us that we have two names f o r the same number. Our s t a t e - I

ment becomes

( a t b ) - + c = a + ( b + c ) .

For what nmbers i s t h i s sentence t r u e ? We have concluded pre-

v i o u s l y that it is t r u e f o r all - numbers. And ao we w r l t e , finally,

For every :lumber a, for every number b, for every nwnber c , ( a + b) +- c = a + (b -t c ) .

!

Some o ther t rue sentences were o f the form I

3 + 5 = 5 + 3 . The property of addition which s t a t e s that a l l sentences following t h i s p a t t e r n are true we called the cornmutatlve property of addi-

t i o n . It waa verbalized as f o l l o ~ v s : If two numbers are added in I

d i f f e r e n t orders , the results are t h e same. In the language of algebra, we say

For e v e r y number a and every number b,

a + b = b + a .

How would you state the associative property of multiplication

i n the language of algebra?

What proper ty is given by the following statement?

For every number a and every number b,

ab = ba.

These properties of the operations enable us to m i t e open

phrases in "other forms". For e x a m p l e , the open phrase 3d(d) 2 can be w r i t t e n in the form 3(d-d), i . e . , 3d , by applying the

a s s o c i a t i v e proper ty of multiplication. Thus, two "forms" of

an open phrase are two numerals for the same number.

Among t h e properties with which we have just been concerned

a r e the commutativity of addition and multiplication. ?my are

we so concerned whether binary operat ions l i k e a d d i t i o n and multi-

p l i c a t i o n a re commutative? Aren't a l l the o p e ~ ~ a t i o n s of a r i t h -

metic carnmutative? L e t us try d i v i s i o n , f o r example. Recal l that

6 + 3

means "6 d i v i d e d by 3" . Now, t e s t whether

E ? 3 = 3 + 6

is a t r u e sentence. This is enough evidence t o show that d i v i s i o n

is not a commutative operation. (By t h e way, can you f i n d some a and some b such that a + b = b a?) Is t h e d i v i s i o n operation

assoc ia t ive? Another v e r y i n t e r e s t i n g example f o r the counting numbers is

the following: let 2 * * 3 be defined t o mean ( 2 ) ( 2 ) ( 2 ) ; and

3 ** 2 to mean ( 3 ) ( 3 ) . In gene ra l , a -:t* b means a has been used

as a f a c t o r b times. Is t h e following sentence t rue: 5 ** 2 = 2 ** 5:'

Do you conclude t h a t t h F s b inary operation on coun t iw numbers is

commutative? Is it a s s o c i a t i v e ?

You may complain t h a t t h i s second example is a r t i f i c i a l , On

the con t ra ry , the * * operation def ined above is actually used in

t h e language of c e r t a i n d i g i t a l computers. You see, a machine

is much happler if you g i v e it a l l I t s instructions on a l i n e ,

and so a "linear" notation was d r v i s e d f o r t h i s opera t ion . B u t

you see that t o the machine the o r d e r of the numbers makes a

great d i f f e r e n c e in t h i s operation. Is there any restriction on

the types of numbers on which we may opera te with

Problem Set 3-12 -- - - I. If x and p are ~ m b e r s of a r i t -m~e t i c , the c lnsure p r o p e r t y

assures us that 3xy, 2x and the re fo re , (3xp)(2x) are numbers

of a r i t h m e t i c . Tiler] tlre asuoclatiqre and commut.ati~e p rope r t*

of r nu i t i p l i c a t i on enable us t o wrik t h i s in a n o t h e r form:

(3xp) ( 2 ~ ) = ( 3 . 2 ) ( x . x ) ~

Write the indicated products in another form as in the above

example :

( a ) ( 2 m ) ( m ) ( d l (&b) (6c )

(b) (5P2)(3d (4 Iloal(l~b) ( c ) n ( ~ n ) ( 3 r n ) (f) (3x)(12)

2 . IT x and y are n-mbers o f arithmetic t nen t h e closure 2

property a l l o w s us t o think of 12x y as a numeral which r e -

presents a s i n g l e number. ?he corn-uta t ive and associative

properties of mulLiplicatFon e n a b l e u s to write o ther nun- E P Z I ~ , ~ f o r t h ~ same number . (4xy)(2x), (?x)(Gxy), a r ~ d

2 l 2 ) 1 ) s r a some of the many ways of. writing 12x y as

i n d i c a t e d products, Similar ly, write three p o s s i b l e i n d i c a t e d

p r o d u c t s f o r c s c k u f tile f ' < ~ l l o w i n g .

( a ) 81be 2 2

( d ) x Y

(b) 7.Y3 2 ? ( e ) 6J1a bc

( c ) 1h:1 ( f ) 2c

Which 3f the Yellowing sentences are true ?or every -:slue of

t h e variabl?'! u p l a i n which of the properties aided i n your decision.

(a) m(2 t 5 ) = ( 2 i 5)m (b) (m + 1 ) ~ = ( Z -I- l ) r n

( c ) ( 2 - 2;lr) .I b = (a -: b ) t 2 y

(d) 3x + y = y t 3x

( e ) ( ?a + C ) d = 2a(c + d) (f) PC + 6 = 6 + PC ( ) .5b(200) - ~ O G ( , ~ b )

( h ) ( 2 u . i ) ~ = PU(VZ)

(j) 3x 2- = x + 3y

'1. The s e t A is given by A = [ O . 1)

( a ) Is A c losed under addi t ion?

(b) Is A c losed under multiplication?

5. ( a ) Is the s e t S of a l l multiples of 6 closed under addition?

(b) Is set S closed under multiplication?

*6. Let us de f ine some blnary operations other than addition and

multiplication. We shall use t h e symbol " 0 " each t i m e . We

might read ''a o b" a s "a operat i o n h " . S i n c e we are g i v i n g

the symbol various meanings , we must de f ine Lts meaning each

time. For instance, f o r every a and every b,

if a b means 2a + b, then 3 5 = 2 ( 3 ) + 5 ;

3 + 5 a + b, then 3 o 5 - -. if a a b means 7 2 '

if a 0 b means ( a - a ) b , then 3 0 5 = ( 3 - 3 ) 5 ;

1 if a 0 b means a i - $I, t h e n 3 0 5 = 3 -t 1 (j) (51;

if a o b means ( a + l ) ( b t 1). t h e n 3 0 5 = ( 3 t 1)(5 + 1).

For each meaning of a o b stated above , write a n u m e r a l f c r

each cf t h e f o l l o w i n g :

( a ) 2 Q 6 ( c ) 6 2

( b ) ($1 O 6 ( d ) ( 3 0 2 ) 0

+7. A r e such b inary operations on numbers as 'chose def ined in

Problem 6 commutative'! In o t h e r words, is I t t r u e that f o r

every a and every b, a 0 b = b 0 a: L e t us examine some

cascs. For i n s t a n c e , if a o b means 2a i b, we see that 3 0 4 = 2 ( 3 ) + 4

0 3 = 2(4) + 3

But " 2 ( 3 ) 4 =. 2(4) + 3'' is a false sentence. Hence, we conclude that t h e opera t ion here i n d i c a t e d by " 0 " i s n o t

commutative. In each of t h e following, decllde whether o r

n o t t h e o p e r a t i o n desc r ibed is commutative:

a + b ( a ) Fo r every a and every b, a o b = - rS C

(b) For every a and e v e r y b, a a b = ( a - a)b ( c ) For every a and every b, a o b = a + h 3

( d ) For every a and every b, a 0 b = (a -t l ) ( b + 1) What d o you conclude about whether all binary operations

are commutative ?

%8. Is the operation " 0 ' associative in each of the above c a s e s ?

For Instance, I f , f o r every a and e v e r y b, a D b = 2a t b ,

13 (4 0 2 ) 0 5 = 4 0 ( 2 0 5) a true sentence? ( 3 0 2 ) 0 5 = 2(2(4) + 2) + 5

= ~(10) + 5 w h i l e

4 O ( 2 O 5) = 2 ( 4 ) + (2(2) + 5) = 8 + 9

Since the sentence ~(10) + 5 = 8 + 9 is false, we conclude t h a t t h i s o p e r a t i o n is n o t associatlve. T e s t the operations

described in problem 7 ( a ) - ( d ) f o r the assoclatlve property.

3-13. The Eistributive Property.

0 ~ r work with numbers in Chapter 2 has shown us a var ie ty of v e r ~ i o n s of the distributive p r o p e r t y . Thus

15(7 + 3 ) = 15(7) -t 15(3) and

are two true sentences each o f whlch follows one of the patterns

which we have recognized. We have seen the importance of this

p r o p e r t y in relating indicated sums and indicated products. We may now state the d i s t r i b u t i v e p roper ty i n the language of algebra:

For every number a, every number b, and every number c,

a ( b -t c ) = ab s ac.

Does this statement agree with our verbalization in Chapter 2?

Since we have s t a t ed t h a t "a(b + c ) " and "ab + ac" are numerals f o r the same number, we may equally well mite

For every number a , every number b , and every nwnber c

ab 4 ac = a ( b - c ) .

We may a l s o a p p l y the commutative p r o p e r t y of muItiplication

t o wri te :

For c v e r y numbel- a , every nwnber b, and every number c .

and :

For every nlmber a, every nwnber b , and every nwbr=r c ,

Nhich of t h e s e p a t t e r n s does the i'irst of G u r a b o ~ e examples

f o l l o w ? the second?

Any one of t h e f ~ u r sentencee above d e s c r i b e s the distributive

p r o p e r t y . A l l fornis are useful in the study of algebra,

B a m p l e - I . >!rite the Indicated p r o d u c t , x ( y + 3 ) as an indicated

x ( ~ -I 3 ) = xy + x ( 3 ) by t h e dlstributfve p r o p e r t y

= XY i- 3~ - Yxamnle 2. Xrite ?X -t 5y as an indicated product.

5~ -1 5~ = ?(x + y ) by the distributive p r o p e r t y

Example 3 . Write the open phrase 3a t- 5a in simpler form - -

3a + 5a = ( 3 -t 5)a by the d i s t r i b u t i v e property

= 8a

Em.rnple 5. :{rite the open phrase, 2x -I 3y + 4x + 6y in s l m p l e r

Y 0 l - m .

2x + 3y -t I ! x - by = ( ~ x + 4x) 4 (3y + 6y) by the commutative and associative p r o - per l t ies o f a d d i t i o n .

= ( 2 4 4)x + ( 3 + 6Iy by t h e distributive p r o p e r t y

= 6x + gy

Problem S e t 3-13a -- 1. Write the i n d f c a t e d p r o d u c t s as i nd i ca t ed sums.

( a ) 6(r 4- s ) ( d l (7 -t x ) x

(b) ( b - + 3)a ( e l 6 ( 8 + 5 ) ( 4 X ( X + 21 (f) ( a + b b

2 Write the indicated sums as indicated products .

(a ) 3x + 3y

(b) am -t am ( c ) x + b x Hint: x = ( 1 ) x

1 1 ( d l TX + 2Y

( e ) Pa =t a H l n t : How is a* def ined? 7

(f) x L + x y

3 . Use t h e associative, commutative, and distributive prope r t i e s

t o mite the foliowing open phrases In simpler form, if

poss ib le :

(a) 14x + 3x

3 (b) $x +-x 2

2 1 ( c ) =p + 3b -t

( d ) 7x + 13y + 2x + 3y

( e ) 4x + 2 y + 2 + 3x (f) 1 . 3 ~ + 3 . 7 ~ + 6 .2 + 7 . 7 ~

The distributive p r o p e r t y stated by the sentence

For every number a, every number b, and every number c ,

a ( b + c ) = ab -I- ac

concerns the t h r ee numbers a, b and c . However, the closure

p r o p e r t y allows us t o apply the d i s t r i b u t i v e p roper ty in many

cases where an open phrase apparent ly conta ins more than three

numerals. For example, suppose we wish t o express the indica ted

p roduc t 2 r ( s 4- t) as a sum. The open p h r a s e contains the four

numerals 3 , r, S, and t. The c losure property, however, allows

us 50 consider 2r as t h e name of one number so we can thinl i in - terms of three n m e r a i s , 2r, s , and t. Thus

2r(s -I t ) = (2r)(s s t )

= (2 r ) s + ( 2 r ) t

= 2rs -t '2rt

Example 1. l:lrlte 3u (v + 3 2 ) as an i nd i ca t ed sum.

By the c l o s w e property we can r e g a r d 3u, 73, and 32 each as t he

name of one number. Then by the distributive p r o p e r t y

3u (v + 3 2 ) = (3u)v + (3u) ( 3 z )

= 3uv + 9uz by t k e commutative and a s soc i z t i ve properties u f multi- p l i c a c i o n

Example - 2. Firite the i nd i ca t ed sum, 2rs + Frt, as an indicated

produ? t . We can do t h i s in three ways:

(1) 2rs + 2rt = 2(rs) i- 2(rt)

= 2 ( r s + rt) (2 ) 2rs + 2rt = r(2s) + r(2t)

= 4 2 s +- 2t) ( 3 ) 2rs + 2 r t = (21')s A - ( 2 r ) t

= P r ( s + t) Although all three ways are c o r r e c t , the t h i r d is usually preferred.

F%an!ple .- 3 . &press the i nd i ca t ed product, 3 (x t- ;r -I- z), as an

indicated sum. 3(x -+ y + 2 ) -7 3x + ?y t 32

P r o b l e m Set 3-13b -- I. slri te each of the indicated products as an i n d f c a t e d sum.

(a) m(6 + 3p) ( d ) (2x :- xy)x

(b) 2k(Ic_ ;- 1) ( e l ( e I- f - g ) h

( c ) 6 ( > s -I 3r + 7q) { I 6 ~ 4 ~ 4- q)

2 Iihich of t h e following open s e n t e n c e s are t rue for every

value of every varia5le. 2

( a ) 2a(a - b ) = 2a + ab

( b ) Lxy 1 Y? = (k

[ s e c . 3-13]

70 ( c ) 3ab -I- 6bc = 3 b ( a 2c)

{d) 2a(b + c ) = 2ab + c r - C ( c ) (Ilx -- 3 ) ~ . = "x 1 3 + x

(I) ( 2 y i- xy) - (2 ,-1 x ) y

3 . Th'rite each of the i n d i c a t e d sums as an l r i d i c ~ t e d product .

( c ) 3~ T 5);':

( d ) ;Ic - 4cd

'> ( e ) 3jr. + 6x'

Hint: th ink of 3x as {3x) (1 )

Hint : 4cd = (2c)(2d)

Another important application of the d i s t r i b u t i v e property is

illustrated b ; ~ the f o l l o w i n g cxmple . Example 1. Writc (x + 2)(x -1- 3 ) as an indicated sum without - -

parentheses.

If vie write t he u i : : t r lbut ive property wiLh the rindicated p r c d u c t

beneath it, we can see which names we must regard as separate

names of nw,bers .

= x' + 2x + 3x + 6 distributive p r o p e r t y

= x2 + ( 2 i- 3 ) x + 6 distributive property

Could you have ysed a d i f f e r e n t form of th? d i s t r i b u t i v e proper ty

to begin y o u r work?

Ekamgle - 2 . Lcr:t;e (a + b ) ( c + d) as an indicated sum without

pa~cntheses. Supply the reason f o r each s t e p . ( a t b ) ( c t d) = ( a + b ) c + (a + b ) d

= ac + bc -+ ad =+ bd

Problem S e t 3-13c -- Write the ind ica ted products as i n d i c a t e d suns without

parentheses.

1 (X $ L)(x :- 2 ) 4. (x -t 2 ) ( y - 7) 2 (X + 1 ) ( x t 5) 5. (m I n){m + n)

, 3 . (x t a ) ( x 3. 3 ) 6 . ( ? ~ + q ) ( ~ + 2 q )

3-14. Summary: Properties - ~f Operations - - on Nwnbers of ArithmeLic,

:Jimbe?s 2.23 be added and multiplied. Ve i.!nTie 1 en:?-led t ' : r t

numbers and t h e i r ope ra t i ons have basic p r o p e r t i e s which we shall

list below and always r e f e r to as the proper t i e s g n ~ m h e r s

1. Closure Property - of A d d t t i o n : For every number a ma every

number b , a s b Is a number.

2 Closure -- P r o p e r t y - of -. M ' ~ 1 t i p l i c a t i o n : F o r every number a and every number b, ab is a number.

3 . Commutative P r o p e r t y o f Addition: For every number a and - - every number b, a -t b = b + a.

1?. Commutative Proper ty o f Multiplication: For e-7er-j number a - and every number b , ab = ba

5. Associative - Proper ty - o f Addition: For every number a and

e v m g number b and every number e l ( a -I b) -I c = .z .+- (b 4- C )

6, Associative Pr,opex4Ly of b l u l t i p l l c a t i o n : For evep; nmber a - and every number b and e v e r j number c , ( a b ) c = a(2c.)

7. D i s t r f h u t i v ~ ! Froper' iy: For e v e r y nmbcr a and every r lur t~bt r r~ -- b and every number c , a ( b 4, c ) = ab + ac

3. Additim . . -- P r o p ~ r t y - , of 0: For eve ry number a ,

a ~ O = a

7. Multiplization Proper ty of 1: For every number a! -- - - a ( l > = a

[ c c c . 2-14]

I ~ l u l t i p l i c a t i o n - Proper ty of 0 : For every number a, - - a ( ~ ) = 0

Problem S e t 3-14 -- %y putring one of t h e signs, +, ::, -. in each of t h e blanks,

and i n s e r t i n s parentheses t o indicate grouping. f i n d the

c c m o u name f o r each number which can be obta ined from

8 3 2

As examples, 8 - ( 3 -1- 2) = 3 , and ( b + 3 ) :. 2 = 22

Sfa te indicated products as i nd i ca t ed sums. and i nd i ca t ed sums

as indicated products:

( 2 ) (1. + 4 5 (h) 3ax =+ 2ay

3 2 (b) F+Ip ( 1 1 ( 5 + YIY + PY

(c) 33(95) + 15(11) (j) (2 -+ a ) b + ( 2 + a )3

Hint: (2 -I E ) is a number , by c l o s u r e

( d ) c -! PC ( 1 ) b -I- 3ab

i 1 (4 (V + $Y (1) 2a(a + b + c )

{f) x ( y !- 1) (m) ( U + 2v)(u + V)

( 3 ) x y + x ( n ) ( a - i - 1) 2

Use the p r o p e ~ t f e s t o write the foilowing in simpler form:

(a) 17x + x ( d ) 1.6a + .7 + .Ira + .3b (b) 2x + y + 3x + y ( e ) by + 2by ( c ) 3(x T 1) + 2~ 4- 7 (f) g x 1 3 + x + ? + l l x

Here you are going t o see how t o t e s t whether a whole number

is exactly d i v i s i b l e by 9. Keep a record, as you go, of t h e

properties of addition and m u l t i p l icatlion v ~ h l c : ~ are used .

Try the following:

2357 = ~ ( 1 0 0 0 ) + 3 ( 1 0 0 ) -t. 5 ( 1 0 ) + 7(1) = ~ ( 9 9 9 -t 1) + 3(99 + 1) 5 ( 9 + 1 ) A r ( 1 )

= ~ ( 9 9 3 ) + 2 ( 1 ) 3(99) + 3 ( 1 ) + 5(:1) 5 ( 1 ) 7(1)

= ( 2 ( 9 9 9 ) + 3(99) + 51")) + (~(1) 3 ( 1 ) + 5 ( 1 ) + ~(1))

= (2(111) + 3(11) + i ( 1 ) ) g + @ + 3 + 5 + 7) = ( 222 + 33 -k 5 ) 9 A (2 4 3 + 5 + 7)

Is 235'1 divisible by 9? Try the same procedure k:irh 358'/a. Can you formulate a genera l rule t o t e l l when a Xhdle number

is d i v i s i b l e by 9? 3 5 . Look for the pattern in the following calculation:

19 x 13 = 19(10 + 3 )

= 19(10) -t 19(3) (what proper ty?)

= 19(10) + (10 4 913

= l i i ( l 0 ) + (10(3) i- 9 ( 3 ) ) (what property?)

= (19(10) - 10(3)) 4 9 ( 3 ) (what p r o p e r t y ? )

= (19 t 3)10 4 9(5) (;;hat properties ?)

The f i n a l r e s u l t indicates a method for "multiplgin~ teens"

(whole nwnbers from I1 t h r o u g h 13): Add t o t h e first number

the units d i g i t o f the second, and multiplp by 10; then add

t o this the product of t h e units d l g i t s of the two numbers

Use the method t o find 15 s 14, 13 x 17, 11 x 12.

Review Problem3 -- 1. ( a ) Write a description of the s e t EI if

H = 121, 23 , 25, 27, . . ., 491. (b) Consider t h e s e t A of all whole numbers greater than

20. Is H a subset of A ? Is A a subse t of H:

( c ) Classify s e t s H and A ( f i n i t e or infinite).

2 . Find the coordinate of a point which lies o n the number l i n e 7 J

between the two p o i n t s w i t h eoordinatzs and 2. ow many p o i n t s arc between these two?

3 . Consider the s e t

T = ( 0 , 3 , 6, 9 , 12, . . . I . Ia T closed under t he opera t ion o f addition? under t he

o p e r a t i o n or' 'kveraging " ?

b . L e t the domain of t he var iab le t be the s e t R o f a l l

numbers between 3 and 5 , inclusive.

( a ) Draw t h e graph of the s e t R.

( b ) Decide whether each of the f o l l o w i n g nmber s is an

admissible value of t: 3 i ~ , q, %, % n. 2 ' C

[ s e c . 3-14]

5. I5 each of t h e follow in^ ser~ tences t rue?

Ikc ide \what pa~tcrn these sentences i'ollow and descr ibe tile

pattern In words. Tlen formulate it in the language of

a lgebra as an open sentence with two variables. Use the

properties of the numbers of a r i t h e t l c as we have discovered

them to t e s t whether the resultim sentence is true f o r a11

v a l u e s of t he v a ~ ~ l a b l e s .

6. In which of A, 3, C, D, E does the sentence have the game

truth set as the s e n t e n c e "x 5 5"? (A) x > 5 o r x = 5 (B) x < 5 and x = 5

(c) 9 5 (D) 4 5 (E) x #' 5

7. Find the truth s e t f o r Each of the following sentences:

( a ) n - 5 - 7 (d) 4 n - 5 = 7 (b) In - 5 = '1 ( e ) Gn - 5 = 7

( c ) 3n - 5 = 7 (f) l 2 n - 5 = 7 8. If rn is a nunber o f a r i t h m e t i c , find the truth s e t of

( a ) r n + m = ~ ( c ) nl 2 an (b) EI r m = ?m ( d ) in -i- 3 < m kklch of the above s e t s Is a subs~t of all the others? !Jhich

Ls a subset of none ocher than itself? I f the domain of m is t h c s e t of counting n u n b e r s , answer questions (a) through

( d ) above.

9. Let T be the truth s e t of

x - : 3 = 5 or x 4 . 1 = 4 .

( a ) Is 3 an elerncnt cf T?

( b ) Is 2 an element of T?

( c ) 1s p a ~ u b s e t of T?

10. If B is t h e truth s e t of x + 1 < 5 a n d x - 1 2 2 ,

draw the graph of S.

11. Consider t h e open sentence

2x 2 1. What is I t s truth s e t if the domain of x is t h e s e t o f

(a) a l l counting numbers? (b ) all whole numbers':'

( c ) a l l numbers of arithmetic?

1 (a ) Is the following sentence t rue?

(b) Do you have to perform any multiplicati~n t o answer

p a r t ( a ) ? Ehplain.

13. Vhich of the following sentences are true?

( a ) 5 ( 4 -I. 2) = (4 + 2 ) 5

( e ) 12 x 8 + 12 x 92 = 1200

(f) ( 5 r %)16 = 500

14. mplain how t h e property o f 1 is used i n per roming the

ca lcu la Lion

15. &plain why

3x + y + 2x + 3y = 5x + ' ~ y

is true for all values of x and y.

6 (a) Write the indicated products

( X + 1)(x -t 1)

(X + 2)(x + 2 )

as indicated sums without parentheses

(b) U s e t he p a t t e r n of the r e s u l t s of p a r t ( a ) to write t h e indicated sum

x2 + 6x + 9 as an indicated product.

Chapter 4

OPEN SEPITEXCES AND ENG LEYH .3 EIITXrZC SS

k-1. ?pen Phl1a?es - an,3 English Phrases

E v e r y J a y YJe are building - ~ p a new language ~f symbols which

is t e r ~ x n i n g n6re an3 more a complete language. 'LJe h a v e use2

! m t h e i ~ ~ t i c a l phrlases, such as 'I% -t- 3y" ; mathematical -:er!r forms,

inc1:;- in^ l 1 -" an3 ")'I ; an3 mathematical sentences, sucrl as ll 7-1 +- 33 = 53".

:!e r e c a l l t h a t a ~ ~ a r i a b l e , such as [In", is the nane of a

r?ef inite b x t ur.specif l e d n;imker. T h e t r a n s l a t i o n of " n" i n t c ~

Engl . l sh ill then m a n r e l a t i n g a n u n s p e c i f i e d nunber t n some-

t!>ir.g of I n t e r e s t ti3 U S , Thus, the numeral "nl' might represent

" t h e nunbe r of p r o b l e m t ha t I worked", It t h e r.lmtjer +f students

at t h e rally", "the number of dimes in Sam's p r 3 c ~ e t " , or "the

number n f feet i n the h e i g h t of t h e s c h o ~ l f 1agp:)le" . '!hat a r e

zoae o t h e r p 3 c ; s i b l e t r a n s l a t i o n s ?

C o n z i d e r t h e phrase "5 + n " . Can we in!!er~t arr English phrase

for t h i s 7 Zuppo~e i:!e use the t r a n s l a t i o n s suzgesteu a t a v e . If

' I n " is the n i l m t e r of' problems I shall be working t o d a y , the r . t h e

phrase "5 t- n" represents "the total number 3f p r o t l e ~ s inc l~ : . , ; l lng

t h e f l v e worked last night"; or, if I h a v e 5 5imes an5 "n" repre-

sents the number of dimes 111 Sam's p o c x e t , then "5 + nl' represent? "the ~ o t a l number of dines, Including my five and those in Sam's

pch:!.:et.'' Notice that the tra1;slation of " 5 + n" depenjs on what:

t r a : ~ s l a t i o n we make of "r," . '?i:ich of the a p p a r e n t l y limitless number of :pans Lar ions 30

we pick? l:'e are rerninrlcil that the var iab le appear in , i:1 the open

phrase, whether "nl' or "x", or "w", or "bl', is the ::ame or' a

num':ier. :. 'hether t h i s is the number of d i m e s , t h e nl.l:nrjer of

stsdents, t h e number ~ l f inches, e t c . , depends: u p c n t h e * l se we p.Lan

"t o K e of t h e t r an : ; l a t lon . The context itself :dl11 f r e q u e r i t l x

suggest or liait t r k n s l a t ions. Thus it wol~ld not mice sense

t o t r a n s l a t e a phrase such as l1;i,5r)CI, 'jF)'j + y" In t e r m . ~ t '

t h e number of zimes i n Sam's p o c ~ e t , h u t it wouid m k e sense to

tnink 3f " y" as represerltirlg t h e number g iTJr ing the p c p u l a t i o n in-

crease i n a s t a t e wli ich had 2,500,000 peraorls at t h e time of t h e

p reced ing census , or as t h e number of additional miles t r a v e l e d

by a s a t e l l i t e which had gone 2,500,000 miles a t t h e t i m e of t h e

l a s t report. S i m i l a r l y , the varlable in t h e phrase l1 .05 + k"

would h a r d l y b e translated as the number o f cows o r s t u d e n t s , bu t

p o s s i b l y as t h e number glving the Increase in t h e ra te ef i n t e r e s t

which had previously been 5 per cen t .

How can w e t rat ls late the phrase "3x + 25"? I n t he absence

of any s p e c i a l reasons f o r plcklng a p r t i c u l a r translation, we

might let x be the number of cents Tom earns i n one hour , mowing

t h e lawn. Then 3x is t h e number of c e n t s earned in 3 h o u r s . If

Tom f i n i s h e d t h e job i n three h o u r s and was paid a bonus of

25 cents, then t h e phrase " 3 x -+ 25" represents the t o t a l number

of cents in Torn's possession a f t e r working three h o u r s . How can t h i s p h r a s e be t r a n s l a t e d if we l e t x be the number of students

in each algebra class, if algebra classes are of t h e same s i z e ?

Or, if x i s the number o f miles t r ave l ed by a c.ar in one h o u r at

a constant speed?

T h e r e a re many Engllsh translations of the symbol "t", in-

d i c a t i n g the o p e r a t i o n of addition of t v ~ o numbers. A few or

t hem are: " t h e sum o f " , ''more than" , " i n c r e a s e d t y " , " o l d e r

t h a n " , a n d o t h e r s . There a re a l s o many English translations of

t h e symbols i n d i c a t i n g t he operat ion of multiplication a f t w o

numbers, Inc lud ing : l l t imes", " p r o d u c t o f " , and o t h e r s . !!hat

a r e English translations of t h e s y m b o l "-" ?

Problem Set 4-1

In Problems 1-6, write Engllsh phrases which correspond t o t he

given open phrases. Try t o v a r y t h e Englfsh phrases as much as

possible. Tell in each case what t h e var iable represents.

1. 7 w ( I f m e bushel of wheat c o s t s w d o l l a r s , t h e

phrase is: " t h e number o f d o l l a r s in the c o s t

n f 7 bushels of whea t . " )

In each of Problems 7-17, f i n d arl open phrase whirh is a t rans-

l a t i o n of t h e g iven English phrase. In each problem, t e l l ex-

p l i c i t ly what the variable represents.

7 . The number of f e e t i n y yards.

(If y is the mimber of y a r d s , then 3y is the number of f e e t . )

8. The number of inches in f f e e t .

9 . The number of p i n t s in k quarts.

10. The number of miles In k feet,

11. The successor of a whole number.

2 The r ec ip roca l of a number. (Tho numbers a r e reciprocals of each o t h e r i f t h e i r p roduct is 1.)

13. The number of ounces in k pounds and t ounces.

14. The number of c e n t s in d do l l a r s and k quarters .

15. The number of cen t s in m do1 lars, k qua r t e r s , m d i m e s and

n nickels.

Id . The number o f i n c h e s in t h e length of a r ec t ang le which is

twice as Ions as It Is v r i d e . (Sl.lggestlon: Dra.sr a f i g u r e

t o help visualize t h e situation.)

1 . The number o f fee t in t h e h e i g h t of a t r i a n g l e if its h e i g h t

is 6 fee t g r e a t e r than its base.

In Prob lems 18-25, write English phrases which a r e t rans la-

t i o n s of t h e g iven open phrases . In each problem i d e n t i f y t h e

v a r i a b l e explicitly.

la. n + 7 n

19. ( 2 r - 5) + 7 2 0 . 3x + (2x + 1) + x

21. 5000 + by

2 y + . M y

* 2 3 . x ( x 4- 3 ) (~int: T h i s phrase might be interpreted as t h e

expression for t h e number oP square u n i t s in

an area. ) 1

'-24. T ~ ( ~ + 5)

~ ( w + 3 ) ( 2 v r + 5 ) (Hint: T h i s ph rase might Se i n t e rp re t ed

as t h e expression f o r t h e number of

c ~ ~ b i c u n i t s I n a volume. )

Cnoose a var iable f o r t h e number of f e e t in t h e length of one

s i d e o f a sguare . Y - y i t e an open phrase f o r t he number of f e e t in t h e perimeter of the square.

Chooze a va r i ab l e f o r the number of inches I n t h e h e i g h t of

a m n t s h e a d . Then write an open p h r a s e f o r t h e number of

i n c h e s in the man 's h e i g h t if i t is known that h i s h e i g h t Is 1 '/ t i n e s t he h e i g h t of h i s head.

-:!rite an open phrase for t h e number of Inches in t h e l eng th

of a second s i d e of a triangle, if it I s t h r e e Inches l o n g e r

t h a n the f i r s t s i d e ,

One s i d e o r a t r l a n g l e is x inches long and a second i s

y Inches long, The l eng th of the t h i r d s ide Is one-half t h e

sum of t h e l eng ths of the f i r s t t w o s l d e s .

( a ) ':!rite an open phrase for the number of Inches in t h e

perimeter of t h e t r i a n g l e .

( b ) !:'rite an open phrase f o r t h e number of inches in t h e

length of t h e t h i r d s i d e .

In a c e r t a i n community t h e r e a re s i x - f i f t h s as many g i r l s as

bo:]s. >!rite an open phrase f o r t h e number of girls in terms of t h e number of boys.

The aclrnisslon p r i c e t o a performance of "The Mikado" is

$2.00 per person. Write an open phrase f o r the t o t a l number

oT d o l l a r s r ece ived in terms of t h e number of people who

b o u g h t t i c k e t s .

If a man can p a i n t a house In d days, wrl te an open phrase

fur t he p a r t of t h e house he can p a i n t in one day. 1 If a pipe f l l l s - of a swimming p o o l I n one hour, wri te an 5

open phrase f o r how much of t h e pool is f i l l e u by t h a t p l p e

in x h o u r s .

34. >/hen a tree grows it increases its r a d i u s each y e a r by a d d i n g

a r ing of new wood. If a t ree has r rings now, wr i t e an open

phrase f o r the number of growth rings In t h e tree twelve years from now.

* 3 5 . A p l a n t grows a c e r t a i n number uf inches per week. I t is now

20 i n c h e s t a l l . Write an open phrase giving the number o f

inches in its h e i g h t f i v e weeks from now.

* 3 6 . Suppose that when a man immerses h i s arms in h o t water, t h e

tenperature of h i s f e e t w i l l rise one degree p e r minute,

beginning at 10 minutes a f t e r h i s arms a r e put in the water. !:'rite an open phrase f o r t h e rise In temperature of t h e lnan ts

feet at any time (more t h a n t e n minutes) af te r h i s arms a r e

immersed.

37. Three sons share in an i nhe r i t ance .

(a) Write an open phrase f o r the number of d o l l a r s of one son's share which is one-hair ~ J T t h e i n h e r i t a n c e .

( b ) Yr l t e an open phrase f o r t h e number ar d o l l a r s of t he

second e o n ' s share, whlch is fifty d o l l a r s more than

one-tenth of t h e i n h e r i t a n c e .

( c ) \:rite an open phrase for t h e t h l r d s o n t s share.

( d ) 'irlte an open phrase f o r the sum of the t h r e e s o n s '

shares .

38 . Choose a v a r i a b l e f o r the number of f e e t in t h e width o f a

rectangle .

( a ) Write an open phrase f o r the length o f t h e rectangle If

t he length is five f e e t less than twice the width.

Draw and l a b e l a f lgure .

(b) \>!rite an open phrase f o r the perlrneter of t he rectangle

d e s c r i b e d in part ( a ) ,

( c ) >!rite a n open phrase f o r t h e area of t h e rec tangle

d e s c r i b e d in par t ( a ) .

4 open Sentences And English Sentences It i s a n a t u r a l s tep from translation of phrases t o t rans la-

tion of sentences.

Example : 4 5 + 3x = 108

How s h a l l w e w r i t e an English sentence f o r t h i s open sentence?

?le might say, " A book sa les ran is paid $ 4 5 a week plus $ 3 f o r

each s e t o f books he se l l s . In one week he was paid $108."

Another t r a n s l a t i o n of this open sentence could be, " A f r e i g h t

shipment consisted of a box weighing 45 pounds and a number of

small c a r t o n s each weighing 3 pounds. The whole shipment weighed

1CS pounds . " :!hat English sentences cou ld you write f o r t h i s same open -

sentence?

Problem S e t 4-2a -- ':!rite your own English sentences f o r the f o l l o w i n g open sentences.

1. n + 7 = 8 2 5. 4n + 7n = 44

2 . 2n = 500 7. 4 k + 7 k = 4 7 3. - - - 17 8. x + x + x + x = l 0 0 4 . x(w + 4 ) = 480 9. ~ + 5 = 5 f Y

1 1 5 . a + (2a + 3a) = (a + 2a) + 3a 10. -p + ?b = 6

Mure often we want to t r a n s l a t e E n d i s h sentences into m e n

sen tence? . ':'e f i n d s u c h open s e n t e n c e s p a r t i c u l a r l y h e l p f u l in

word problems srhen the English sentewe is about a guantity which

we are i n t e r e s t e d in finding. Exanpie 1. "Carl has a b o a r d 44 inches long. He wishes t o c u t

it i n t o two pieces s o t h a t one p iece will be three inches longer

than t h e o t h e r . How l o n g shou ld t h e s h o r t e r piece be?"

!>#-e may sometimes see more e a s i l y what our open sentence

s h o u l d be i f we guess a number f o r t h e q u a n t i t y a s k e d for i n

t h e problem.

I f t h e s h o r t e r piece is 18 inches long, then the longer piece

is (18 + 3) i n c h e s long , Since t h e hhoie t loard is 44 inches long, we then have the sentence

J 18 + (18 + 3 ) = 4 4 .

Although t h i s sentence is not t r u e , it suggests the p a t t e r n w h i c h

we need for an open sentence. No t i ce t h a t t h e q u e s t i o n in t h e

problem has pointed out our va r i ab le . Ide can now say:

If t h e s h o r t e r piece is K inches l ong ,

t h e n the l o n g e r piece is ( K + 3 ) inches long, and the sentence I s

K + (K + 3 ) = 44.

I We say t h a t this sentence is f a l s e when K is 18. T h e r e

probably IS some v a l u e of K f o r whlch t h e open sentence I s true.

If we wanted t o f i n d the l eng th of t h e s h o r t e r piece, t h i s c o u l d

be done by finding t h e t r u t h s e t of t h e above open sen tence .

Perhaps you feel an urge t o f i n d a number whlch does nlake tihe

above sentence t r u e . If so, go ahead and try. For the p re sen t ,

however, o u r object ive is p r a c t i c e in writlng t h e open sen tences .

1 Later we s h a l l b e concerned w i t h f i n d i n g t h e t r u t h s e t s of s u c h

setltences and thils answering the questions in t h e problems.

In t h i s example we t r i e d some p a r t i c u l a r numbers f o r t h e

quantities involved t o h e l p see a pattern f o r t h e apen sentence. You may sometimes see the open sentence irnmedlately w i t h o u t having

t o try p a r t i c u l a r numbers,

Notice t h a t t he English sentences are o f t e n about inches o r

, pounds o r years or d o l l a r s , b u t t h e open sentences a r e a lways

ahou t numbers only.

N o t i c e a l s o t h a t we a r e very c a r e f u l i n describing o u r

va r i ab l e to show what it measures, whether I t is the number of

inches, the fiumber o f d o n k e y s , ur t h e number of tons .

Example 2. "Two cars start from the same p o i n t a t t h e same time and t r a v e l in the same d i r e c t i o n a t constant speeds of 34 and 4 5 mlles per hour, respec t ive ly . In how many h o u r s w i l l t h e y b e

35 rnlles apart?"

If they travel 4 hours , the raster ca r goes i!5(4) m i l e s and

the s lower car goes 3 4 ( 4 ) miles. Since t h e faster car s h o u l d

[ s e c , 4-21

ther. be 35 miles f a r t he r from the s t a r t i n g po in t than the slower car, we have t h e sentence

4 5 ( 4 ) - 3 4 ( 4 ) = 35, which is f a l s e , It suggests, hodever , t h e following:

If t l l ey t r a v e l h hsurs , t hen the faster

c a r goes 45h miles and t h e s lower c a r

goes 34h miles, and

45h - 34h = 35.

Example 3. " A nan le f t $10,500 f o r hls w j d o w , a son and a d a u g h t e r .

T h e ;didow received $5,000 and t h e d a u g h t e r rece ived twice a s much

a s t h e s o n . Haw much d i d the son receive?" l f t h e son received n do l l a r s ,

then t h e d a u g h t e r r ece ived ';in

d o l l a r s , and

n + 2n + 5000 = 10,500.

Problem Set 4-2b -- ?!rite open sentences t h a t wn3uld help you s o l v e prcblems 1-13,

being careful t o g l v e t h e meaning of the ~ ~ a r i a b l e f o r each. Your

wurk rrlay be shown In the form indicated in Example 3 a b o v e . IC

is n o t necessary t o f i n d t h e t r u t h s e t s o f the open sentences.

1. Eenry and Charles wer-e uppuslng cana lda tes in a class e l e c t i o n .

Henry received 30 v o t e s more than Charles, and 516 m e m b e r s of

the class ?iroted. How many v o t e s d i d Charles rece ive?

(~int: If C n a r l e s received c vo tes , write an open phrase f o r

the number o V v o t e s Henry rece ived . Then write y o u r open

sentence. ) 2 . A rec tangle is 6 times as long as it is wide. I t s per imeter

is 144 i nches . H0.4 w i d e I s t h e rectangle? ( ?emember to d r a w a f i g w e . )

3 . The largest angle 3s a triangle is 20° more t h a n t w i c e t h e smallest, and the t h i r d angle is '70~. The sum of the angles

of a triangle 1 s 180~. IIOW large i s t h e smallest angle?

5 -

LO.

A br id j ie has three spans, 3ne cf which is 100 reef longer thar!

each of t h e o t h e r t w o , If the b r i d g e is 2500 f t . . long, how

lans 1s each of t h e o h o r t c - spans?

A class of 4 3 students was separated i n t o t w o c l a s ses . If theye were 5 more students in Mr. Smith ' 3 class than i : ~ Miss

Jones's class, h o ~ m a ~ y students were in each class? : Can

y u u d o t h i s one in t w o ,days? If there were y students i n

Miss Jones 1s class, f i n d two ways t c say how many were in M r . Smith's c l a s s . )

The l e n g t h of a rectangle I s 5 inches more than j . ts w i d t h .

What is the l eng th of t h e rectangle if I ts area Is 594 square inches ?

Jnhr , is three tfmes as old as Diclc, Three years ago t h e sum

of t h e i r ages was 22 y e a r s . How old is each riuw? ( ~ r l n t :

Find a phrase for t h e age of each three years ago in terms n f C S c k l s age now.)

John has 1.65 In h i s pocket , all in nickels, dlmes, and

quar te rs . H e has one more q u a r t e r than he has d i m e s , and

t h e number of n i c k e l s he has I s one more t h a n t w i c e t h e

lumber of dlmes. How many dimes has he? ( ~ i n t : If he has

d d i m e s , wr i t e a phrase for the value of all h i s dimes, a

phrase f o r t h e v a l u e of all his quarters , and a phrase for

the - ~ a l u e of a l l h i s nickels; then write your open s en t ence , ) I bought 2 3 postage stamps, some of them a-cent s t a m p s and

some 7-cent stamps. If the t o t a l c o a t was 1-19, how atany

o f each kind d i d I buy?

A passenger t r a i n t r a v e l s 20 miles p e r hour fas ter than a

f r e i g h t t r a l n . A t t h e end of 5 hours t h e passenger t r a i n has

t r a v c l e d 100 r i l e s rar t9er t h a n the freight t r a i n . How f a s t

30e3 t h e T r e i g h t t r a i n t r a v e l ? (~lnt: F o r each t r a i n f i n d

a pl.lrase f o r the number of r.iles it has t r a v e l e d . )

A s t o r e has 39 quar t s of m i l k , some i n pint car tons and some

in h a l f - p i n t c a r t o n s . There are 6 times as m n y p in t ca r tons

as h a l f p i n t ca r tons . Haw many half-plnt c a r t o n s a?e there?

12. Mr. Ilro~.rn is employed at an i n i t i a l salary of $3600, w l t h

a n annual increase of $390, while Mr. Wkllte starts at the same time at an initial salary of $4500, with an annual

increase of $200. After how many years w l l l the two men be

ea rn ing t he same salary? 13. A t a b l e is t h r e e times a s l o n g as i t is wide. If it were

3 feet s h o r t e r and 3 feet wider , it would be a square. How

long and how wide is I t? raw two p i c t u r e s of the t ab le t o p , )

14. ' { ! r i te y o u r own problem f o r each of the following open

sentences.

(a) 5n + 10(n -I- 2 ) + 50(an) = 450 ( b ) a ( 3 a ) = 300

( c ) .60x+ 1.10(85 - X) = 78.50 ( d ) a + (a + 3 ) -+ (a + 6 ) - 69 ( e ) b = 2(1R - b) (f) 3 0 h + 4(5 - h) = 57

1 1 ( P I $ + T + x = 1 I f . Translate t h e f o l l o w i n g into an open sentence. Then t rans-

l a t e back from the open sentence to a d i f f e r e n t problem in

English.

"Three hundred sight-seers went on a trip around the

bay In t w o boats. The capac i ty of one boat was 80 mare

paszengers than that o f the o t h e r and b o t h b o a t s were f u l l .

How many passengers were in each boat?

- 3 . Open Sentences Involving Inequalities

Our sentences need n o t a l l be equalities. Problems concern-

i n g " g ~ e a t e r t h a n " o r " less than" have real meaning.

Suppose we say, "Make a problem f o r t h e sentence d + 2 ) 5." The word praulem could be , "If I added two d o l l a r s t o what I now

have, I would have more t h a n f i v e d o l l a r s . Row much do 1 have now?I'

P r o b l e m S e t 4-3a -- ':'rite ppot; l e n s for Lhe f u l l a w in2 open senten8:es.

L. a > 3 5 . a + ca + ,?a> 4,? 2 . n + 1 ( 1; , . a + l a + 3a = 5;' u , i15qi175 * y t 3 < *I, ar,d y + ?, , 5 4. 5 ( n + 5) < 10Q 1 $I. -k < ' 2 i

5. p + 10,000 > 160,000 * lo . s > 2j12- ::)

As -:;ith equations, ;t idill someti!nes help to f i r 1 5 a n open

seatence 1rl p r~?~ le rns a k o u t i n e q u a l l t l e s iT xe t r y a p a r t , i c u l a r

number ' i r s t ,

Zxarnple 1. "In s i x months Mr. A d a m s earned rnljre than $7<:'3'3.

How much d i e he e a r n pe r month?"

If he earned $11~0 per month, I n 6 months h e ~ 0 ~ 1 3 e a r n

6 x 1100 dollars, The sentence wi3ald t ,hen be 6 x l lo i j > - iOQ{ j .

T h i u , of course, is not true, but it suggests ?!hat we s h d u l d d o .

If r4r. Aaams earned a g o l i a r s per month , in b months he

would ea rn 5a d c l l a r s . Tnen

ha > 7000. Example 2 . "The d i s t a n c e an o t J e c t f a l l s d u r i n g tne f i r s t second

is 32 feet l e s s t h a n t h e distance it falls during the s e c o n d

second. During the t w o seconds i t falls 48 feet or l e s s , depending

on t h e air r e s i s t ance . How far does it fall d u r i n g t h e 9econ3

second?

If t h e ob j ec t f a i l s 42 feet d u r i n g the second second, then it f a l l s (42 - 32) r e e t during t h e f i r s t s e c o n j . S l n a e the t o t a l

distance f a l l e n is l e s s t han o r equal t o 48 feet., o u r sentence is

(42 - 3 2 ) + 42 I 48. T h i s s u g g e s t s how to w r i t e t h e cpen sentence. If t he o b j e c t

f a l l s d f e e t d u r i n g t h e second second, then it f a l l s ( d - 3 2 )

f e e t d u r i n g t h e ffrst second, and

( d - 32) + d 4e.

[ s e c . 4-31

Exar-iple 3 . "T.:.<j s i de s of a t r i a ~ g l e have lecgths r?f F ir-ches ar.,!

i r l r h e a . '::hat Is t h e length of' t h e t h i r d s ide9"

Yo2 :my ha7e drawn m n y

triangles in t h e past and h a v e be- n cc:ne a-..rare of' t h e f a c t t h a t t h e

len,:r.h of any s i d e l ~ f a t r i a n g l e aust

t e l e z s than t h e ?urn of t h e lengths

of t he ~ t h e r cwo s l d e s . Tnus If t h e

thlr3 s i d e of this t r l a 1 2 ~ l e is n

i n c h e s l ong ,

n < 54- 6. A t t h e same time the s i x I n c h s i de must be less in l e n g t h t h a n

t h e sum of the l e ~ g t h s of t h e o t h e r two; thus

d < n + 5. Since b o t h of t h e s e conditions m u s t h o l d , the open sentence f o r

o u r p rob lem i s

n < ? + b a n 3 5 < n 4 - 5 .

P r o b l e m S e t 4-3b --- Write open sen tences f o r prot lems 1-13, being c a r e f u l to g i v e

t h e meaning of t h e var iable f o r each.

1. One t h i r d of a number added t o t h r e e - f o u r t h s of t h e same num-

b e r is equal t o or grea te r t h a n 29. What Is t h e nutnber?

2. B i l l i s 5 years o l d e r t h a n Norman, and the sxin oi' their ages

is l e s s than 2 3 . How o l d is Norman?

3. A square and an e q u i l a t e r a l t r i a n g l e have equal perimeters. A s ide of t h e t r v i a n g l e is r i v e irlches l r ~ n g e r t h a n a s ide of

t h e square . )!hat is t h e l e n g t ! ~ of t h e s ide of t h e s q u a r e ?

D r l w a f i g u r e .

h. A b o a t , t r a v e l i n g downstream, goes 12 miles per h o u r fast .er

than the rate of the c u r r e n t . I t s v e l o s i t y uuwnstream i s

l e s s t h a n 30 mlles per k~c;ur. What Is t h e r a te of the c u r r e n t ?

5. Joh:-~ s a i d , "It .dill take me more than 2 h o u r s t o mow t h e lawn

and I must nc t spend more t h a n 4 h o u r s on t h e j ob or I won4t

b e able t c go swimming." Yow much time can h e expect t o

spend on t h e job?

6. OR a h a l f - h o u r TV show t h e a d v e r t i s e r i n s i s t s there m u s t be

at l e a s t t h r e e minutes f o r commercials and the n e t w o r k Ln-

s i s t s there must be less than 12 minutes for commercials.

Express t h i a in a mathematical sentence. How much time

m u s t t h e program d i r e c t o r p r o v i d e f o r material G t h e r t h a n

a 3 v e r t i s ing?

7. A t e a c h e r says, "If I had 3 times as many students in my

c la s s as I d o have, I would have at Least 26 more t h a n T now h a v e . ' I How many sturlents does he han/e i n h i s c l a s s ?

8. The amount of $205 is t o be d i v i d e d among Tom, Dick and H a r r y . D i c ~ is t o have $15 more t h a n H a r r y a n d Tcm is t o

have twice as much as D i c k . How must the money be d i v i d e d ?

9 . An amount between $205 and $ 2 2 5 , inclusive, is t o be d i v i d e d

among three b r o t h e r s , Tom, C i c k and H a r r y . Dick is to have

$15 mare than H a r r y , and Tom is t.o h a v e twice a s much as '

Dick. H o w much money can H a r r y expect?

*10 . A s t u d e n t has t e s t grades of 75 and 82 . What m u s t he s c o r e

on a t h i r d t e s t t o have an average o f 88 o r h i g h e r ? If

150 is t h e h i g h e s t score possible on t h e t h i r d t e s t , how

h i g h an average can he achieve? What is the lowest average

he can a c h i e v e ? 11. Using t w o va r i ab l e s , w r i t e an open sentence f o r e a c h o f the

following E n g l i s h sentences .

( a ) The enrollment I n S c o t t Schoo l i s g r e a t e r than t h e

enrollment in Morris School .

( b ) The enrollment i n S c o t t School is 500 grea te r t h a n t h e

enrollment in Morris S c h o o l .

R e v i e h Problems --.

1 . Wrl te E r i q l l s h phraoes whj ch are rea l i s t i c translaciuns o f

the f o l l o w i n g open phrases. In eack problem be c a r e f u l t o

i d e n t i f y t h e variable explizitly.

( a ) x + 15

) 3p + 3 ( p - 1) ( c ) 613 - lot ( d ) 9 ( l g r - t - 251

( e ) i b ( 3 k - 4 )

In problems 2 , 3 , 4, 5, write open phrases which are translations

of t h e word phrases . l n eactl projlern be c a r e f u l t o indicate what the v a r l a b l e r ep r e sen t s , If it is n o t a l ready given. In c e r t a i n problems you ray use more than one va r i ab l e ,

2 . ( a )

( b ) ( 4 ( d l

( @ I 3 . ( a )

( f l '1 . ( a ,

( b )

A number dirnlnished by 3 .

Sam's age selerl years from now.

Mary ' s age t e n years ago.

Temperature 2 0 degrees h i f i h e r than the present

tempera tJu re. Cost of n p e n c i l s at 5 cents each

The amount of money :n my pocket : x dimes, y n l c k e l z ,

and 6 pennies.

A number increased by t i ~ i c e the numter .

A number Increased by t d l c e anothe? number.

The nunber of days i n w weclcs.

une mllZlvn more t h a n twice t h e populat ion of a c i t y

In .?knsaz.

Annual sa lary equivalent t o x d o l l a r s p e r month.

One d c l l a r nore thar twice B e t t y Is allowanze. The d l s t a n c e t r ave l ed :n h h ~ r s a t a constant speed

o f 40 m.p. h,

The rea l e s t a t e tax on property h a v l n g a v a l u a t i o n of

y d o l ; a r s , t h e tax r a t e being $25 .00 per $1000

va;ua:lon.

F o r t y pounds rnore than E a r l I s r ~ e i g h t ,

( e ) Area of a rectangle hav ing one s i d e 3 inches longer

than a n o t h e r ,

5. ( a ) C o s t of x pounds of steak a t $ l . 5 ~ , per pound.

( b ) Catherine 1s earnings f o r z hour s at 75 cents an hour.

( c ) C c s t of' g gallons of gasol ine at 33 .2 c e n t s a g a l l o n ,

( d ) Cost of purchases: x melons at 2 9 cents e a c h and y

pounds of hamburger at 59 c e n t s a pound.

( e ) A t v l o - d i g i t number whose te11s' d i g i t is t and whose

u n i t s ' d i g i t is a. 6. !#!rite E n g l i s h sentences which are t r a n s l a t i o n s of t h e open

sentences .

( a ) ? r < , Y $ O

( b ) y = 3609

( c ) z > 10C,000,000

( d ) u + v + w = 180

( e ) z(z + 18) = 363

In problems 7 , 8, 9, wri te open sen tences corresponding to t h e

word sentences, u s i n g one var iab le in each. In each problem

b e c a r e f u l t o s t a t e ifhat t h e v a r i a k l e represents, if this is

n o t already ind i ca ted . 7. ( a ) Mary, who is 16 and has two b r o t h e r s , i s 4 years o l d e r

t h a n he r s f s t e r ,

( b ) B i l l bought b bananas a t g c c n t s each arid paid 54 cen tz .

( c ) If a number is added t o twice the number, t h e sum i s

less than 39. 3) Arthurts allowance is one dollar more than twice

B e t t y I s , b u t Is two d o l l a r s less t h a n 3 times B e t t y t 5 .

( e ) T h e distance f rom Dodge C i t y to Oklahoma City, 26;)

miles, was t ra7 je led i n t hours a t a n average spee.1 o f

43 miles an hour .

(f) The agto trip f rom St. L t ~ u i s t o Memphis, 333 miles,

wae n u d e in t hours at a maximum speee of 55 mile:

an hour .

8. ( a ) P i k e ' s Peak is more t h a n 14,N10 f e e t a w v e sea level.

( b ) A book, 1.4 incb.es t h i c k , ha? n sheets; eacn shee t is

I 0.003 inc:hes t h l c k , al?d each c o v e r is - i n c h e s t h l c k 10 an3 is ' c l u e .

( c ) Tdo r n i l l i v n I s more ti-an twice t h e p o p u l a t i o n ST any

c l t y in Colorado,

( d ) A square of s i d e x has a Lhrger area t h a n has k

rectangle of s i d e s ( x - 1) and [ x + I ) .

(e) The tax on rea l e s t a t e is ca l cu l a t ed a t $24.00 per

$1000 v a l u a t i o n . The tax assessment on pr3pert;l valuea a t g d o l l a r s is $346.00.

( f ) IT E a r l added 4 3 pounds t o his weigh t , he ;dould s t i l l

r.3t vlelgh mre t h a n 152 puunds.

9. ( a ) T h e s u m of a whole number and i t s successor i s 515. ( b The s L n or a wklule number and it^ S u c c e s ~ o r IS 576.

( c ) T h e sLm of two numbers , t h e s ecand g rea te r t h a n the

f i r s t b y 1, is 57G. ! d ) k board 16 f ee t long is c u ~ In two pleces such t h a t

u t l e p'ece 13 one foot l o n g e r t h a n twice tk.e o t h e r ,

[ e ) Catherine ea rn s$ ' 2 . 25 baby-sitting f o r 3 h o u r s a t c c n t ~ an h o u r .

10. A two-digit number is 7 more than 3 ~irnes the sum of t h e

d i g i t s . R e s t a t e t h i s by an open sentence. (Hint:: Express

t h e nmber uy means of twc v a r i a b l e s , as in Problem 5 ( e ) , )

11. The sum of t w n numbers is 42. If the f i r s t r~umber 2s

represente ' j by n , wr i te ar, expression f o r the second number

u s i n g t ,he var iab le n.

1 . ( a ) A number i s increasec by 17 a n d t h e sum Is multiplle3

by 3. ldrl5.e an open sentence s t a t i n g t h a c the resulting

product eq,~als 192. ( b If 17 i s a3ded yo a number and t h e sum I s multiplied

by 3 , t h e resu1;ing p r o d u c t is less t han 192. Restate

t h i s a s an open sentence.

13. One number is 5 tfmes a n o c h e r . T h e sum o f t h e two numbers

is 15 more t h a n 4 t i n e s t h e smaller . Express t h f s by an

open sentence.

1 4 . Sue has 16 more books than S a l l y . Write an open sentence

showing t h a t toge ther they have more than 26 books.

15. (a) A farmer can plow a field in 7 hou r s w i t h one of his

t r a c t o r s . How much of t h e f i e l d can he plow in one hour w i t h that t r a c t o r ?

( b ) With h i s o t h e r t r a c t o r he can plow the f i e l d in 5 hours. If he had b o t h t r a c t o r s going f o r 2 hours ,

how much of t h e f i e l d would be plowed?

( c ) How much of the field would t h e n be left unplowed?

( d ) Write an open sentence which i n d i c a t e s t h a t , If b o t h

t r ac to r s are u s e d f o r x hours, t h e f i e l d w i l l be

completely plowed.

+16. If you fly f r o m New York t o Los Angeles, you gain three

hours. If t h e flying time is h hours, when do you have

t o leave New York in o r d e r t o a r r i v e in Los Angeles before

noon? Write an open sentence f o r t h i s problem.

17. Mr. Brown l a reducing. During each month ror t h e past FJ months he has l o s t 5 pounds. H i s weight is now 175 pounds.

Wt was his weigh t m months ago if rn < 8? Vrite an open

sentence stating that rn months ago his weigh t was 200

pounds.

Write open sentences for problems 18 to 2 3 . T e l l c l e a r l y what

t he varlable represents, but do no t find t h e tmth s e t of the

open sentence.

18. ( a ) The sum of a whole number and i t s successor I s 45. What are the numbers?

(b) The sum of two consecutive odd numbers is 76. What

are the n-umbers?

13. Mr. Barton p a i d $176 f o r a f reezer which was so ld at a discount of 12% o f t h e m r k e d p r i c e . What was the

marked price? 20. A manfs pay check f o r a week of 48 h o u r s was $166.40. He

1 is paid at the r a t e o f I- times h i s normal r a t e f o r a l l 2

hour s worked in excess of 4 0 hours , What is h i s h o u r l y

pay rate?

1 A Inan r ' lres a r i f l e a t a t a r g e t , Two seconds a f t e r he fllres

i t ha hears the saurid of' the b u l l e t striking t h e target . If

the speed cf sound J s 1100 f c e t per seccrid and t h e s p e e d of

t n e bullet is 1700 feet per sezond, how f a r away i3 t h e

target? 2 . The ztudents a c t e n d i n g Linc3ln High School nave a h a k i t of

c u t t i n g across a vacant l v t near t he school i n s t e a d of

f o l l o w i n g the side-walk around the corner. Tkie lo t ; 1s a

rectangular l o t 200 f e e t by 300 fee t , and the s h o r t - c u t

f c l l o w s a straight l i n e from oce corner of t h e l o t to t h e o ~ p o ~ i t e corner. How long 4 s t h e s h o r t - c u t ? (Hlnt: Use

the Pythagorean Theorem. )

23. Dne end of a 5C-fo3t wire is a t t a c h e d to t h e t o p of a

v e r t i c a l telephone pole. The wire is pulled t a ~ t and t h e

l o ~ e l - e n d is a t t ached t o a concrete b l o c k on t h e ground.

T h l s h l a c k is 3,'3 f e e t f r v m the base of the te le7hone pole,

on le,;el ground. ;:hat 3s the h e i g h t cf t h e po le?

24. ( a ) A t an a ~ t o parkir,g l o t , the charge Is 35 c e n t s f o r t h e

f i r s t hou r , o r f r acc ion of an hour, and 20 c e n t s f o r

each succeedlcg (whole o r p a r t i a l ) one-hour period.

If t i s the number of one-huur pe r iods p a r k e d a f t e r :he

i n i t i a l h o w , write an o p e n phrase for t h e parklng fee.

(b) ' J i t h the same charge f o r parking as in the preceding

prublem, if h is ;he t o t a l number of one-hour periods,

parked, w r i t e an open phrase f 'gr t h e parking f e e .

2 Two qua r t s of alcnhnl are addcd to the water I n t h e

r ad i a to r , and t he mixture then cantains 2C per c e n t a l c u i i ~ l ;

t h a f is, 20 p e r cent of the mixture is pure a l c o h o l . {trite

an oger~ sentence rar t P . l s English s e ~ ~ t e n c e . [ ~ l n t : '?!rite an opeL phrase f o r the riurr~bber of' quar t s of a l c o h o l In terrns

o f the number of qua r t s of waser o r i g i n a l l y I n t h e r a d i a t o r . )

2 . ( a ) Two water-p:pes are bringing water i n t o a r e s e r v o i r .

One pipe bas a capaci;y of' 100 gallons pe r minute, and

t h s se(;wnd 110 g a l l o n s pe r minute , I f water f lows f rom

t h e f i r s t pipe f o r x nin~tez a n 3 t h e s e c ~ n 3 for

y minu tes , write an open phrase f o r t he t o t a l f l o w In

g a l l o n s .

( b ) In t h e preceding problem, if t h e flow f rom t h e f i r s t

p i p e is stopped a t t h e end of two hour s , wr i t e t h e

expression f o r the t o t a l flow i n gallons in y minutes,

where y is greater than 120.

( c ) 3 1 t h t h e data in part ( a ) , write an open sentence

s t a t i n g that t h e t o t a l flow is 20,000 gallons.

27 . ( a ) P l a n t A grows two inches each week, and I t is now 2 0

inches t a l l , +!rite an open phrase f o r the number of

i n c h e s in i t s he igh t w weeks from now.

(b) P l a n t E grows three i n c h e s each week, and it is now

12 i n c h e s t a l l , Write an open phrase f o r t h e number of

inches in its h e i g h t w weeks f r o m now.

( c ) I n t he course of some weeks, t h e p l a n t s w l l l be equally

t a l l . Express t h i s by means of an open sentence.

2 , A man breathes 20 tiraes per m i n u t e at sea level and takes

one ext ra breath per minuke f o r each 1500 f e e t of ascent.

( a ) 'Write an open phrase f o r the number of b rea ths he takes

each minute h f ee t above sea l e v e l .

( h ) A t y fee t above sea l e v e l a man breathes 24 times per

minute. With the i n f o r m a t i o n obtained in (a), w r l t e

an open sentence stating this f a c t .

2 9 . Use each of t h e verb symbols =, f, <; d , ), $, I, 2, in x and g raph each of the open sentences which you ge t . - 2'

3 G . Find open sentences whose graphs are t h e following: I -

( a , I A 1

1

1 2 3 4 5 6 7

31. Graph the following open sentences:

( a ) x = 2 or x > 5 (b) x = 2 a n d x > 5

( 4 x > 2 o r x < 5

( d ) x < 2 and x < 5 " 3 2 . Find an open s e n t e n c e whose graph w i l l be:

You w i l l prcbably need $ 0 make a compound sentence u s l n g

" a n d " a n d " o r " , as well a3 inequalities.

33. A man, w i t h f i v e d o l l a r s in h i s pocket, s t o p s at a candy

s t o r e on n l s way home wlth t h e intention of t ak ing h i s wife

t w o pounds 01' candy. He f i n d s candy by the pound box selling

f 'or$1.69, $1.95, $2.65, and$3.15. I f he leaves t h e store

x i t h t w ~ one-pound boxes of candy,

(a) m a t is the smallest amount of change he cou ld hat:e?

(b) \&!hat I s t h e greates t amount of change he could h a v e ?

( r ) What s e t s of two boxes can he n o t afford? * 3 4 , At t h e en, j of Chapter 3 (~xercises 3-14, P r o b l e m 5 ) you

? l s r , n 7 ~ e r e d a "mile for m d t i p l y l n g t e ens t1 . Using a and b,

r e s p e c t i ~ r e l y , to s t and f o r t h e units d i g i t s o f t h e two

n;~mbers , you s h o u l d now be able to write an open sentence

which expresses the product p in terms o f a and h. erhen you

h a * ~ e w r i t t e n your sentence, use t h e distributive p r o p e r t y to

v e r i f y t h e correctness of your c h o i c e .

Chapter 5

TKE REAL liUMBERS

5-1 . T& Peal Iiurnber Line

As you w o r ~ e d wlth t h e number l i n e , you n12y have ' o ~ e n

c u r i o u s about severa l things. For one t h i n g , a iine extends

witklout end t o the left as wcll as Lo t h e rlgnt. We have , how-

ever, labeled only those p o l n t s on t h e r i g h t of 0 . This r a i ses

a question which we shall answer in this s e c t i o n : How shall we

l a b e l the points on the l e f t ' ?

In Chap te r 1 you were t o l d t h a t there are r a t i o n a l numbers

t o be a s soc i a t ed w l t h p o l n t s on the l e f t h a l f of the number l i n e ,

b u t meanwhllc you have dealt only v; i th rational numbers on t h e

right h a l f . .For ano ther thing, you were t o l d t h a t some p o l n t s

on t he number l i n e do not correspond to rational numbers. Thiz

raisss a second question: Where are some of these points on t h e

number l i n e w h l c h do not correspond t o rational numbers , and

what new nunzhers a r e associated w i t h them?

L e t us r e t u r n t o t h e f i r s t of these ques t ions : H o w s h a l l

we labe l t he p o i n t s on the l e f t of O? There is no d o u b t t h a t

the l l n e contains i n f i n i t e l y many p o i n t s t o the l e f t o f 0 . It

is an easy matter t o l a b e l such p o i n t s if w e follow t he p a t t e r n

we used t o t h e r i g h t o f 0 . As b e f o r e , w e use t h e i n t e r v a l

from 0 to 1 as t h e u n i t of' measure, and locate points equally

spaced a l v n g the l i n e to the l e f t . The f i r s t of t h e s e we l a b e l ---

- -1, the second 2 , e t c . , where t h e symbol "-1" is read "nega t ive - 1 , 2 is read ' 'negative 2 " , e t c . What is the coordinate of' t he

point which is 7 units t o the l e f t o f O?

Proceeding as before, we can find additional p o i n t s to t h e

left of 0 and l a b s 1 them with symbols similar to those used

f o r numbers to t h e rlght, with an upper dash co indicate - t h a t t h e

number is to t h e left of 0 . Thus, for example, -$ is the same

d i s t ance from 0 an the l e f t as 3 I s on t h e r i g h t , e t c .

The s e t of - a l l numbers associated with p o i n t s on t h e number

line is c a l l e d the s e t of real numbers. The n u m b e r s t o the l e f t

of zero are c a l l e d the negative real numbers and those to the

right are c a l l e d the positive real numbers. I n t h i s language,

the n m b e r s of arithmetic are the non-negative real numbers . - The s e t of all whole numbers {o, 1, 2, 3 , . . . I combined with - -

the s e t [-1, 2 , 3, . . . I is c a l l e d t he se t o f i n t e g e r s -

[ ..., -3, 2 , -1, 0 , 1, 2, 3 , ... 1. The set of a l l rational nlmbers of' arithmetic combined with t h e negat ive rational

n m b e r s i s cal led t h e set of rational numbers. (Certainly, a l l

rational numbers are real numbers.)

Hemember tha t each r a t i o n a l number I s now assigned t o a

p o l n t of the number l i n e , but t he re remain many points to whlch

rational numbers cannot be assigned. The numbers a s s o c i a t e d

with these points are called t h e irrational numbers. (Thus, a l l

irrational numbers a re a l s o real numbers.) Hence, we can regard

t h e s e t of rea l numbers as t h e combined s e t of rational and

irrational numbers.

- F o r example, ell i r ~ t e g e r s , such as 4, 0 , 2 , are rational

numbers; f i nd examples of r a t i o n a l num-nerz which are n o t i n t e g e r s . -

i FurLhernore, a l l rational numbers such as f, 0 , 6 , a re real -

i numt;cr;.

Thi' second questlon remains: Imert- al'c some of the po :n t s

I on t n e n u r n k r l i n e which do n o t co r r ezpond t o rational numkers?

1 I t wili be proved I n a later chapter t h a t , f o r example, t h e

/ n u r n h e ~ f i ii an i r r a t i o r l a l number . L e t u s l o c a t e she

points with coord ina tes f i and "&, respectively . First of a l l , we r e c a l l chat is a number w h ~ s e square

/ is 2 . YOU may have learned t h a t the l eng th of a d i a g o n a l of o

I s q u a r e , whose s ides have l e n g t h 1, is a number whose square is

2 . (DO you know any f a c t s about, r i g h t triangles which will

help you v e r i f y th is : ) ) In orde r to l oca te a p o i u t on t h e number

l i n e f o r a, a l l we have to do is c o n s t r u c t a square with side o r length 1 and t ransf 'er the l eng th of one of' its diagonals t o

o u r numi~er l i n e . T h i s we can do, as in t h e f i g u r e , by drawing

a c i r c l e whose c e n t e r is at t h e p o i n t 0 on the number line and

vyhose radius is t h e same length as the d iagona l o f the square.

T h i s c l r c l e c u t s t h e number l l n e i n t w o p o i n t s , whose coordinates

a r c the real numbers and -&, respectlve1:r .

Later you wlll p rove t h a t the number x2 is not 3 r a t i o n a l

number . Maybe you bel ieve t h a t f i i s 1 . 4 . Test f o r yourself

whether th is is traile b y squaring 1.4. Is ( 1 . 4 1 ~ t h e same

number as 2: In t h e same way, test whether f i is 1 . i l l ; 1.4111.

The square ~f each of these decimals is closer t o 2 than the pre-

ceding, Lut there seems to be no r a t i o n a l number whose square i s

2 .

[ s e c . 5-11

There a r c many more polnts on t n e ro,l'L number l i n e which

h a v e coordin?tes whlch are n o t r a t l o n a l n ~ i n b e r s . Do you think

is such a p o i n t ? 3 t hky? 2

Problem SEL 5-1

? . Draw t he graphs o f the following s e t s :

-1 (4 {o, 3 * -5, T I

2 Of the two points whose coordinates a r e given, which is to t h e

right of the o y h e r ? -

( 3 ) 3 , - 4 (4 5. 0 (h) -4, &

(b) 5, -4 -5, -10 -16 -21 (f) 2 -q 7, ( c ) -2, -4 -1 1

(d 0 , 3 (j) 7, 2 (d) -A-, 1

The number T is t h e r a t i o of t he clrcurnference o f a c i r c l e to

its di3meter. Thus, a c i r c l e whose diameter is of l e n g t h 1

hac a circumference cf l e n g t h ;7 . Imagine such a c i r c l e r e s t i n g on the number l i n e at the poFnt 0 . If t h e c i r c l e I s

rolled on t h e line, w i t h o u t s l i p p i n g , one complete r e v o l u t i o n

50 the right, It will s t o p on a p o i n t . What is t h e coordinate

of t h i s p a l n t ? Tf r o l l e d t o the l e f t one revolution it ~ 1 1 1

s top on what p o i n t ? Can you l o c a t e t h e s e p o i n t s approximately

on t h e real number l i n e ? ÿÿ he real number i~ , l i k e a, is n o t a r a t i o n a l nunber. )

4. 2 ) Is -2 a whole number? An integer? A rational n-amber?

A real number? - 10 (b) Is - a whole number? An i n t e g e r ? A r a t i o n a l number? 3

A real number?

( c ) Is -fi a whole nunber? An In teger? A rational number?

A real number?

j. ' ihich o f i h e following s e t s are the sane?

A is Lhe set of whole n u ~ b e r s , 3 i s the s e t of positlve

i nzege r s , C is the set of non-negacive integers, I is cne

set of I n t e g e r s , Pi is t h e s e t of counting nuntors.

5 - 3rder on the Z e a l N u ~ b e r Line ---- 3ow dj.d wc d e s c r f b e o ~ d e r f o r the positive reel nunbers?

S i n c e , f o r exm~ple, "j is 5 0 the lef'l; of 6'' or, the rumue? line, ant s i n c e " G , I s l e s s than 6", we agreed t ha t t h e s e t w o sen tences

say <he same t h i n g a b o ~ t 5 and 6 , We wrote this as the rrue

sefit cnc e

5 r 6 . Thus , f o r a p a i r of positive real numbers, "is to the left of"

on Lne n c ~ S e r line and "is less than" describe the same order.

\;l-2; s h a l l we mean by "is l e s s than" f o r ar-y ~ ,wo real

n ~ n b e r s , whether they are positive, negative, o r O? Our answer

is simply: "is to the l e f t of" on the real n m b e r l l n c .

L e t us l o o k f o r a j u s t l f i c a t i o n in common experl ence . 411

of ..is a y e :m i l i a r with thermometers and are aiqare t n n t scaies on

t h e ~ . o x e : e r s z s e nuncers above C and numbers b e l ~ v i 0 , as well

as 2 i:seli,. We know tk,zt the coole r t h e wee ,~he r : fhe iower

on tfie s ca l e we read t h e temperature. If Ke p l a c e a themorneker

in L horLzontai position, we see thaz It resembles p a r t of our 1 1 real n m s e r l i n e . hken we say is less thant' ("I ,s a lower tem-

pt'r'a+cure t h a r , " ) , t!c Jean " i s t o t h r :eft, of1 ' on t h e t,herrnr;lmeter - -

s c a l e . On t h i s s c a l r , , +dkljch num:ser 1s the lesser, 5 or lo?

TIXIS l/!e extens o x l ' c rner meaning o f ' ' i s l e s s t h a n " t o t h e

1::hole zct of r e a l nmsers . We agree that :

1 1 .. Is l e s s t h a n 1 ' f o r r ea l n m h e r a

means " i s to the lefr; of"' on t h e

rcal n . m b e r line . I!' a and b

?-re real n - m b e r s , " 3 i.5 less

t1:an D" is wrir~cn

a < L .

(!:gw and in ';hc ,I:'utur4c a variable rLs understood to have as

i t s dolrxin t h e set hi. real numbers, unless ~5herwise s t a t ~ d . )

Can you give a mean in;^, for "is grc:iccr 5han1' f o r PC-al

nurncers? A s k e l ' o r e , use t h e symbol ">" f o r "is greater t -han" .

In the s m c :<>::, exslain the N I P ~ ? ~ I ~ Y ~ ~ S of " < " , - ")", - "g"', "$" for real nwfbcl-s .

1. 90r ezcn cf zhc f o l . i o v ; i n ~ sentet -dcs, determine which qre t r u e

and wh:sh f a l s e .

( c ) - 4 p 5.5

(f') - 4 # 3.5 -

( R ) -6 > 3

(11) 7 . 5 < -I: -

(i) -3 < 2.8 -

) 7~ { - 2 . 8

2 . Consider the fol lowir ,g p a i r s of real numbers and tiell which

of che sentences below each p a i r are t r u e and which are

f a l s e . For example, f o r he pair -

2 and 22-2: 2

-2 < t r u e ;

- is f a l s e ; -2 - 2

-2 > is f a l s e .

- - 5 A. ' (a) 3.14 and -3: ( b ) 2 and 2: ( c ) 2 i ~ r l d 2 ~ 2 :

- ( d ) O.OC1 and

. b~?.v: the p a p h of the t r u t h s e t o f cach of the f a l l o w i n g

sentences . For exarnpl? : -

x > 1, I 1 1

4 - 3 - 2 7 0 1 3 -

- (f) c ( 2 a n d c > 2

- - ( g ) a . r 3 and a 2 7

(h) d < -1 o r d > ? - (i) a (6 a n d a < -2

. F o p each or' t h e f 'o l1o~; ing s e t s . wri te an opEn s e n t e n c e

i n v o l v i n g t h e varLable x which has t h e g i v e n s e L as its

' I r u L h s e L :

( a ) A is t h e s e t of a l l real numbers no t equal t o 3 . ( b ) 2 Is t h e s e t of all r e a l numbers i e ss than or e q u a l t o

- - ( c ) C i s the s e t of 311 rea l numbers n o t iess than 5 ,

; ::fioose afiy p o s i t i v e real nurn3er p : chco'e zr-j n c g a t i v e ~ e z l

numter Y:. h h i c h , i!' any , of' t h e ?o i lowlr .g zer,te!lces aril

: Let t h e dotnain of' t h e varlable p be she s e t oi' i n t ege r s . T h e n f i n d t n e r r u t h s e t oi

- (a) 2 < p a n d p < 3 .

(b) p ( -2 and - $ < p .

( c ) p = 2 o r p = - 5 .

- i . ( a ) During a c o l d Cay the LemperaLure r l ses 10 degrees from

-5'. Whht is the final t empera tu re?

i t ) On a n o t h e r day t h e temperature r i s e s 5 degrees f rom - l o 0 . How hi@ does it go? -

( c ) During a January thaw the Cemperature r lses f r o n 15' to 35U. How much d l d it r i s e ?

6 . I n t h e slanks Gelow u s e one of =, <, ) to makc a t r u c ::en-

t e n c e , if p o s s i b l e , In each case .

(d) -9 -= 2 9

- - (i.) 2.

5 2

There are c e r t a i n simple b u t h i g h l y impor tant f a c t s about LF.e o r d e r of the real numbers on t he r e a l number l'ne. If we

choose ar.y two cif f 'erent rea l numbers , we are s u r e chat t h e firs;

1s l e s s than t h e second or the second is l e s s than the first, b u t

n o t both, S t a t e d in tP.e language of a l g e b r ~ , t h i s p r o p e r t y of

o r d e r for r e a l numDers becomes the comparison proper ty :

If' a is a rea l number and L

is a real number, then exac t ly

one o f the f o l l o s i n g is t r u e :

Problem Set 3-2b -

1. For each of' t he f 'ollowing p a i r s of numbers v e r i f y t h a t t h e

ccrnprlson proper ty 1s Lrue by deberminlng whlch one of Che

th ree possibilities a c t u a l l y h o l d s between the numbers: - -

( a ) -2 and -1.6 ( d ) 11, and -

(b) O and 2 ( e ) 12 and (5 i 2)(3 x y)

( c 1 5 4, - (f) ? a n d 2

2 . Malie up t r u e sen tences , usLng <, involving t h e followi3g

p a i r s :

(i) -&, -1 .5

(j) & + 7 , 1 . 5 + 3

3 . Tilt: ccmparison p r o p e r t y s t a t e d in t h e t e x t is a statement

i n v o l v i n g " < " . Trjr t o formulaLe t h e copresponding p r o p e r t y

i n v o l v i n g ">", and t e s t it with t h e p a i r s of nwnters in

+ Tr.g to s t a t e a comparison proper ty involving ">" - .

4 'dhich is lpss t h a n t h e ocher , - 5

OF g? YOU can f i n d o u ~ by

a p p l y i n g the nultiplicacion proper ty o f 1 to each number to ge:

; I 11 6 24 - - = - - x r = - a n d 2 = > k 5 = Then - 4 < i;, 5 because - o b 5 3 0 ' ;k is t o 5 5 0 3 0 9 25 the l e P t oi' or) Lhe n u n - ~ e r 1 i n e . 30

You s h o u l d nov~ u e a b l e to compare any two r a t i o n a l numoers.

lim: would yo^: decide which i s the l e s se r , I - ? (DBSCPIDB 11; 55 the p r o c e s s ; do not a c t u a l l y c a r r y it o u t . )

167 i e r i l ~ g s YOU notlced, in comparing @ and - 2 5' th3t < 3

113 167 i . , < and j < - ( i - e m , 5 5 57 } Could :TOU

167 d e c l d e about; t he o r d e r of and - w i t h o u t writing them as 113 55

f r a c t i o n s with the sm,e denominator? How c o u l d you find o u ~

' O or Or suppose that x and y s i ~ n l l s ~ i y whicn i s l e s s e r , - 27 2 - -

are r e a l numuel-s and t h a t x < 1 and -1 < Y. Again using the

number line, what c ? n you say a t o u c the order of x and y?

The p r o p e r t y of' ~ r d e r used in t h e se 'ast t h r ee examples we *

c a l l t h e t r a n s j . t i v e p r o p e r t y :

If a, t, c are rea l numbers

and il' a r b and i~ ( c,

t hen a < c .

Problem S e t 5-2c --

1 . In eacn of t h e foilowing groups of t h r e e r e a l numters, det,er-

n i n e t n e ir order:

F~~ example, $, 2, -5 have the orde r : - g C $, $ ( $, - I 3

-= - < f. 7

.-- :cotnoto d

k'ron: tiie Lat i . r l , t r a n s l r e , t o go across.

[ S P C . 5-21

- (b) T T , V , and -&,

1 1 2 (i) 1 + F , 14- {+12, (1 + F) .

2 S t a t e a transitive property f o r ">", and illustrate this p r o p e r t y in t w o of' the e x e r c i s e s in Problem 1.

3 . A r t and Bob a r e seated on opposite ends of a see-saw (teeter-

t o t t e r ) , and Artts end o f the see-saw comes slow12 t o the

ground. Cal gets on and Art gets o f f , a f t e r which Bubls

end of the see-saw comes to t he ground. Who is h e a v i e r ,

A r t o r Cal?

4 . Is there a t r a n s i t i v e p roper ty for the relation " = " ? I f s o ,

give an example.

5 . S t a t e a transitive p r o p e r t y f o r ">", - and give an example.

6 . The s e t of numbers greater than 0 we have c a l l e d t h e

p o s i t l v e rea l numbers, and t he s e t of numbers less than 0

the negative real numbers. Describe t h e

( a ) n o n - p o s i t i v e r ea l numbers,

( b ) non-negative r e a l numbers.

7 . F ind the orde r of' each o f t h e following p a i r s of numbers: - -

(a) g -d g- - -1 04 ( c ) % a n d

5-3. Opposites

Itken we labeled p o i n t s to the l e f t of 0 on Lhe real r:~cn;bc~

line, we began b y marking off successive unit l e n g t h s t g the l e f t

of 0 . We can a l so t h i n k , however, of p a i r i n g off p g i n k s at -

equal d i s t a n c e s from 0 and on o p p o s i t e s l d e s o f 0 . Thus, 2

I s at the sane distance from 0 as 2 . What nurnber is at t h e 7

cwl? dlstance f rom 0 as -? If you choose any p o i r ~ c on t h e r? nwnber l l n e , can you find a p o i n t at t h e same distance rrom 0

and o n t h e opposite s i d e ? Lhat about t h e p o i n t 0 i t se l f ' ?

S lnce t h e t w o numbers In such a p a i r are on opposite s i d e s

o f 0 , it is na tu ra l to c a l l them opposites, The opposite of a

non-zero real number is the o ther . real number which ic a t an equal

dlstance from O on he real number line. i fhat is the o p p o s i t e

of' O?

Let u s c c n c i d e r some t y p i c a l r e a l numbers. Write them In

a co lumr~ . Then write t h e i r opposites in ano the r column; then

study the a d j a c e n t s ta tements . - 2 is t h e o p p o s i t e o f 2 ,

- 1 . 1 - 1s the opposite o f - 2 2 '

0 , 0 ; 0 is the opposlte of 0 .

The ststements themselves are cmbersome t o w r T t e , and xe need

a r;ymool meaning " t h e o p p o s i t e 3f1'. L e t u s use the loser ~-irlsh 1 1 11 - to msan "the opposite of" . With t h i s E ~ I ~ G ? . ;he thllee s t a x -

ments become t h e 'crce sen tences :

0 = - 0 . (~ead t h e s ~ sen tences E ; I P ~ ~ ~ L I ~ ~ z ' . )

We c a n l e ? r n t ~ 3 thl~gs from t h e s e s c n t c n c e s . FLr:;t. i - *

apgoars th-L " - 2 " a n i " - 2 " are e i i ' f e r en t nmes i'or Lhc 3smc

n u m t e r . Tnat - s , "n:gative 2" a r ~ d "the oppasitc of 2" r e~rese i - , t

t h e same number. Hence, i t makes no d i f i 'erence at ?:hat he' &';

:he dash is d r a m , s i n c e t h e rneanlng is the same f o r i k e u p p e r

an5 lovrer c!ash, TRls being the c a s e , we do not; need bc,th symLoJ s . V.?iicl~ s i ~ a l l vTe ye t a in ' ? The upper dash ref'ers o r l l y tu ncga-

t i v ? numbers, whereas the l o w r dash may a p p l y L O an;r ~ c a l

nunae r . hot;^ t h a t the oppositc of' the p ~ s l t i v e : lumber 2 is - t h e - 1 neg2tive nunoe r 2 , and t h e opposlte cf' the negative r l u m b c r ;;

1 c!

is :he pos:tive ncmber - . ) Hence It is natural to r e t a i n 2 the "opposi te cf" symbol to mean either "ne&ative" or "apposite

o f " when the numPer in quest-Ion is posi , t lve . Now the s m t c n c e s

may b e w r i t t e n

-2 = - 2 , ( r e a c "nega t ive 2 is the o p ~ o s i t e o f 2 " )

1 1 1 - = - - ) (read 2 is t h e opposite of nega t ive 9)

2 2

The second of' tnese sentences can be road a l so a s :

S ~ c o n d , we obse rve in g e r ~ e r a l tha', t h e opp~slte 01' t h ~ -- o ~ y c s i t ; e of a n m b e r is t h e number itself; in zymbols : -- --

For every real number y,

-(-Y) = Y.

m a t is the o p r o s i t e of' the opposlte of Lhe opposite of a number?

Wnat; is ttie qpposite 01' the o p p o s i t e of a n e g a t i v e nuntbell? 1 1 11 bherl we a t t ach t h e dash - to a variable x we a r e perform-

i n g on x the u p z ~ ~ a t i o n of "deccrmining the opposite of x". Do no1 confuse t h i s wiih t h e b i n a r ~ operation of subtraction,

rqhictl is perl'ormed on L W O numbers, such as 3 - x, rneaning "x - subcracteu : 'rom j . I T I',?lat k i n d of number is -x if x is a

positive number? 11' x is a negative nunber: If x 1s O?

Wc s h a i l rzad " -xf' as t h e " c p p o s i t e of x" . Thus, if x is a nunber to the right of' 0 (positive), then - x is to t h e

lef t : (negative); if x 1s LO the Icr't of 0 (negative), then

- x i a c o Lhe rj&t ( p o s i t i v e ) .

Problem S e t S-3a - 1. Form the o p p o s i t e of e s c h of the fo l lowing numbers:

(a) 2.3 ( c ) - ( -2 .3) ( e ) - ( a 2 x O )

( b ) -2 - ) ( d ) - ( 3 . 6 - 2.4) ( r ) - 0 ~ 2 t- O )

2 . \illat kind of nulnter is -x if x is positive? i f x is

negative? if x I s ~ e r o ?

. Khat kind of number i s x if -x is a positive number? if

-x is a negative number? if - x is O?

4. (a) Is e v e r y r e a l n7mber t he opposite of some real number?

( b ) Is the s e t of a l l o p p o s i t e s oT real numbers t h e same as

the s e t of a11 real numbers?

( c ) Is t h e s e t of a l l n e g a t i v e numbers a subseL of t h e set

of a l l opposites of r e a l numbers?

(d) Is the set of a l l opposites of real numbers a s u b s e t of

the e c t of a l l negative numbers?

( e ) Is e v e r y opposite of a number a negative number?

The crdering o f numbers on t h e r,?al nurnker l i n e n p e c i i ' i e s 1 1

hat - - is less ~ h a n 2 . Is t h e opposite n f - - less t h m t r l e 2 2

o p p a s i t e of 2? Piake up o t h e ? similar exarn?les 01' pairs 31'

n u m b e r s . After y ~ u have d e t e n m i n~i.l :hc orciering ol ' s pair, then

i'iiicl t h ~ orderi.ng ai" ",heir opposites. You w i l l see that here

I s a gencra l p-oper ty fo r opposites:

F o r r e a l r.umLc:Ys a a n i L ,

Lf a ( L , then -b ( -a.

Problem Se t 5-3b -

1 . For each of the following pa i r s , determine whlch is t h e g rea te r

number :

( a ) 2 . 9 7 , -2.47 (4 -1, 1 ( I 0 , -0

(b) -12, 2 ) - 0 , 2 (h) -0.1, -0.01

( c ) -355, -762 (f) 0 . 2 , 0 . 4 (i) 0.1, 0.01

2 . Write t r u e sentences f o r the following n-mbers zrtd thelr I 1 II o p p u u l t e s , us ing zhe r e l a t i o n s < or ' I >" ,

Exampl?: For the nuxbers 2 and 7 , 2 ( 7 an6 -2 > - 7 . 2 1

( a ) 7. - 6 4

(d) 3 (7 I 21, 3 ( 2 ~ + 8)

8 + 6 ( b ) , - 7 7 (4 -2

22 (4 7 7 , 7 (i) - ( 3 + 7 0 ) , - ((5 + 013)

3 . Graph the t r u x h se", of the following open ~entences:

( H i n t : In p a A s ( c ) and ( d ) use order p m p e r t y of o p p o s i t e s

before graphing. )

(a) x > 3 ( 3 ) - x > 3

(b) x > -3 (d) -X > -3

. D e s c r i b e t h e t r u t h s e t of each open sentence :

( a ) -:i # 3 ( c ) x < 0 ( e ) -x > - o ) - x + - 3 ( d ) -x < 0 (11 -X 2 0

5 . Por t h e following secs , give two open sentences e a c h o f

whose t r u ~ h se ts is the g iven set ( u s e opposites when

convenient):

( a ) A is t h e s e t of a l l non-negative real n u m t c r s .

( b ) 3 is t h e s e t of a l l rea l numbers not equal to - 2 .

( c C is t h e set o f a l l real numbers n o t greater ~ h a n - 3 .

( d ) 6 . ( e ) E 1,s the s e t of a l l r ea l numbers.

G . Wr5.te an open sentence f o r each cf the fallowing g r a p h s :

7. For each ol t h e following numbers wrl te I t s o p p o s i t e , and

then c5oose the grea ter of t h e number and its opposite:

( a ) 3 (f) -0.01

(b) 0 ( g ) - ( - 2 )

( d ) -7.2

(E) -&

( I *6. L e t i s w r i t e " +I1 f o r the phrase is further from O than1'

on the real n>mber l i n e . Does I ' >I1 h a v e the comparis~n

p r o p e r t y en-joyed by ">" , t h a t I s , if a and c are d i f :e rent

real nurfbers, is it t r u e t h a t a S b o r rJ h a but not

bo th? Does " b l ' have a transitive property? For w h l c h SUE-

sez of T h e s e t of real numbers u o " + " and ">" nave the

same meaning?

9. Trans1 a t e the following English senyences into open s c t ~ t e n c e s , describing t h e variable used :

(a) John's score I s grea ter t h a n n~gative I C O . 'dhar. is h i s

score?

L ) I know c h a t I dont t have any money, but I am no m o w

than $200 In debt.. What: is m y f l n a n c - i a l condition?

( c ) Paul has p a i d $10 of his bill, b u t s t i l l owes more than

$25. What was t he arr,ount of' Faults kill'?

10. Change t h e numerals " - 2 and l 1 ;.5+,

- ~ 9 ' t c l Corms tu.itJh t h e same

l2 denominators . (~int : F i r s t do Lhis for ~ arrd q. l5 ) l,llhat

15 is the order oi' - and - q. ( ~ i n t : Knowine t;he order

of and 5, what i s t h e order o f t h e i ~ oppos: t r ? ? ) Nov;

s t a t e a general r u l e f o r d e t e ~ r n i n l n ~ y the o r d ~ r of t w ~

nega t ive ratlonal numbers ,

5 A b s o l u t e Value

We now want t o dei ' ine a new and v e r y u s e f ' u i operation on a

single real n ~ u n b e ~ ' : t h e operation o: t a k l n ~ its a b s o l u t e va.il:e.

The abso lu t e v a l u e of a non-zero

real number iz t h e grea ter of' t h a t

number and Its opposite. The

a b s o l u t e v a l u e of C is 0.

The absolute v a l u e o f 4 is 4 , because the g y e a t e r of 5 and

-11 is 4 . Tne a o s o l u t r va lue of - 2 i s $. ( x h y ? ) lmat is 2

the a b s o l u t e value of -17? WhJ.ch is always t h e grent;er of a

non-zero number and its o p p o s i t e : t h e posi:.ive or t h e rleaativr

number? E x p l a i n why t h e a b s o l u t e v a l u e of' m y rc.zl number is

a p o s i t i v e number o r 0 .

As u s u a l , we agree on a symtsol to i n d i c a t e the ope ra t i on .

hie write

In1 LC mean the 3 b s o l u t e v a l u e ~ . f t h e number n. For example,

1

4 = 4, 1-31 = - I = &, 1121 =?2.

I d f i l , ~ : thqt. e a c h el' these Is non-negat ive .

If' you look a t these numbers and ; he i r absolute t a l u e s on

t h ~ nwnser l ' n e , what can you conclude about Ll le distance between

a ? ~ u ~ b e r = and O? You no t i ce t h a t the d i s t a n c e between 4 and 3

is 9 : tetucen - 3 and 0 is , s t c . Not i ce tnaL ttie d i s -

t ance between any two po in t s of t h e number line is a non-

negative r ea l n m t e r .

The distance between a r e a l number and 0 O n the Y e k l

numb3r l i n e is the ~bso1c ; te

v a l u e of thaL number.

I . F i n d the abso lu te values of t t e f o l l o w i i ~ g nurr~sers:

3 ) -7 (4 ( 6 - 4 ) ( e ) -(14 -t 0 )

( b ) -MI ( d ) 1 4 x 0 ( f ) -(-(-m 4

2 . ( a ) What k ind of number is -; what k i n d o f number is 3

(Pion-nsgative o r negat l ve? ) If x is a no^-negative re31 number, what k i c d of

number is 1x12

2 ( b ) mat kind of number is - T; what kind of number is 7

1 - $1 ? on-negative or n e g a t i v e ? )

If x is a n e g a t i v e r e a l number, v r h a L k ind of' number

is Jx\ ?

( c ) 1 1x1 a nun-negative number f o r every x? Explain.

3. iJor a rdega t ive number x, which is grea ter , x cr Ixl?

4. Is the s e L (-1, - 2 , 1, 21 c l o s e d under t h e o p e r a t i o n o f

t ak ing aasolute v a l u e s of I c s e lenents?

We n o t c t h a t i ' o r a non-negacive number, ehe greater oi' the

number and i t s opposite is the number I t s e l f . T h a t is:

F o r every real n u m b e r x

which is 0 o r p o s i t i v e ,

1x1 = X.

h a t can be sald of' a negative number a.nd i t s a t . so lu te v a l u e ?

't!rit;e corm,on numerals f o r the f o l l o w i r ~ ~ ; pairs Yor yourself.

1 (!&at kind e l numbers are -5, - F, - 3 . 1 , -467?) You found t ha t

1 - 3 4 = - ( - 3 . 1 ) ~ 1 1 - = - ( - /-'t67( = - ( - h t 5 ~ ) ,

Is it now clear t h a t we may say, h he a b s o l u t e v a l u e of' a negative number is t h e oppos i t e of the nega t lve number":' That is:

For every negative r e a l number x, 1x1 = -X.

i . Wnick of' :he f o l l o w i n g sentences are ; r u e ?

( a ) 1-71 <: =I (a ) 2 d 1-31 ( g ) - 2 1-51 ( b ) 1-21 2 1-31 ( e l 1-51 { 14 [ti) Id571 > 1-41

( c ) ]!+I < 111 ) -7 4 17 (i) 1-212= 4

3 X h s t is Lr,e t ruck: s e t o f each open ser,tcncc? *

( 2 ) Ix, - 1, (b) 1x1 = 5 , ( c ) 1x1 + I = 4, (a) 5 - ,x l

= 2 .

. ?.;hLch ci' Ci;h i 'ol loving open senLerlces are tru? f a r a l l r e a l

nunbers x ?

(Finz: G I x a p o s i t i v e v a l u e ; t h c n g lve x a n e ~ a t i v e

v a l u e . Kow come L o a decis~on.)

5. U e s c r l l e the v a r i a ' r l l ~ 2 used and ';ranslate ^into a:? open - cen tenzc : J o n n kcs l e s s money t h a n ;, 2nd - h u e l e ~ s

r 0 . Hob: rnucn none;: does 20hn have?

Graph t h e t r u t h s e t s of t h e following sentences:

( a ) 1x1 < 2 ( b ) x > -2 and x < 2 I.) 1x1 > z (d) ; < -2 o r x > 2

Compzre ;he graphs of t h e sen tences in prcbicns 6(n ) and ( ~ 1 . 1~ 6 ( c ) 2nd ( c i ) .

S:?o\i t h a t : S x iz, a negative r e a l n u r b e r , $hen x Ls

the oppozite o f t h e a b s o l u t e va lue 01' x; tha t is, :f

x ( 0 , t h e n x = - 1x1 . (Hint : i k n t is t h e a p p o s i t e of' the

op~osite of a number?)

Graph t h e set of l n t e g e r s l e s s than 5 whose abso lc te

v a l u e s a r e grea te r than 2 . Is -5 an clement of this

s e t ? Is 0 an element cf' t h i s s e t ? Is -10 an element

of this s e t ?

If R is the s e t of 311 real numbers, P t h e s e t o f all

positive real numbers, and I the s e t of a l l i n t e g e r s ,

write chree numbers:

(a) in P t u b not in I,

(b) In R but not in F, ( c ) i n R b u t not in P nor in I ,

( 6 ) in P b u t not i n R .

Trans la te i n t o an ogen sensence: Tie t e ~ n p e r a u r e today

remailled w i t h i n 5 degrees o f 0 .

Compare the truth s e t s of t h e two eentences

5-5. Summary

1. Poi~ts to the left of 0 on the number l i n e are l a x l e d with

n e g a t i v e numbers; the se t of real number:: c o n s i s t s of' a l l - numbers o? arithmetic and t h e i r o p p o s l w s .

2 . Many p o i n t s on the nmber l i n e a rc n c t ass igned r a t i ~ n a l

n w b e r coordinates. These po in t s are labe led with irrational

numbers. Tne s e t of real numbers consists of all r a t i o n a l

and irrational numbers.

3 . "Is loss than" f o r r e a l numbers means "to the l e f L o f " on

chc number l i n e .

' Cornparlson Proper ty . If a Is a real nunbet> and b is a - . real nmber , then exactly one of the following I s t r u e :

a < b , a = b , b < a .

5. Transitive Froper tx . If a, t, c are r e a l numbe-rs and if

a ( b and o ( c, then a ( c .

6 . The opgosite o f 0 is 0 and the o p p o s i t e of any o t h e r real

n m b e r is the o t h e r number which is at an equal distance frcm

0 on the r e a l number l i n e .

7. The a b s o l u t e value of 0 is 0 , and the absolute v a l u e of

any o t h e r real number n is the greater o r n and t h e

o p p o s i t e of n .

8. If x is a positive number, then -x is a nega t ive n u r n b ~ r .

If x is a negative n u n b e r , then -x is a positive nuntber.

9 . The a b s o l u t e va lue of t he real number n is aeno ted by In] . A l s o , In1 is a non-negative number which 1 s t h e d i s t a n c e

between 0 and n on t h e number line.

10. If' n > 0 , then In1 = n; - If n ( 0 , then (nl = -n.

1 . ' t h l c h of tihe Yollowir;~: sen turxes are t r u e ?

1 6 + 1 ( a ) -2 < -5 ( d l - 2 - ( I.:+

( b ) - ( 5 - 7) = - ( 2 ) ( e ) -2 / i 7 ,-;I] ( c ) 4 5 - 3 ) < -1-4 ( t7 ) 1-21 ,:-3i

2. Which o f t h e fo l lo -d ing s e n t e n c e s are i ' l1.s~: 4 5 1 and - ? < - 6 (a) - - + 5 > - 2 ) + 6 ( d ) 7 .C

3. Draw t h e graph of each of' the f ollowinc sen tencez :

(a) v 2 l a113 v c 3 ( s ) x r 4 or x > 2

(lj) I F ! = 2 (d) Jx[ = x

4. Describe the t r u t h s e t of' each s e n t e n c e .

(a) Y ( 3 and Y > 4 (d) 1x1 = -x

(b) - Ju[ c 2

( c ) -3 < x < 2 3 . Considcr the open sentence " 1x1 c: 3 " . D r a w the g r aph of i t s

: r u t h s e t if the domaln of x is the set of:

( a ) ~ e a l numbers

( b ) i n t ege r s

( c ) nor . -negat ive rea l numbers

( d ) nega t ive in tegers

6 . Descr ibe t h e v a r f a b l c s used ar.d t ra r -~s?atc each of the f'o1Lov:-

i n g into &n open sentence:

(a) Or, Thursday the average t.emperature was h0 loncr chan on 0

Friday, and on Friday it was !;elow -10 . What was tne sverage temperature on Thursday?

r3 (b) On Sunday t he average temperature remained within 6

0 of -5 .

7. If R is t h e s e t of a l l real numbers, P the s e t o f a11

posiLive r ea l numbers, F the s e t of a l l rational numbers., I the s e t of a l l in t ege r s , which of the following are t r u e

statements?

( a ) F FS a ~ubset of R .

(L) Every elenent of I is an element of F.

( c ) Them are elements of I which are not. elements of X. ( a ) Every clernenb of I is an element of P .

( e ) mere are elcrnents o f R which are n o t elements of F.

8. D r z w t h e graph o f the s e t of i n t ege r s l e s s than 6 whose

absolutc v a l u e s are grea ter t h a n 3 . Is -8 an element of t h i s s e t ?

9 . When a c e r t a i n in teger and its successor a re added, the

r e s u l t is the successor Itself.

(a) Write an open sentence translating the E n g l i s h sen tence

above.

(b) Find t h e t r u t h s e t of t h l s s e n t e n c e .

10. The perimeter of a square is l e s s than 10 i n c h e s .

(a) \ h a t do you know about t h e number of units, s, in t h e

side of' this square. Graph t h i s s e t .

(b) mat do you know about t h e number o f unlts, A, in t h e

area of this square. Graph t h i s set.

Chapter 6

PROPERTIES OF ADDITION

6-1. Addltion - - of .- Real Nunbers

E v e r s i nce first grade you have been adding numbers, the non-

negative numbers o f arithmetic. Now we are deal ing with a larger s e t of numbers, the r e a l numbers. Our l o n g experience with add-

ing non-negative numbers, b o t h in arithmetic and on t he nunber

l i n e , should quick ly give us a c lue as t o how we add real numbers.

L e t us c o n s l d e r the profits and losses o f an imaginary i c e

cream vendar durlng h i s 10 days in business. We u s e p o s i t i v e

numbers to r e p y e s e n t p r o f i t s and negative numbers for losses .

L e t us d e s c r i b e his business a c t i v i t y , and then mal ie two columns.

one f o r t h e a r t t h e t i c of computing h i s net income o v e r two-day

p e r i o d s , and the other for picturing the same on the number l i n e .

Etrsiness Arithmetic I l u~be r Line . -

Mon.: Profit o r $7 7 + 5 = 12, I 7 - q - 5 - +

Tue.: Profit of $5 n e t income - - I - 0 7 12

b!ed. : P r o f i t of $6 - 4 -

mu.: Lozs of $4 6 + (-4) = 2 , + 1

p-+ 6 - 4 L-

(Tire t r o u b l e ) n e t income O 2 6

Ri , : Loss of $7 (another t i r e )

I

Sat. : P r o S i t o f $4 (-7) !I = -3 -7 - 3 0

Sun.: Pay of res t Mon. : Loss of $3

0 1

( c o l d day) 0 i- ( - 3 ) = - 3 - 3 0

Tue. : Loss of $4

( c o l d e r )

Wed. ; Loss of $b + I-- -----?- I 7 I

(gave UP) (-4) i- ( - 6 ) = -10 - I0 - 4 0

These ~ c c o u n t s Illustrate almust evcry possible sum o f yea1

numbers: z p c a i c i v e p l u s a p o s i t i v e , a p o s i t i v e p l u s a negative,

a n e ~ a t i v e P ~ U E a positive, 0 plus a negat ive , and a negative

plus a negative. One of t he purposes of this s e c t i o x l wLll be t o learr: how t o

t r a n s l a t e i n t o the l ang -~agc of algebra operat ions wh:ch we f i r s t

describe ~;ometr5cally. Addition on t,he number line is such an

operat ion; we sh~ll try to define it in t h e langusg? of algebra. If we take 7 i . 5 and p i c t u r e t h i s additior, on t he number

I l n e , bi,e f i r z t go from CI to 7, and thcn from 7 \?re move 5 more units tu the r i g h t . If w e c o n s i d e r (-7) + 4, we f i rs t go f ro r .

0 to {-7), and then frcm ( -7) move h i t s to the r i g h t . These

~ x a m p l e s remind ~s of something we d r e a d 7 know: To add a positive

num5er, WE mcve t o :he r i g h t 3n $he nurr:bei~ l i n c . It should now

be clear fron: ow. 0:her examples a b o v e what; h3ppens on t h e nun- ber l i n e wher we add a negative mmber. hhen we added ( - a ) , we moved 4 units t o th? l e f t ; when we added (-61, we moved 6 units

to the le?t ; . We have one more case t o cor.sider: If we add 0 ,

what motign, if any, r e s u l t s ?

We have now d e s c r i b ~ d t h e m~tion i n all cases; l e t us see

if WF! can learn to say a l g e b r a i c a l a how f a r w e move. F o r ~ e t --- f o r tke moment the direction; we .just wan: t o know now f a r we -- g o when rie go from a t o a + b. kU~t.11 'o 13 p c s i t i v e i.le go to

the r 'ght . Yes , 'out h o w f a r Y ~ @ go J u s t b ur.its. the-n b

is nega t ive , we g o to the I 2 f t . How far? ble go (-b) units. (Remember ( - b ) is p o s l t l v e if b i s n e g a t i v e . ) If b is 0, we don ' t go a t all. f i a t symbol do we kr,ow whll:ti near,s ''b if b

Is p o a i t l v e , -b if b is n e g a t l v c , and 0 If b is CI"?

"1bj1', of courae. And s o we have learned t h a t to f ind a + b on

the number Iin~, we start from a and move t h e d i s t a n c e Ib ( t o t 5 e r i gh t , if b is positive;

t o the l e f t , i f b is n e g a t i v e ,

Problem Set 6-1 -- 1. Do the following problems, using p o s i t i v e and negative numbers:

( a ) A f o o t b a l l team l o s t 6 yards on the first play and

gained 8 yards the second play. ?,bat was t h e n e t yai-d-

age on the two p lays?

( b ) John pald J i m the 60jf he owed, but John c o l l e c t e d the

5% A 1 owed him. lrlhat is the n e t r e s u l t of John1 s

zwo transact ions ?

( c ) If a thermometer registers -15 degrees and t h e ternpera- ture r ises 10 degrees, what does the thermometer then

register? What if t h e temperature had r i s e n 30 degrees

i n s t e a d ?

( d ) Miss Jones l o s t 6 pounds d ~ r i n g t h e f i r s t week of her

d i e t i n g , lost 3 pounds the second week, gained 11 pounds

the t h i r d week, gained 5 pounds t h e last week. '&at

was her net gain or loss? 2. Perform the ind ica ted operhatlons on real numbers, using the

number l i n e t o a i d you:

(a) ( 4 + (-6)) i ( - 4 ) ( e ) 2 + (0 1 ( - 2 ) )

( b ) 4 + ((-6) + ( - 4 ) ) ( f ) ( ( -31 + 0 ) + (-2.51 (4 -(4 + ( - 6 ) ) (d 1-21 + (-2)

( d l 3 + ( ( - 2 ) + 2 ) (h) ( -3 ) + (1-31 + 5) 3. T e l l in your own words what you do to t h e two given numbers

t o f i n d t h e i r sum:

(a ) 7 + 10 ( f ) (-7) + (-10)

(b) 7 + (-10) ( g ) (-7) + l o ( c > 10 + ( - 7 ) ( h ) (-10) + 7 ( d l (-10) + (-7) (i) (-10) + o (4 10 -t 7 (j) 0 + 7

4. Ln whlch -parts of Problem 3 d i d you do t h e addition jus t

as you added numbers in arithmetic?

5. What cou ld you always say about t h e sum when bo th nwnbers

were negat ive ?

- 2 Definition of f l d d i t i o ~ . - b l ~ not; ~ r a n t kt.o use vr i?at rr:c haire ;usC. I v a r n e d about a d d i t i o n

on t h e n w n b ~ r l i n c to sa-5 f i r z t in F,y.l lsh and t'ner~ Ir. the l a r ~ g -

uage of algebra, rhat i!e mean by a - b Tor all rea l nmbers 3

and h. The accountc of fhe ice cream vendor ha-J-e intrc>duced

exper4ence how add a and b i f b o ~ h a re non-negative ~ ~ w b e r s

So let us conside:. arlo:..tler ~ x m p l c , n,Unely, s r .c~at irre p l ~ s a

necati3re. :.kzf:, I s

(-9) d ( - 6 , ) ~ Is'c have fcurld, on t he number 1 Snc, t h a t

(-q -r (-6) .. (-I@). Our presen; j o b iz t o t h i n k 2 b i t more ca , re fu l ly about .just hcw

wc rAeached (-13). W begin b y moving from 0 ko (-'I). Where

is (-:+) on the nurrher linc? It i3 t o t.t-ir: lqft of ' O. 30w :3rCL

"Distance between a nwrtber and 0" was one of t h e mcanirzn of t h e

absolute velue o f a. nunbe-. Thus t h e distar.ce betvreer. 0 and ( - : c )

i ! ( O f course wc r e a l i z e tt~at; it is eas ie r ta rmite i L t h a n

1 - ! I I , but the express5on ! -i: 1 remFnds us t h a t t.rr l.&rc;.e t h i n k i n g

of "disLance from 3 " , and t h i s 1s worth remembering at p r e ~ e n t , ) ( - 4 ) IE thus 1 - 4 ] tu the left oY 0. \ h e n we now c o n s i t e r

(-4) + (-6), we move another 1-61 t3 tke l e f t . iphere are we nows A t

- ( 1-111 .:- 1-61). Thus our t h i n k i n g about dis tance Cram 0 and about d l s t anc t . moved

,In the n m b c r line has l e d us t o recogn tze >hat (-4) + ( - b ) = - ( 1-1: 1 + 1-61

is a true sen tence .

You can rea3onably ask at this p o i n t what we have accomplished

by a12 t h i s , We have taken a simple expression like (-!-) + (-61, and mad2 jt l ook more complicated! Y e s , but :he cxpresaion

- ( ) - , $ I .' 1-61), complicated as It l o o k s , nas one gr~at advantag?.

I t c o n t a i n s only op"ations which we &:now hos t o do Prom p r e v i o d s - .- - - - - - - experience? Both I - > 1 and 1-61 arc posltive numbers, you s c e , -- and xe know how to add positiv~ numbers; and - ( 1-21 I 1-61 i s

t h e ~pposite of a number, ~ n d we know how t o flnc that. Thus

we haye succeeded in expressing the sum of two nesat ive numbers

f o r vihich sum we p r e v i o u s l y had jus t a picture on t h e number

l i n e , in terns of the l a n g u a ~ e of algebra as we have built it

up thus f a r .

Think through (-2) + ( - 3 ) for yourself , and see t h a t by t h e

same reasoiling you arrive a t the true sentence

( - 2 ) +- ( - 3 ) = - ( l - z l 3- 1-31]. $?om these examples we see that the following de f ine s the

sum of twc l n e g a t i v e numbers in terns of operations i.rhich we

already know how to do:

In English: Tha sum of t w o nexat ive numbers is negative; t h e a b r o l u t e value of t h i s sum is the sum of the absolute values

of t h e numbers.

In the language of algebra:

If a and b are b o t h negative numbers, t h e n

a t b = - ( l a ! ; I‘u I )

Problem Set : -')a

1. Use t h e definition above to find a corrmon name f o r each of

the f o l l o w i n g indicated sums, and t h e n check by us ing i n t u i -

t i o n concerning gains and l o s s e s , o r by usfng the number l i n e .

B a m p l e : by definition

( - 2 ) -+ ( - 3 ) = - (1 -21 -t 1-31] = - ( 2 - 1 - 3 )

= -5 Checlc: A i03s of $2 followed by a loss of $3 i s a net loss

of $5. ( a ) ( - 2 ) + (-7) ( d l (-25) + ( - 7 3 )

1 2 (4 (-33-1 -k

2 . Find a common name for each of the f~llowing by any method

you choose:

(a ) (-6) -t ( - 7 ) ( c ) - ( 1-71 + 1-61] (b) ( - 7 ) 4- ( - 6 ) (a ) 6 -t (-4)

126

( e ) (-4) I 6 (h) - ( 1-31 - 101)

(f) I 61 - I - J I (I) 3 i- ( ( - 2 ) 4 C)

id 0 1 ( - 3 )

3 . F i n d t h e truth s e t of each of the following zente!lces:

( 3 ) ( -5 ) (x) = - 0 - 5 1 + 1-31]

Ib) x 4 ( - 5 ) = - ( I - 5 1 t- 1-31)

( 2 ) ( - 5 ) - ( X I = -(I4 1 -:- 1-51]

( a ) x -: (-5) = -41-31 + 1-51)

. mink a ~ a 4 i 1 ol Problem 3 i n Problem S e t (5-1. hhen one umber

is positive and o n e is negative, how f a r Is thulr sum f rom O?

5. When one number i s posicive and the other is negative, how

do you know whether the sum is p o s i t i v ~ or neqa t i -~e ' :

+6. Is t h e I"ol1owlng a t rue sentence for all non-negative values or x?

( - 1 ) + {-x) = -(l-ll + 1 - X I ) .

So f a r , we have considered t he sum of two non-negat ive num- b e r s , and t h e sm of t w o n e g a t i v e n~unbers . Next we c o n s i d e r the sun of tv:o numbers, of which one L s p o s i t i v e - and the o t h e r is

nega t i i7e .

L e t us l o o k again a t a f e w examples of gains and l o s s e s :

T r o f i t of $7 and l o s s o f $3; 7 t- (-3) = 4; 1'7 1 - 1-3 1 = i % \

Profit of $3 and loss of' $7; 3 + ( - 7 ) = - - ; 1-71 - 131 = 1-111

LOSS ~r $7 and p r o f i t of $3; - 7 ) - 2. = - 1-71 - 131 = 1-41

Loss of $3 and profit of $7; ( - 3 ) + 7 = 2 . , 17 1 - 1-3 1 = 1'1 1

Lo35 of $3 and P r o f i t of $3; ( - 3 ) 4- 3 = 0 ; 1 3 1 - 1-31 = l o ] Conslder these examples on the nmber l ine , and a l s o thlinic

again about the ques t ions in Problems 4 and 5 in Problem S e t 6-2a.

From these it appears t h a t the sum 01, two numbers, of which one

is p o s i t i v e (or 0 ) and the o t h e r is negative, is obta ined as

f o l l o w s :

[ s e c . 6-21

The absolute value of the sum I s t h e d i f f e r ence

of t h e zbsolute v a l u e s of the numbers.

The sum is positive if t h e p o s i t i v e number has

the greater absolute value.

The sum is negati-.re ir the negat ive number has

the greater a b s o l u t e ualue.

The sm Is 0 if the positive and negat ive num-

bers have t h e s,me abso lu te value.

T i ' a - > @ and b < 0 , t hen :

a + b = la1 - Ib(, if la1 2 lbl and

a t b = - ( ( b l - I a l ) , if ( b l .> l a / .

If b - > 0 and a < 0, then:

a + b = l b l - l a l , if lbi 2 ] a ] and

a + b = - ( I31 - l b l ) , ~f la] > l h l .

Problem Se t 6-2b -- 1. I n each of the following, find t h e sum, first a c c o r d i x t o

t h e definition, and t h e n by any o t h e r method you f i n d con-

venient , .

( a ) ( - 5 ) - 3 ( e ) 18 -I- (-14)

(b) (-11) -:. (-5) (f) 12-7.;:

8 ( c ) ( - T ) I 0 2

(9) (-$ i - 5

( d ) 2 A ( - 2 ) (h) (-35) + (-65)

2 . Is the s e t of all r ea l numbers closed under t.he o p e r a t i o n

of addition?

3. Is the s e t of a l l nega t ive r e a l numbers c losed under add i -

tion,,? Justify your answer.

4 . In the course u f a week the ~Yariations in mean temperature

from the seasonal normal of 71 werq -7 , 2 , -3, 0 , 9, 12, -6. mat were the mean temperatures each day? h h a t is rhe sum

of the v a r i a t i o n s ? [ s e e . 6-21

5. For each of the following open sentences, f i n d a real number

which will make the sentence t rue : (a) ~ + 2 = 7 (f) c + (-3) = -7

(b) 3 + y = -7 ( 9 ) Y + 5 = - 6 2j 5

( c ) a + 5 = o (h) $X + (-4) = 6

5 ( e ) (-6) + x = 5 "6 ( J ) ( 3 + ~ ) + ( - 3 ) = -1

6 . Which of the f o l l o w i n g sentences are true?

(a ) ( - 4 ) i- o = 4 (b) - ( I -1 .51 - 101) = -1.5 ( c ) ( - 3 ) + 5 = 5 + (-3)

((d) (4 + (-6)) + 6 = 4 +

7. Translate the following English sentences I n t o open sentences.

For example: Bill spent 60# on Tuesday and earned 4% on Wednesday. He couldn't remember what happened on Monday,

but he had 30# left on Wednesday n i g h t . What amount d i d he

have on Monday?

If Bill had x cents on Monday, then

x + ( -60) + 40 = 30 .

This can be written x 4 ( - 2 0 ) = 30,

( a ) If you d r i v e 40 miles north and then d r i v e 55 miles sou th , how far are you from your starting p o i n t ?

( b ) The sum of ( - 9 ) , 28, and a third number is ( - 5 2 ) . What is the t h i r d number?

( c ) A t 8 A . M . the temperature was -2 ' . Between 8 A . M . and

129 noon the temper~ture increased 15'. Between noon and

4 P.M. the temperature increased 6 ' . A% 8 P . M . the

temperature was - 9 O . :That was the temperature change

between 4 P .M. and 8 P.M.?

( d ) If' a 200-pound man lost 4 pounds one weelc, lost b

pounds che second week, and at the end of the third

week weighed 195 pounds, how much d i d he gain in the

t h i r d week?

( e ) A stocl; which was listed at 83 a t closing tlme Monday

dropped 5 p o i n t s on Tuesday. Thursday m o r n l r q it was

listed at 86. :&at was thc change on Wednesday?

6 3 Proper t ies of' -- Addition

'We were caref 'ul to describe and l i s t the properties of a d d i -

t i o n when we d e a l t with the rmtnbers of arithmetic, Noh; that itrC

have decided h o w t o add real numbers, we want to v e r i f y t h a t

these p r r ~ p e r t l e s o f a d d i t l o n hold true f o r the real numbers gen-

e ra l ly . We lzn~! .~ that our definition of a d d i t i o n includez the usual

a d d i t i o n of numbera of arithmetic, bu t we a l so want t o be a b l e

t o add as s imply az xe could b e f o r e . Can :.IF s t i l l add r e a l

numbers in any order and group them in any way to sult o u r con-

venience? In o t h e r words, d o t h e co~mutative and a s s o c i a t i v e

p r o p e r t i e s of addition s t l l l hold t r u e ? If we are akle to

satisfy ourselves t h a t these p r o p e r t i e s - do c a r r y o v e r :o the r ea l

numbers, then we are assured t h a t t h e structure of n m b e r s i s

main ta ined as we movc from the numbers cf a r i t hme t i c to the r ea l

n m b e r s . Similar questions about multiplica~ion will come up

l a t e r . Consider the following questlons: Are ': -1 ( - 3 ) and ( - 3 ) + ':

names for t he same number? Check t h i s G n the number l i r re . Do

similar ly f o r ( - 1 ) -1- 5 and 5 + (-1); a n d i'or (-2) + (-6) and (-6) -t (-2).

Do t h e above sentences cover cvc ry p o s s i b l c Ease of addition

of real numbers? I f n o t , supply examples of t h e missing case:.

130 It appears khat t h e sum of any two real nmbers is the same

f u r ' either order of n d d i t i o n . TIlis is the

Commutative Property - or Add i t ion : For any two real numbers a and b,

Next, compute t h e following pairs of sums:

(7 + ( - 9 ) + 3 , and 7 + ( ( - 9 ) + 3 ) ; (8 + ( - 5 ) ) + 2, and 8 i- ( ( - 5 ) +- 2) ;

( + 5) + - 6 , and 4 + ( 5 1 (-6)) .

Wha: d o y ~ u observe about t h e results?

We could l l s t many more examples. Do you t h i n k the same resul ts would always hold? We have the

Assoc ia t ive of Add-n: For any real

numbers a, b, and c ,

( a + b) + c = a + (b + c ) .

Of course, if the associative and commutative properties hold

true in several instances it is not a proof t h a t they will hold

t r u e in every 'nstance. A complete proof of' the >roper t i e s can be given by applylng t h e precise definition of addition o f r e a l

nurr.bers t o every poas lb le case of t h e properties. They are long prcofo , eapcc ia l ly of the associative property, because t h e r e

are many cases. We sha l l not take t h e time t o g ive the proofs ,

but perhaps you may want to try t h e pmof f o r t he commutati~re proper ty in some o f t h e cases.

The associative property assures us tha t in a sum o f t h r e e

r e a l numbers it doesn't matter which adJacerit pal r we add f i r s t ;

it is customary to drop t h e parentheses and,leave such sums in

an unspecified form, such as 4 + (-1) + 3.

Another property of addition, which is new f o r rea l numbers and one t h a t we shall find useful , is obtained from the def in i -

t i o n of addition. For example, the definition t e l l s us that 4 + (-4) = 0; t h a t ( - 4 ) + (-(-4)) = 0. In genera l , the sum of

a number and j t a oppos i t e is 0. We state t h i s as the

rsec . 6-21

Addi t ion Property - of Opposites : PGT every real number a,

a t- ( - a ) = 0.

One more property tha t stems d i r e c t l y from fhe d e f i n i t i o n

is the

A d d i t i o n Prope-ty of 0: For every real num- -- - - b e ~ ? a,

a t O = a .

Make ~p several examples t o illustrate t h l s and the p r e c e d i x

p rope r ty .

Problem S e t 6-3 -- 1. Show how the properties of addition can be used t o explain

why each of t h e following sentences is true:

The l e f t nmeral is

5 4 ( 3 i ( - 5 + (-5)) +- 3 a s ~ o c i a t i v e and commutative prc - p e r t i e s of addition.

addition property of opposi tes .

= 3 + 0 conmutzt ive property o f add l t l on .

Tne right nqmeral is 3 + 0.

( e ) ( - 2 ) + ( 3 -1 (-4)) = ( ( - 2 ) + 3 ) + (-Q)

2. Consider various ways t o do the following computation5

mentally, and f i n d t h e one t h a t seems easiest ( i f t he re is

one). Then perform the a d d i t i o n s in t h e easiest way.

5 5 ( a ) + 28 + (-=I

(f) 253 + ( - 6 7 ) + (-82) 4. (-133)

( h ) (x - 1 2) -t ( - X I -I ( - 3 )

(i) w -t (w + 2 ) I- (-w) 4- 1 + ( - 3 )

3. Using the associative and commutative proper t ies of addition,

write a simpler name f o r one phrase of each of t he following

sen tences , and f i n d the t r u t h s e t of each:

(a) x = x + ((-X) + 3 ) (b) s + (7 + (-m)) = rn

( c ) n t ( n + 2) -t ( - n ) t 1 + ( - 3 ) = 0

( d ) ( y + 4 ) -4 (-4) = g + (-4)

4 . Use the d 2 f i n i t i o n of a d d i t i o n f o r n e g a t i v e nmbers to show t h a t If a < 0 and b < 0 , then a + b = b + a .

+5. Use the definition of addition to show t h a t a + O = a for

a l l real numbers a.

+6. U s e t h e definition uf a d d i t i o n t o show that a (-a) = C f o r

all real numbers a . (Hin t : Separate out the c a s e s a = 0

and a # G . If a # 0, one o f a and -a is positive, the

o t h e r negative (Why 3 ) .

6 - -- The Addition - P r o p e r t y - af Equality.

Tlyere i s a 1 1 7 t ~ e r " = r t e b 7 u t r d d i :.I -,n $7 : J ,~L , I :.,e mu?{; rL e

attentLon. ic'e l i 1 7 0 ~ t h a t

3 (-5) = (-1).

Elis mean: t h a t 4 -, (-5) and (-1) are two names f o r one number.

L-e t US add 3 to t h a t number. Ther (4 + (-5)) -I 3 and (-i) -1 3 are again 5iro naues Lr~r ene number. Thus

Alsc, f o r e x a . m p l ~ ,

S i m i l a r l y , Since

7 = 15 + (-81, -7 + ( - 7 ) = (15 + (-8)) + (-7)

This ovggcsta t:?e

Addi t ion Property - cf Equality: F G r any real

numbers a, b , c ,

if a = b , t h e n a d c = b + c

In w o r d s , if a and b are two names f o r one nwnber, then

a 1 3 and b + c are t w o names f o r one nwnber.

L e t us use the previously s ta ted proper t ies of addition,

and ;he above p r o p e r t y of equality i n some examples. b a m p l e 1. Determine the t r u t h s e t of t he open sentence

3 x + - = -2. 5

Can you guess numbers which make tY.is sen tence t rce? If you

don't see it easi ly , could you use p r o p e r t i e s o f addition to

h e l p ? Let us see. We do n o t r e a l l y know whether there g any number making t h i s sentence true. If, however, there is such

a nwnber x which ~ a k e s t h e s e n t e n c e t r u e ( t h a t is, if t h e t r u t h 3

s e t is not empty) , then x + 7 and -2 are t h e same number. 3 Let us add -- t o this number; then by the a d d i t i o n p roper ty 5

of equality we have

3 3 3 (x + -) I ( - 1 ( - 2 ) + (-9. 5 5 3 F;hy did we add --'? Because in 5hIs case we wish t o chafige 5

the left numeral so it will contain the numeral "x" a l o n e . iqatch

t h i s happening in the next few l ines .

Con t inu ing , we have

1;; Rlus, !:re a r r i v e at the acw CF?n senter:r:e x = ---. Ii' 2 number !I

x rn;ikes thc o r i g i r l a l ser?tqnce t r u e , i t 2 lsc rna,:es this nev,

ser?tcnce true-. 01' tni3 we s r e de13ta in br.r,ausc :.ie a p p l i e d pra-

p e r t i e s which hgl5 sruc for 311 r e a i :lwnbers. This t e l i r i u s i 2

thar -- Ix t h c o n l y p b s z i b l e truth value 01, t h e original sen- J --

C ;cnce. Bu5 T!, daez n o t guarcnte? that it is n ruth : ~ a l u c . - -. , -

&I-e we have d i ~ , - o v e r ? d a v e r y Important idea about scntenc~s

nal,:ing t h r ~ r i g i r i a l scnkencc true, then the only number which 7 -, 1 ? 12 1 J x can be TLZ -7. The nlruute wc check and flnd t h a t -- 5 -- does

make Lhe serk?.ence t r u e , l o ! have t ' ~ u n d the one an3 only number

~ t i i c h beLur.gs t;? Sne ruth set.

The scnt.encc i n thc- p r e v i ~ u s example is an equation. il!c

shall often call t h e t ruth s e t a f an cquation i t s solutiorl. s e t , -- --

and i t s members :;elutions, rind we s h a l l w r i t e f ' solvc-" i n s t e a d -- of "determine t h e t r d t h s e t of'"

Exmiple - 7 , o l v c t h e equaxion

I I' 5 T - - L' - X ? ( -A) 2

Is t r u e for some x,

3 tnen ( 5 2 -) 4- - :- 1 2 2 c is t rue f o r the sane x;

? + 2 = x . i - O is t rue f o r t h e same x;

7 = x is t rue l ' o r t h e same x.

the right s i d e is: 1 7 4 , 1 7 + +j = - - 2 2 , bF)

1 Heme t h e t r u t h SPY. 1 s [7 j .

EL-oklern Set 6-4 -- - - 1 S o l v e each of LA^ followjng e q u a t i o n s . I + l h l t l ? your worlc in

t h e form shov~n ffor 3 a m p l ~ 2 abo~re ,

1, x - 5 = 13 2 . {--GI 7 ? = ( - 3 ) 7

3 . (-1) -!- 2, :- ( - 3 ) -, LL : X + ( - 5 )

4. (x - 1 - 2 ) x = ( - 3 ) - 1 x 5 5 . ( - 2 ) - 4 x i ( - 3 ) = x + (-I!

-5. The A d d i t i v e Inverse Two numbers whose sm 1s 0 a r e r e l a t ed in a very s p c ~ ial

way. Fcr exam2le: what number when aaded t~ 3 y i e l d s the sum Cis

mat nun;her vrfien added to - h y i e l d s CT In general if x and

y are r ea l n ~ u l b e r s such that

x -4- y - 0,

we say t n l t y is an addi t ' ,ve -. invers;. - o:' Y,, Under th i : d c f ' i n i -

C o n , i s x tk.sn als2 an a d d l t i v e In-.lerse ,f y'

JJow i e t u~ L h l n ~ abouc any number z wkich is ari a d d i t i v e

i n v e r s e of, S R ~ , 3 . C f course we LrLC:;r one such number, r .me ly - 3 :

fcir by the sc'di_tio;l property af n p p o s ' t e z . 3 + ( - 3 ) - O . (:an

t h e r e be any o t h e r number z such t h a t 3 + Z = O ?

A l l of our experience with nuqbers tclle us "No, t he r e is no

other such number". Put how can we be absolutely sure? We

can s e t t l e t h i s question with the use of o u r p r o p e r t i e s of addi-

t i o n , j u s t as we d i d in &ample 1 Ir, the preceding sec t ion . If,

f o r some fiumber z ,

3 4 - z = O

is a true sen tence , then ( - 3 ) + ( 3 + 2) = ( - 5 ) T Q

i:., a l s o a +:?-de sentence, by t h e addition p r v p e r t y of cquallty.

(:ii-l:; did we add -3 ? ) Then, however, ( ( - 3 ) - 3) - z = - 3

i.: t r u e f o r the same z by the associative p r ~ p e r t y of addition

2nd the C property oS addition. This f l n a l i y tells us that

2 = - 3

mist a l s c be true; we have , f o r this l a s t z tep , used the addition

p r o p e r t y of opposi. 'scs.

:;Rat have -de done here? ',Je started o u t by ck,ooslng z as

a number which is an additive i n v e r s e of 3: we found out t h a t

z had t o equal - 3 ; that i s , that -3 is n o t jusc an a d d i t i v e

i n - ~ e r s e of 3, b u t also :.he only additive i nve r s e of 3 .

Is there anything s p e c i a l about 37 Do you t h i n k 5 has more

ti-ia.~ one addir,i ire Znvcrxe ? How about (-6.3) $ We c e r t a i n l y au:;Sz

:i.t., 2nd we can show thac they do not by the same l i n e o f r e ~ s z n i n g

2s the above. Can :.re, however. check ali numbers ', ihhat we necd - i.s a r e s u l t f'or any rea l n m b e r >r , a ~esult iv~hich is supposed

GO tc11 u s something l i k e t h e fo l l owl r i g : ille Imow that I -x) is onc a d d l t i ~ ~ s inverse of x; ~2 doubt if t.here is any other, and - '-.i;is is h ~ l r ; we p r o v e there is none . Let us parallel t h e reasonini.:

::e c;:;eci in the s p e c l a 1 yase in vihich x = 3 , and see If' vie ;:an

sr-~.~.:? at t h e c o r r e s p o n d l i ~ g conclusion.

SL~-ppose z is any additive invcrse of .x. t h a t Is, any . .1-:j*?!i;.ey s3jl:h that

X -! z = C.

.,?:zf, corresponas to t h e f i r s t s;ep in our prev5OuS spec t a l case'

I sec . 6-51

'dc use 5 2 1 ~ add i s i sn p rouy ryy c f equa;ity t c wrlte

(-A) ( X T Z ) = ( - X I I 0.

'Je t h e n have that

( -1- X ) i C = -x.

'vhas arc t . 1 ~ two r e a ~ o n s we !la-&,e used i n a r r i v i n g at This 123t

smzsnce.:

7'11 P n

[) - = -x ( ;;.n;, 7 $)

3rd f i n a l l y

z - - -X. (-,thy 4 , ) ' a j ~ . have Z U C C C P ~ P ~ ~ I J c a r r y i n g a t o u r prc-gran, n o t j u s t xhen

x = 3 bu t l o r m y x. F.ach numbcr x has a tu~Lque ( m s a r i l ~ g li . j u s t o n e " ) additive inverse, namely -x.

You probably have all kinds of q~;~lrrts and questions a t t h i s

pain:, a d t h e s e sxae to b e expected s i n ~ e this is t h e ;'ir~t proo:

k;.h.le!l ycu have se?n :r t n i s courorl . 3- at we have d3ne is 'io

use <acts :-..?ich t:e ha-a? p r e > a i o u > l y Icnot.m abont. a l l ?pal numbel-s - ir! order to argue o u t 2 fiel:~ - fa^; about; ali r c a l numbers, a new f a c t wnich :rou ~ e r - t a l n l y expected trl, bc t r u e , but. ~ h i ~ h R C V C P -

t n e l e s s took t h i s 2,inrl of' checlri.ng. in!? s l l a l l do a nunker .;f

p-oofs in tnis course , and yqu xi11 bemme nc3re and marc s c c u s -

torned Yo t h l s k l n d of reasonin? as you prucress I r the meantime,

l e t uc ma!cc u ~ l c moFe cc~mmcr?t about t h e prcof jusr c z m p l 3 c d .

Tne S U C . ? , P S S ~ ~ : ~ 3 t ~ ~ : ; iire e_>o:,; xcre of ~ o t i r s e cnclsen qul'ce d e l l h e r -

aT.ely j.n order to make the proof' su:ceed. m i s might give you

cr.e impression ;hat t h ~ p-GO? ?:as ''zig~ed~', t h a t it cou idnT t, ccmc o u t ar-ty oc; lcr way+ Is 'chis f a i r " Y e s , it; is. 2nd in f a c t e,,.jery p r c o 3 5 '"r:yEed" in t k c o,?nse <fiat we take on1.y s teps t ~ 3

nelp us toliar3s o u r $ o a l , a ~ r l do not Cake s t e p s v!hich f z i l \(J

do 11s any ~ , o o r i . !Thib~-n pie 2 tar tcd f'rorn 3 {, z = 3,

we chcsc Lo use K I P addition 3 r o p o r t y of equality to add ( - 3 ) ;

we zou ld ..lave added any otl-ler punrber i n s t ~ > d , b u t j . t wouldnt t

have ne lped us. And so we didnl t add a d i f f e r e n t number, bu:

added - 3 .

Statements o f new f a c t s or properties, which can be shown

t o f o l l o w f rom previously es t ab l i shed properties, are frequently

( b u t n o t always!) c a l l e d "theorems". Thus the prope r ty about

addltive inverses ob ta ined above can be stated as a theorem:

Theorem o-5a. Any real number x h a s e x a c t l y one additive i n v e r s e , namely -x.

An argument by which a theorem is shown t o be a consequence of

other p rope r t i e s is called a proof of the theorem.

Problem S e t 6-5a -- 1. For each sentence, f i n d its t r u t h s e t .

(a) 3 + x = O

(b) ( - 2 ) + a = o ( 9 ) ( - ( 2 + +)) t a = o

( c ) 3 -I 5 1- y = 0 ( h ) 2 + x + ( -5 ) = 0

1 (d) x + (--) = 0 2 (i) 3 + ( - x ) = 0

( e ) 1-41 -I- 3 -E ( - 4 ) + c = 0

2 . Were you able t o u s e Theorem 6-5a to save work in so lv ing

these equa t ions?

L e t us look at another example f o r t h i s technique of show-

ing a general property of numbers. Of course we cannot p r o v e

a general property of numbers u n t i l we suspect one; let us find

one t o s u s p e c t . Recall t h e picture of addition on the number

l i n e , or the d e f i n i t i o n of addition if you p re fe r , to see t h a t

( - 3 ) 4" (-5) = - ( 3 + 51, Another way of writing t h a t ( -3 ) + ( -5 ) and - ( 3 -t 5) are names

f o r the same number i s that

- ( 3 + 5) = ( - 3 ) + ( - 5 ) . T h i s might lead us to suspect that t h e opposite of' the sum of

two numbers Is the sum of t h e opposites. Of course we have

checked t h i s only f o r the numbers 3 and 5 , and it is wlse t o

! check a f e w more cases. Is

! - ( 2 + 9 ) = ( - 2 ) ( - g ) ? IS

-(4 + ( - 2 ) ) = ( - 4 ) + ( - ( - 2 ) ) ?

(idhat is another name f o r (-(-2)) ? )

Is

I Our hunch seems to be t rue , at least i n a l l the examples we

have t r i e d . L e t us now, instead of checking any more examples

by arithmetic, s t a t e the general property which we hope to prove

as a theorem.

Theorem 6-5b. For any real numbers a and b

-(a I- b) = ( - a ) -t (-b).

Proof. Ide need to prove t h a t ( - a ) + ( - b ) names the same

I number as -(a c b). Let us check t h a t ( -a ) + (-b) acts l i k e : t h e opposite of (a + b). We look at (a 4- b) + ((-a) + (-b)),

f o r If t h i s expression I s 0, ( -a ) + (-b) w i l l be t he oppos i t e

of (a + s ) . (a + b) + ((-a) + (-b)) = a + b + (-a) (-b)

= (a + (-a)) i (b + (-b)) ( ihy?)

And so we find that f o r a l l real numbers a and b,

( -a ) + ( - b )

is an additive inverse of (a -t b ) , and, s i n c e t he re is only one

additive inverse , that

- (a + b) and ( -a ) t (-b)

name the same number.

Problem Set 6-5b -- 1. Which of the fo l lowing sentences are t rue f o r a l l r e a l nwn-

bers? Hint: Remember that t h e opposi te of the sum of two nwflbers is the sum of t h e i r opposites.

2. In t.ilc following ?roo:' supp ly t k e r eason for each s t e p :

For a l l rllmbers I:, y ana z,

( - x ) + (y - (-z)) - y + (-(x -L zj) . Proof .

-

( - x ) -1 (;y -1 c - z g = ( - X I - ((-z) + :{) = - i i - r j ) - y

= f ( x ; z ) ) + y

' y - ( - .I). IS - ( 3 4 6 - - : 5 - ( - 3 ) - (-6) -1- -I .; (-5)? >ghat: do

you t h i n k is true f'or the o p p o s i t e of' t he sum CI' more t i ~ a n

t k l 0 I ~ U I I ~ L ~ ~ I ~ ~ ~ :

T e l l w h i c h of tae f o l l o ? ~ i n g sentences are true.

(a) -((-2) - 1 6 (-5)) = - (-6) L 5

(L?) a - (-b) + (-29 = 3a -- b 4- 2

( c ) - - (--b) + (-5c) 4 .7d) = (-a) i b i 5c + (-.7d)

( d ) - ( 3 ~ i Zy + ( -Pa ) i ( -3bO = (-2~) + 2y -: (-:'a) i ( -3b) d

c:; ' Gii;e an ergunent I'or t h e :vnclusion you made f o r the second

question In Vroblern 3 .

~ 5 . Fruw tile f o l l o w i ~ ~ p rope r ty cf additrion:

For any real nurnber a and any real number b

and any r ea l rlumber i3.

if a + c = b + c , then a = b .

6-6. Summary

We have defined additLon of real numbers as fo l lows : The sum of two p o s f t i v e (or 0 ) n<mberhs is familiar

from aritk~etic . The sm of two negat ive numbers is negat ive; t he absolute

value of this sum i s the swn of t h e absolute val.ues of

the numbers.

The swn of two numbers, of which one is p o s i t i v e ( o r U)

and the other is negative, is obtained as follows:

The absolute value of the sum is t h e difference

of t he absolute values of t h e numbers.

The sum is p o s i t i v e if the p o s i t i v e nw;l.ber h a s t h e

grea ter absolute value . The sum is nega t i ve if the negative number has the

g r e a t e r abso lu te value. The sw, is 0 if the p o s i t l v e and nega t ive numbers

haire the same absolute va lue .

We have satisfied ourselves t h a t the following properties

hold for addition of real numbers:

Commutative Proper ty - of Addition: For any two rea l numbers

a and b:

a + b = b t - a .

Associat:ve Proper ty of Addition: For any r e a l numbere a , - -- I b, and c,

( a t b ) + c = a -t (b + c ) .

Addikion Property 01' Opposites: For every real number a, - - -- i

a + ( -a ) - 0. i

Addition Proper ty of 0 : For cvery real number a , -- a 1- O = a.

Addicion Proper ty o f Equality: For any real numbers a , -- - -- b, and c ,

if a = b , then a +- c = b + c ,

We h q - - e used t h e addition property of equallty to determine

the t r u t h sets of open sentences.

!$e have proved that t h e a d d i t i v e i n v e r s e i s mique - t h a t i s ,

that each n m b e r has exac t ly one a d d i t i v e i n v e r s e , v;hich we e a l i

its opposi te .

[ s e c . 6-61

V:e have d i s c o - ~ e r e d and proved the fact that t h e oppos i te of the swc of twu r l ~ ~ n b e r z is the same as the Fum toi ' their ~ 1 ~ p ~ s i t e s .

Fe',; lev^ Problems - 1, Find a comnon n m L e for eech of the foilowin?:

3

( 2 ) 3(8 I - (-6)) ( a ) a ) - 1 5 u

( b ) ( - 2 ) -i- 3 i 3 (e) 1-61, 131 (-3)

2 A 7 .! (-11: 1 (f) G ( I -I I - A ! ] " ) f i l ch oi" 5l7c z ' o l l ~ : . i i ~ sentences ape t?U?:'

( a ) 2 -1 (-8) = (-8) - 3

( b ) 1 - 8 1 - I - ( - 8 ) = O

3. ShO>i :?.2:s ',he pi70ker',ies of addlt:orr ,:an be u;3ad ;u csplaln

;ah-$ e a ~ h of t h e f o i l ~ : 4 i n g szn tences Is true:

I . Fir-1- the tr~tki set 09 each 01' t h e fol;oxin~.:

6. Two nwnbe~s are added. What do you know about these ~liunbers if

( a ) t h e i r sum Is negat ive?

(b) t h e i r sum is O?

( c ) t h e i r sum is positive?

7 . A salesman earned a basic salary of $80 a week. In addition

he rece tved a commission of 3 % of h i s f o t a l sa les . During one week he earned $116. \ h a t was the amolult of h i s sales

for the neek? Write an open sentence f o r t h i s prabl~m.

8. A f i g u r e has f o u r sides. Three of them are S f e c t , LO feet,

and 5 f e e t , respectively. How l o n g is the f o u r t h side?

(a ) Write a compound open sentence f o r this problem.

(b) Graph the t ru th s e t o r the open sentence.

9 . If a, b, and c are numbers of arfthmetic, write each 01'

the indicated sums as an indicated produc t , and each of the

i n d i c a t e d p r o d u c t s as an indicated sum:

( a ) ( 2 b + c ) a 2 ( e ) x y + zj.

(b) Fa(b + c ) (f) 6a2b i 2ab 2

( c ) 3a I- 3b ( ab ( a c t 3b)

( d ) 5x I- loax ( h ) 3a(a + 2b -1 3 c )

3 Given the s e t {-5, 0 , T , -.75, 5 )

(a) Is this s e t closed under the o p e r a t i o n of taking the

opposite of each element of the s e t ?

( b ) Is this s e t closed under the operation of taking the

absolute va lue of each element?

( c ) If a set is closed under the operation of taking the

opposite, is it closed under the op~ration of t ak lng

the absolute value? $hy? 3

Given the s e t {-5, 0, f , 5,. 7 )

(a) Is this s e t c l o s e d under t h e operat ion of t ak ing t h e

absolute value of each element of t h e s e t ?

(b) Is this s e t closed under t h e operation of t ak lng the

opposite of each element:

( c ) If a s e t is closed under the operat ion of t a k i r g the

a b s o l u t e value, is it closed under the operatlon of

taking t h e opposi te '. Ihy?

12 Two automobiles start from t h e same city travelling in the

same direction. Iv'rite an open phrase for the t i m e it Lakes

t h e faster ca r to g e t rn miles ahead of the slower ca r , If

t h e rates of the ca r s are 3 0 m.p.h. and 20 m.p.h .

Chapter 7

PROPERTIES OF MULTIPLICATION

7-1. 1~ lu l t ip l i ca t ; ion -- of Heal. Nuntbers

N o w l:t us decide how we s h o u l d multlply tKo real n~~rnl:c-.s to

obt,a:n noth her real nwrbber. A l l t h a t we can say at p r e s e n t . 'is t h a t

we know how t o multiply two nm-nega t i ve numbers .

Of primary i lnpuratsrlce here, as In t h e definition 3f adc? l t i on ,

I s t h a t we rn?int .~Ln the "str~~zture" of the number s y s t e t n . b!e know

t h a t if a , b , c are 2.n:~ n.mbers - o f a r i t h m ~ t . l c , then

ab = b a ,

( a ~ ) c = a(bc),

a 4 1 = a,

a . 0 = 0,

a ( b - c ) = ab + a c .

(ma: namez dd-id we g i v e to t h e s e prcpe r t i e s of' ~ n u l t i p l i c a t i c n ? j What~ tvc r rneani~g w e give to the produc t u f two real numbex, we

must se sure t h ~ t it agrees w i t l ' ~ t h e pl-oduc ts which we hl rveady have

f o r non -nega t i ve r e a l numbers and that the abov? --- p r o p e r t Yes - of

multlplicatlon still hold f o r a l l r ea l n-m5ers. ----- Consider s o n ? possible produc t s :

( 2 } ( 2 ) , ( 3 } ( 0 ) . (L):(O), { - 3 } ( ~ ) , ( 3 ) ( - F ) , ( - 2 } ( - ? 8 ) .

(DO these incl l ldi ; examples of e ~ e r y 2 a E c of mult:pl.ica t i o n of ? o s i -

t i v e ar.lri negat ive number's and z e r o ? ) Notice that t h e f t r s t t h r e e

p rod~c t : : i n ~ o i v e o n l y ncn-n~gative r l - n h k r s and are t h e r e f o r e

already de termined :

Now l e t us :ry to scc what the reniairllrlg tLree products will h a v e

t o be in orde r to preserve t h e L a s i c p r o p e r t i e s of m u 1 t l p l l c a t l ' ~ n

l i s t e d abovs . 1:1 the f i r s t p l a c e , if w e want t h e r n ~ , l t i p l i r : . : t ! : ~ n

p r o p e r t y of 0 t o hold for a l l real numbers, then we must have

( - 3 ) ( 0 ) = 0 . The other two products can be obtained as follows:

3 = ( 3 ) I o )

0 = ( 3 ) ( 2 + ( - 2 0 , by writing 0 = 2 + ( -2 ) ; (~otice how t h i s introduces a negative number i n t o t h e discussion.)

0 = ( 3 m ) + ( 3 ) ( - ~ ) , if t h e d i s t r i b u t i v e proper ty 1 s t o ho ld for rea l n m b e r s ;

0 = 6 s ( 3 ) ( - 2 ) , s ince ( 3 ) ( 2 ) = 6.

We know from uriqueneas of the additive i nve r se that the only real number whlch y ie lds 0 when added t o 6 is the number - 6 . Therefore , if the proper t ies of numbers are expected to hold, the

only possible va lue f o r (3)(-2) which we can accept is -6 .

Next, w e t ake a similar course to answer the second question.

0 = ( - 2 ) ( 0 ) if the multiplication property of 0 is to hold f o r real-numbers;

0 = ( - 2 ) ( 3 + ( - 3 b , by writing 0 = 3 + ( - 3 ) ;

0 = (-2) ( 3 ) + (- i) ( - 3 ) , if the distributive proper ty is t o hold f o r real numbers ;

fi = ( 3 ) ( - 2 ) 4- ( - 2 ) ( - 3 ) , if the commutative p rope r ty is to hold f o r r e a l numbers;

0 = ( - 6 ) + ( - 2 ) ( - 3 ) 9 by the previous r e s u l t , which was ( 3 ) ( - 2 ) = - 6 .

ROW we have t o come t o a point where ( - 2 ) ( - 3 ) must be t h e o p p o s i t e

of - u ; hence, if we want the p r o p e r t i e s of multiplication to hold

for r e a l numbers, then ( - 2 ) ( - 3 ) must be 6.

L e t us think of these examples now in terms of absolute v a l u e .

Reca l l t h a t t h e p r o d u c t of two p o s i t i v e numbers is a p o s i t i v e

n w b e r . Then w h a t a re t h e va lues of 1 3 ) ( 2 1 and 1-211-31? How

do t h e se compare, respec t ive ly , with ( 3 ) ( 2 ) and ( - 2 ) ( - 3 ) ? Compare

( - 3 ) ( l l ) and - ( 1-31 141) ; (-5)(-3) and 1-51 1-31; ( a ) ( - 2 ) and 10) 1-21

Thls is the hint we needed. If we want the s t r u c t u r e of the n ~ m j e r system t o be t h e Eame f o r real numbers as it was f o r the

1 nurnbers o f arithnetic, we m u s t d e f i n e t h e p roduc t r,f t w c rea l

/ numbers a and kl as, follows:

I f a and b a r e b o t h nega t ive or

bo th non-negative, then 3,'o = 131 1 b l .

If one o r t h e numbers a and b <s

non-negative 2nd t he other is negative,

then a b = - ( l a [ ( b ( ) .

It is impor tant t~ recognlzc that la1 and (bl m e numbers

of srittmetlc f'or any r e a l numbers a and b; and we alre-djr know

the p r o d u c t lallbl. (why?) Thus, th? product lallbl is a post-

t i v e n v n b e r , and we obtain the p r o d u c t ab as either la / 1 b ( or

iTs opposite. Agaln we have used o n l y the o p ~ r a t i o n s with whlch

we are already familiar: rnu l t ip iy ing p o s i t i v e num'rlers or 0, and

tak ing o p p o s i t e s . It w i l l h e l p you to remember t h o definition by

comple t ing t h e santences: (Supply the words "positive", "negative1',

o r "zeroT' . )

The produc t of two positive numbers is 2 n : ~ nb e rv . The product of t w o n e g ~ t i v e numhcrs is a numb2r. The p roduc t of a n e g a t i v e and a p o s i t i v e n a n ~ e r is a

n ~ m b e r . T h e p roduc t of a rea l number and 0 is

S i n c e the produc t ab is e i t h e r l a ] l b ( cr its oppos:te and

since (a1 lbl is non-negative, we can s t a t e the following p r o p e r t y

of multiplicstion:

F o r any r e a l numbers a , b ,

lab1 = la1 l b ( .

Problem S e t 7 -1

I . Use t h e d e f i n i t i o n o f multiplication t o c a l c u l a t e the follow-

ing.

Examples: (a ) (5)(-3) = - (1511-31) = -15

( b ) I - 5 ) ( - 3 ) = (1-511-31) = 15 ( c ) (-5)(3) = - t 1-51 131) = -15

( c ) 1 ( - 3 ) ( 2 1 1 ( - 2 1 (fl l - 2 1 ( - 3 ) + I - ~ D 2 . Calculate t he following:

1 (a) ( - $ ( - 4 1 (f) ( - 3 ) ( - 4 ) + 7

Find t h e values of the following f o r x = -2 , y = 3, a = -4: (a) P x + 7 y

(b) 31 -x) + 0 - 4 ) ~ + 7(-a))

( c ) x2 + 2(xa) + a 2

( d l ( x + a ) 2

x2 + (&I + (-4) ~yi)

(f) Ix + 21 + ( -5 ) I ( - 3 ) + a l 4. Which o f t h e following sentences are true?

( a ) 2x t S = 12, f o r x = -10

(b) P ( - ~ ) + 8 = 28, f o r y = -10

+ 8 # 20, f o r x = 2

+ 30) < 0 , f o r b = 2

( r ) [I+ 31 + ( - 2 ) ( 1 x + ( - 4 ) ] ) 2 1 , f a r x = 2

th?: r graphs . Exarnple: Find the t r u t h s e t of' ( 3 ) { - 3 ) - c = 3 ( -11 ) .

I 1' ( 3 1 1 4 ) + 1: = 3(-4) is true f o r some c ,

t hen -9 + c = -12 is t r u e f o r t h e s a m e c ;

(-5 + c ) + 9 = -12 -t $ is t r u e f o r the saxe c ;

If' c = -' 2 3

then the l e f t member is ( 3 ) ( - 3 ) + ( - 3 ) = -12,

and the r i g h t member is 3(-4) = -12.

Eer-ce. the t r ~ t h s e t is I - 3 3 .

(a) x 6 ( - 3 ) (-4) = 8 (b) 2(-2) + y = ? , ( -? I (c) x + 2 = 3(-6) + ( - I l ) ( - 8 )

( d ) x + (-5)(-6) = ( - ? ) ( 2 )

( e ) x - (-5)(-6) + \ - 2 ) ( 3 )

( f ) x > ( - q ( - 2 ) + (-5)(7) '2 (a ) lx l = (- -) J ( ' T I + ( - 1 1 (-51

6 . G l v e n t h e s e t S = [ I , -2, -3, f i n d the s e t F of all

products o f p a i r s of elements 03 o b t a i n e d by multiplying each element of S by each element cf $ .

7. Given the s e t 3 of all rkalnurnbers , f i n d t h ~ s e t Q of 211

p r o d . ~ c t s o f pairs of elements of R. Is Q t h e sane s e t 2s A? Can ycu conzlude t h a t R is closed under multlpllcatinn?

8. Given the s e t N of a l l n e g a t i v e r ea l numbers, f i n d the s e t

T or' all. products of p a i r s nf elements of N. I s t h e sei; of

negative r e a l numbers c losed unCer multiplication? I

9 . Glven t n e s e t V - {I, - 2 , -3, 4 1 , flnd Lhe s e t K o f all

posj Live numbers o t t s ined as prcduc ts of peirs of elements nf Y .

7 0 . P r o v e that the a b s o l u t e value of t h e p r o d u c t ab is the

product lalwlbl o r the abso lu t s v a l u e s ; t h a t is,

11. MiaL tail y o u szy about t w o rcal n u m b e r s a and k I n e a c h

of the following cases?

( a ) ai, is p o s : ~ t i v e .

( b ) a b is n e g a t i v e ,

( c ) ah is positive and a is p o s i t i v e .

(d) a h is positive and a i s negative.

( e ) ab i s negat ive and a Is p o s i t i v e .

( 1 a b is nega t i ve and a is n e g a t i v e .

1 2 . G l v e the reason f o r esch numbered s t e p in t he proof o f t h e

following theorem.

Theorem. If a and b are numbers such t h a t b o ~ h a and

ab are p o s i t i v e , then b i s a l s o positive.

P r o o f . W e assunie t h a t 0 < a and 0 < ab. Then:

1. Exactly one of t h e foll~wing is t r u e

b < G , b F O , O < b .

2. I f ' b = 0,then ab = 0 . I t Therefore , b = 0" is fa l se , s i n c e ab = 0 and 0 < a5

are contradictory.

3 . If b < 0 , then ab = -(la]lbl>.

4 . If b < 0 , then ab is n e g a t i v e .

heref fore, "b ( 0" is f a l s e , s ince ab is negative and

O < ab are contradictory. Therefore, 0 < b; t h a t Ls, G is positive, nincc this is

t h e on ly remaining possibility.

7 - 2 . P r o p e r t i e s - cf Multiplication - The d e f i n i t i o n of multiplication f o r r ea l numbers g iven I n t h e

preceding section was suggested % the s t r u c t u r e properties which

we wish to preserve f o r all numbers. On the o t h e r hand, we have

not actua1;y aaourned these properties, since t h e d e f i n i t i o n could

have been given at t he o c t s e t w i t h o u t any reference to the proper-

ties. H o w e v e r , now that we have stated a d e f i n i t i o n f o r

multiplication, it becomes impor tan t to s a t is f y ourselves that this

d e f i n i t L o n r e a ? l y l eads tu the des i r ed p r o p e r t i e s . In o t h e r wcrds,

we nead t3 -- p r b v e that m u l t Y P i i c a t i 3 n s o defined do5s nave the

prcpe r t l e ; . S i n c e the definLtlon is s t a t e d i r r t e r m ~ of operations

on p o s i t i v e numbers and 0 arld 9f t a k l n q opposites, these clperations

are the on ly ones avaFlable t n us In the p r o o f s .

% . i u l t l p l i c a t l o n p r o p e r t y o f 1: Far any re21 n:: n L ? r a, -.-- - - a.1 - a .

P r o a f : If a is p o s i t i ~ ~ e GI? 0, we know t h a t a ( 1 ) = a.

If a 1: negative, our d e f i n i t i o n of ! ~ i ~ l t i p l i c a t i o n s t a t e s that

Try LC zxp l a in t h e r e a s G n why eacn s t ? p In t h e above p roo f is

t r u e .

W- .~ l t ip l i ca t :on p r o p 2 r t y of 0: F 3 r any real n w a e r a, -- - - 3.0 = 0.

Write o - ~ t t h e ?roof of t h i s p r o p e r t y f o r y o u r s e l f .

Col:uiil~ta t i s re pr3perty - qf m~ltiplication: F o r any real numbers

a and b,

a5 = ba.

P r o f If one o r Loth t h e numbers s , b are zero, then ab= ba.

( h h y ? ) If a and b are both posi t i .ve or bo th nega t i ve , then

ab= l a [ l b ] , and 03 = ]b[(al . Since ] 3 I b ] a r e numoers sf arithnetlc,

la1 I L ) - l b l 121.

Her.cr-,

ab = ba

fmlr t h e s e ~ W G C Z E ~ S .

If one of a and b is p o s i t i v e or O and the o t h e r Is

negative, then

ab = - ( la1 lbl) and ba = - ( It1 l a l ) . S i n c e

la1 l b l = I b l l a l J

and s ince i f numbers are equal t h e i r opposites are equal,

- ( 1.1 l b l ) = - r l b l l a l ) . lienee,

for this c a s e a l s o .

Here we have given a complete proof of t h e commutative pro-

p e r t y f o r all r e a l numbers. We have based this proof on t h e

precise d e f i n i t i o n o f multlpllcat?on of real numbers.

Problem Set 7-Pa -- I . Apply t h e p r o p e r t i e s proved so f a r t o compute the fcllowing:

,

2 Illustrate the proof of the commutative proper ty by replacing

a and b as follows:

RsscrLative p r o p e r t y of mu:tiplication: Pclr any r e a l n.mbers - - .

a , b, and c ,

Proof': The proper ty must be shown t o be t rue f o r one negative,

twc nega t i ve s , o r t h r e e negat ives . T h i s i s leng thy , b u t we shall

be able to s i m p l i f y it by observing that

and

Thus I(ab)cl = l a ( b c ) l f o r all r e a l n u m b c r s a , b, c .

This reduces the proof of the as soc i a t ive pro-per ty of n ~ u l t i -

p l i c a t i o ~ ~ to the problem of showing t ha t ( a b ) c and a(bc) a r e

e i t h e r both p o s i t i v e , b o t h ze ro , or b o t h neg2 t i ve .

For example, if both ( a b ) e and a ( b c ) a.re n e g a t i v e , then

I ( a b ) c ] - - (a (bo) ) and ( a ( b c ) i = - ((ab)c) . Thus -(a(bo)) = - ((able)

and hence a ( b c ) - ( a b ) c .

If one o f 2 , b , c is zero, then ( z b ) c - 0 snd a ( b c ) = b.

(why?) Hence, f o r t h i s case ( a b ) c = a(bc) . I f a, b , and c are a l l d i f f e r enL from z e r o . we need to con-

sider e i g h t d i f f e r e n t cases, depending on which ilunbers are p o s i -

t i v e and which are negat ive , as shown in t h e t a b l e b e l o w .

One colwnn has been f i l l e d In for p a s i t i v e a, and negatiT\le b

and c . In t h i s case, ab is negatille and bc is p o s i t i v e . Hence (ab) c is p o s i t i v e and a ( b c ) is p o s i t i v e . Therefore ( a b l e = a ( b c )

in this case.

The assoc ia t ive pmpertg s t a t e s tha t , in mul t ip ly l r lg t h r ee

numbers, w e may f i r s t form t h e product of any ad jac-.nt pzLr. The

e f f e c t of a s s o c i a t i v i t y alGng w l t h c o n u n ~ t a t i v l t y is to allow u a

t o wri te products of nmbers without grcuping symbols and t o per -

form the i r ~ u l t l p l i : a t l o r , in any groups ar.d any 01-ders .

Problem S y t 7-2b -- 1. Copy t h e t a b l e g iven a b o ~ e , a r d complete it. llse it to check

the remaining eases and f i n i s h the proof .

2. In each of the roliowirlg v e r i f y t ha t t h e two n u m e r a l s name t h e

same number.

(f 3 l. 2 and ( 3 ) ((-2) ( 0 ) )

3 . Exp la in how the associative and connrutatLtre p m p e r t i e s call

be used to perform the following r n u l t i p l l c a t i o n s in t h e

easiest manner.

(a) ( - 5 1 (17) (-20)(3)

155

Another p roper ty is one which ties t oge the r t he ope ra t i ons o!'

1 a d d i t i o n and r n u l t i p l i c a t i a n .

Distributive p rope r ty . F o r any rea l numbers, a , b, and c ,

a ( b + c ) = ab + a c .

We shall consider o n l y a few examples:

( - 3 ) = ? and ( 5 ) ( 2 ) + ( 5 ) ( - 3 ) = ?

+ ( - 3 ) = ? and ( 5 ) ( - 2 ) + ( 5 ) ( - 3 ) = ?

( - 5 2 ) + ( - 3 = ? and ( - 5 ) ( - 2 ) + (-5)(-3) = ?

The distributive p r o p e r t y does hold f o r a l l r e a l numbers. It

c ~ u l d be proved by applying the definitions of multiplication and

addition to a l l p o s s i b l e cases, but t h i s is even more t ed ious than

! the proof of associativity.

F I Problem Set 7-2c -- - - I Use the distributive proper ty , if necessary, t o perform the indi-

j cated opera t ions w i t h t h e minimum amount of work:

: 1. ( - 9 ) ( - 9 2 ) + (-9H-8) 3 Q. ( - 7 ) ( - T ) -+ (-7)(i)

2. ( . 0 3 ) (6) + (-1 - 6 3 ) ( 6 ) 5. ( - 6-93) + (-79

2 2 6 - (-7)(3) + (-5)(5)

We can use t h e d i s t r i b u t i v e proper ty t o prove another u s e f u l I , property o f multiplication.

Theorem 7-2a. For every r ea l number a,

I

To prove t h i s theorem, we must show that ( - 1 ) a is the opposite i of a, t h a t is, that I

a + ( - l ) a = 0. I , [ sec . 7-23

Here we 5a:re shown that (-l)a is an addit: ~ v e inverse of a . since -,.> ,hL a l s ~ know t h a t -a is an add i t i ve Inverse of a 2nd

t h a t tho additive inverse is unique, w e have proved t h a t

Prob lem S e t 7-2d -- Use Theorem 7-Pa t o prove the following:

1. For any rea l ncnbers a and b, ( - a ) ( b ) = -(ah).

2. For any r e a l numbers a and b, (-a)(-b) = ab.

3 . Write t h e common names f o r the p r o d u c t s :

( a ) (-5)(a~) (4 ( - i c ) ( $ )

(b) ( - 2 4 ( - 5 4 2 ( e l ( 4 c ) ( - h a ) Y

( 4 (3x1 ( - 7 ~ ) ( f ) (-0.5d)( 1.2~)

7 I - 2 Use oi' t h e Multiplication Prcpe r t l e s ---

I:Tp h a . : ~ seen in the , ? 'x?ve problems t h a t w e rr.a;r now write

In fact,

-ab = - ( a h ) = (-a) (b) = ( 2 ) (-b) .

No;( t k a t we can rr.ultiply r ea l numbers and have at our disposal

the pro3ert ies o f m u l t i p l i c a t i o n of real n m b e r e , we have a s trong

b a s i s f o r dealing with a v a r i e t y uf sltuatioas In algebra.

Problem S e t 7-3a + --

1. Use the dlstrikutive proper ty to u r i t e t h e f3llowing as indi -

2, In doing Problem 1, you probably used t h e p roper ty : For any

real numbers a and k , ( - a ) (-b) = ab.

Find the parts of t h e problem i n whlch you used it.

3. !Jse the d i s t r i b u t i v e p r o p e r t y t o wr i t e t h e Sollowing phrases

as indicated p r o d u c t s :

( c ) 12 + 18 i d ) 3~ i- 33' f 3 2

( e ) icm + kp

(h) ( &)a2 + (-6)b 2

i c a + c b + - c

(j) 2a + ( - 2 8 )

4. Apply the d i s t r i b u t i v e and other prope r t i e s tu t h e following:

Example: 3x + 2x = ( 3 + 2 ) x = 5x

(a) 12t+7t ( g ) ( 1 . 6 ) b + ( 2 . 4 ) b (b) 4% + (-15a) (h) ( - 5 ) x + 2x + Ilx ( c ) ( - 5 ) ~ + 1 4 ~ (1.1 3 a + 7 ~ ( ~ ~ r e f - ~ l I ) (d) 122 -t z (Hint : z - 1.2) ( 3 ) 4p + 3p t 9p ( e ) (-3m) + I-6m) ( k ) 8r + (-14r) + b r

1 (1) 6a + (-ha) + 53 c 14b

158

h i a phrase which has tk? f n r m of an irzdlcat;ed swn A + E, 1 and R a r e cal led -. term c f the 3hrese; in a phrzse of the form A - B c C , A, B, and C a r e ca l led terms, e t c . The d i s t r i b u t i v e

property is v e r y h e l p f u l in simpllfylng a phrasc. Fhus we fo:lnd

that

ga + 8a - (5 + 8 ) a = -.3a

is a p o s s i b l e a ~ d a d ~ s i r 2 b l e s imp1 i f i ca t :on . However, i n

5x + by

no suzh simplification is p o s s i b l e . Why?

We may s o ~ ~ e t i n e s be able to a p p l y t h e d i s t r i b u t L v e p r o p e r t y

t o some, bu t not a l l , terms of an exprescion. llhlls

2 ( 2 6x + ( - 9 ) x + 11x2 + 5jr = bx + (-9) + 11)~' + 3y = bx t 2x - 5y.

We s h a l l h a > ~ e frequent occasion t o do t h i s kind of s l r . p l i f i c a t i o n .

For conveni~nce we shall ca l l . it c 3 l l e c t i n g terns o r combi r l l :1~

t e rms . We s h a l l usua l ly do t h e mlddle s tep mentally. T h u s - 1 5 ~ + ( - 3 ) ~ = Gw.

Prablem Set 7-30 - - -

I. Collect t e r n in the fol lowing phrases :

(a) 3x + 10x ( h ) & + & (b) k 9 b + ( - 4 4 ( 1 ) 5p + 4p + ap

( c ) Ilk + ( - 2 ) k (j) 7x t ( - l o x ) 4 3x ( d ) (-27b) + 30b (k ) 12a + 5c I { -2c ) + 3c2

( e ) 17n J (-16)n ( 6a 4 45 + c

( f ) x f 8x (~fnt: x = lx) (m) QP + bq 4- ( - 3 ) ~ + 79 (d ( - 1 5 2 ) + a ( 2 ) 4x + (-21x2 + (-5x) + 5x2+1

2 . W k t other properties of r ea l nuvbers bes ides the d i s t r i b u t i v e

p r o p e r t y d i d you use in Problem 1, parts (f), ( g ) , ( k ) , and

(m)?

3 . F ind the t r u t h se t of each of the fo l lowing open Eentences.

(mere p o s s i b l e , c o l l e c t terms in each phrase . )

( a ) G x + y x = 3 0 (f) (-3a) +- 3a t 5 = 5 ( b ) ( - 3 a ) + I - ~ a ) = 40 ( g ) x c 2x + Sx : 42 ( c > x c 3x = 3 + 6 x ( h ) x -t- .> - 20

( d ) 3y + 8y -t 9 = -90 (1) ~ ~ = y + 1

(e) l4x + (-14)x = 15 ( j ) 1 2 = 4y t 2y

7-11 . F u r t h e r Use of the Mul t ~ p l ~ c a t i o n Propert ies ----- We have seen how the distributive p rope r ty allows u s to

c o l l e c t terms of a p h r ~ s e . The p rope r t i e s o f m111 t . l p l i c a t i o n a r e

h e l ~ f u l a l so in c e r t a i n t e c h n i q l ~ ~ s of a lgebra r e l a t e d t o p r o d u c t s

involving ?hrases. 2 3 Example 1. "(3x y)(7ax)I1 can be more simply w r i t t e n as "2lax y . "

G i v e the reasons f o r each of the f o l l o w i n g s t eps whicli show t h i s

is true. 2 (3x y)(7ax) = 3 . x a x . y . 7 * a . x

= 3.7 .a.x.x.x.y

= ( 3 . 7 ) a ( ~ - x - x ) ~

( ~ o t i c e t h z t we write x.xSx as xs.)

While i n p r a z t i c e we do n o t wr i te d o w ~ a l l these s t e p s , we

must contilrue to b e awa?e of how t h i s simplification depends 3n

our b a s i c p r o p e r t i e s of mu1 t i p l i c a t i o n , and w e s h o u l d be p repared

t o expla in t h e in te rven ing c, teps at any t i m e .

[sea. 7-41

Problem S e t 7-4a -- Si r .p l iPy fhe following expressions and, i n Problem 11, w r i t e the

s t e p s w h i c h explain the simplification.

We can combine the method of t h e preceding exercises w i t h the

d i s t r i G u t L v e property t o perform multiplications such as the

following:

( - 3 a ) ( 2 a + 3b + (-519 = (-ha0) + (-gab) + 15ac.

Furthermore, s i n c e we have shown in Sec t ion 7-2 tha t

we may again w i t h the help of the distributive property simplify

expressions such as

- ( + - 7 ) - ( - 6 = (-1)(x2 + (-7x) + (-6)) 2

= (-x ) + 7x + 6.

Problem S e t 7-4b -- ' Write in the form ind l ea t ed in the examples above:

As you will remember from some of the work we did in Chapter 3,

sometimes the d i s t r i b u t i v e p roper ty is used seve ra l times in one

examp 1 e .

I Example 1. ( X + 3 ) ( x + 2 ) = ( x + 3)x + (X + 3 ) ~ = x 2 + 3 x + 2 x + 6

/ Example 2 . (a + ( - 7 ) ) ( a + 3 ) = (a + ( - 7 ) ) a + (a + ( - i ) ) 3

= a2 + (-7)a + 3a + (-21) ! = a* + ((-7) + 3 ) a + (-21) i

! = a2 + (-4)a + (-21)

(X t y + z ) ( b + 5) = (x + y f z ) b t (x f y + Z ) F i 1 = bx + by + bz + 5x + 5y + 5 2 .

Problem S e t 7-4c -- 1. Per form the following m u l t i p l i c a t i o n s .

( a ) (x + 6 ) (x + 2 ) ( d ) ( a + 2)(a + 2) ( 4 (7 + ( -3) ) (Y + (-50 ( e ) (x + 6) (x + ( c , [ga + ( -5 l ) ( a + (-2)) (f) (Y + 3 ) (Y +

2. Show tha t f o r real numbera a, b, c, d,

(a + b ) ( c + d) - ac + (be + ad) + bd. (~otice that ac is the product of the first terms, bd l a the product of the second t e r n , and (bc + ad) is the sum of the

remaining products . 3. Multiply the following:

(a) (a + 3Ha + 11 (b) (2x + 3)(3x +. 4) ( e ) (m + 3 ) ( m + 3)

(4 (a + c)(b + dl (f) (2 + 4 ( 7 + 2 )

4, Multiply the following:

( a ) ( 3 a + 2 ) ( a + l ) (d) (2pq + (-8)) ( 3 ~ q + 7)

(b) (x + 5)(4x + 3) (el (8 + ( - 3 ~ ) + (-y2))(2 + (-11) ( c ) t l + n ) ( 8 + 5 n ) (f) ( 5 ~ + ( -2x) ) (3~ + (-4)

7-5. Multiplicative Inverse. We found in section 6-4 that every real number has an addit ive

inverse. In other words, for every real number there I s another real number such tha t the sum of the two numbers fe 0. Since a given real number remaim unchanged when 0 ie added to it (why?),

the number 0 is called the identity element - for addition. Is there a corresponding notion of multiplicative Inverse f o r

real numbers? Flrst, we must have an identity element for rnultl-

p l ica t ion . Since a given real number remains unchanged when it is multiplied by 1 (why?), the number 1 is called the identity

element - f o r multiplication. For a given real nwnber l a there another real number auch that the product of the two numbers is l?

Consider, f o r example, the number 6. Is them a real number whose product with 6 is l? By experiment or from rour knowledge

1 of arithmetic, you will probably say that 5 is such a number,

because 6 - t = 1. Find a number whose p r o d u c t with -2 i s 1. Do 1 3

t he same f o r - 7 and f o r q. Before going any f u r t h e r , l e t us write

down a prec ise definition of multiplicative inverse.

If c and d are real numbers such that

then d is c a l l e d a multiplicative i nve r se of c .

If d is a multiplicative inverse of c, then is c a multi-

p l i c a t i v e i nve r se of d? Why? Does every real number have a

m u l t i p l i c a t i v e inverse? What is a multiplicative inverse of O?

We can observe something of the way these i nve r se s behave by

looking at them on t h e number l ine . On the diagram below, some

numbers and their inverses under multiplication are joined by

double arrows. How can you t e s t to see t ha t these p a i r s of numbers

r ea l ly are multiplicative inverses? Can you v i s u a l i z e the p a t t e r n

of t h e double arrows t f a great many more pairs of these i nve r se s were s imi lar ly marked?

How about t h e number O? With what number can 0 be paired? Is

there a number b such that 0.b = l? What can you conclude abou t

a multiplicative inverse of O?

As you l ook a t t h e double arrows in t h e above dlagram, you

may g e t t he impression t h a t the r e c i p r o c a l of a number must be i n a form obta ined by Interchanging numerator and denominator. What

about A? c ~ e r t a i n l ~ , (4 A ) . ~5 = +( J~.J;;) = i . 5 - 1, s o t n a t 1 5 JS and 7 are r e c i p r o c a l s .

The property toward which we have been working can now be

s t s t e d . It really is a n e w property o f the real nurnbel-s , s ince

It ca rmo t be d e r i v e d as a consequence of t h e p rope r t i e s which we

have s t a t e d up t o t h i s point.

Existence of m u l t l plieative inverses : For --

every real number c d i f f e r e n t frorn 0, t he re

e x i s t s a real numter d such t h a t cd = 1.

That t h e real rlumbers actually have t h i s p roper ty , if it i s

n u t already obvious t o you, will become c learer as we do more

problems. It is also obvious f rom experience t h a t each non-zero

numbzr has exactly - one m u l t i p l i c a t i v e inverse ; t h a t is, t h e multi-

p1ical;ive inverse of a number is u r ~ i q u e . We s h a l l assume unique-

ness, a l t h o u g h it could L c proved from th2 other properties, j u s t

as we did f o r the additive inverse . (see Problem Set 7 - 6 3 . )

Problem S e t 7-5 -- 1. Plnd the irlverses under multiplication of the following numbers:

2. D r a w a number l i n e and mark off w i t h double arrows the numbers 1 1 3 3 , 7, - 3 , - T , T , 7 , and t h e i r multiplicative i nve r se s .

3 . D r a w a number l i n e and mark off w i t h d c u b l e a r rows the numbers 1

3 , T , - 3 , - 7 , and t h e i r additive inverses . How does t h e J F' TJ

pattern cf double ar rows d i f f e r f rom the p a t t e r n in Problem 27

4 . If b Is s multiplicative i n v e r s e of a, what va lues for b

do we o b t a i n if a is l a rge r than l? What v a l u e s of b do

we obta in if a is between 0 and l? What is a t n u l t i p l l c a t i v e

i nve r se of l?

[ s e e . 7-51

! i 165 !

I j 5. If b is a multiplicative inverse of a, what values f o r b

t do we g e t i f a is l ess t h a n -l? If a < O and a > -I? 1

What is a multiplicative inverse of -l?

[ 6. F o r Inverses 'under n lu l t i p l i ca t i on , what values o f t h e Inverse b do you o b t a i n if a is positive? I f a is negative?

i

j 7-6 . - M u l t i p l l c a t l o n P r o p e r t y - o f Equality

i In the p rev ious chap t e r , we stated t h e addi t j -on proper ty of

I e q u a l i t y . Can we f ind a corresponding multiplication property? Consider t h e fo l lowing s t s tements :

I s i n c e ( - 2 ) ( 3 ) = - 6 , then ((-2) (3) ) ( - 4 ) = (-6) ( - 4 ) .

I since ( - 5 ) ( - 3 ) = 15, then ((-5)(-3)) (3) = (15) ( i1 '

N o t i c e that 1 ' ( - 2 ) ( 3 ) ' t and "-6" are different names f o r the

same number, and when we multiply ( - 4 ) by t h i s number we ob ta in

2 (3) ) ( - 4 and " (-6) (-4) l1 as d i f f e r e n t narlies f o r a new number.

In general, we have the

Multiplication proper4y of e q u a l i t y . For any real - -- numbers a , b, and c , if a = b, then ac = bc.

F t I Problem Set 7-6 --

1. Explain the statement, "Since ( - 5 ) ( - 3 ) = 15, then / ( ( - 5 , ~ - 3 d ~ f 1 = ( 1 5 , ~ + , 1 l 1 ~ ~ ~ ~ ~ ~ 1 ~ t ~ ~ ~ - - ~ ~ - ~ ~ ~ ~ ~ ~ ~ ~ ~ .

2 , hihich of the following statements a r e tn le?

( a ) rf 2x = 6 , then 2x($) = 6(;).

i 1 (b) If p = 9, then $(3) = 9 ( 3 ) .

1 I 1 1 5 (c) If p = 12, then p ( 4 ) = 12{~).

t 2 3 1 ( d ) rf y = 16, then $($) = 1 6 ( ~ ) . '* 3 3 3 3 ( e ) If 24 -+-I, then 24( ) = 5tr1(~). 5 3

3. Find the t r u t h set of each of the following sentences. !

Example: Determine the t r u t h set of $ = 60.

If

then

-& = 60 is t&e f o r some x,

2, = 60($) is true f o r the same x; 245 (Why did we mul t l p ly by 9)

(@$$ x = 24 is true for the same x;

If' x = 24,

then the l e f t member is ~ ( P V ) = 60, and the right member I s 60 .

Sa ' $ { 2 4 ) = 60" is n truc sentence, and the t r u t h set is (241.

(a) 12x = 6 ( d ) 15 = 5y ( 9 ) 3=1 (b) 7x = 6 ( e ) 5 = 5 Y 2

(h) ;z = 5 ( 3 ) 6 x = 6 (f) 2 = 5~ 2 3 (0 y = H

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

(a) 7a = 35 1 (4 5 = p

I (b) -X = 5 3 7 ( f ) 3 + x = - - 5

5. (a) In the formula V = i ~ h , f i n d the va lue of B if you know t h a t V is 84 and h I s 7 .

(b) If P = 5, V = 260, and v = 100, use the formula PV = pv to f ind the value of p .

7-7. Sol~tions - of E q u a t i o ~ i s

I n the p a s t , you fowld possible elements of t r u t h sets of

c e r t a l r : s en t ences , such as the eqvntion

and ther. checked ttiese possibilities . Prow we a r e prepared to

so l iTe s:;ch equations b y 2 a o r e general p r o c e d ~ z e . (To " s o l v e "

aeans to find the t r u t h s 2 t . )

F i r s t , we know t h a t any vall.le o f x f o r which

I s true is 3 l s o a \ d u e of x f o r which

(3x + 7 ) + ( + - 7 ) = (x + 15) + (( -x) -I (-7)) /

is t r u ? by t h e a d d i t l o n p r o p e r t y of equa l i ty . The n m e r a l s in

each member ~f t h i s scntence can be simplified t o ~ + . v e

(3x + (-x)) + (7 + ( -7) ) = (x + (-x)) / + (15 + (-70,

PX = 6 .

Here we added the r e a l n m b e r ((-s) + ( - 7 ) ) to each 7,ember o r t h e t i sentence and obtained the new sentence "x = 8. T h a s , each nwn-

bcr of t h e t r u t h s e t of "3x + 7 = x + 15" is a number o f the truth s e t of "2x = 8," because the addition p r o p e r t y uf equality holds

f o r a l l reel numbers.

Lext, we a p p l y the multiplication p ro3e r ty of equality to

0bt31:1

1 1 (2x1 = (8) 9

Thus, each nunber of the truth s e t a f "2x = 8'' is a nlxnber of t h e

t r u t h s e t ~f ''.I; = 4." We can now deduce that eirery s o l i l t l u ~ ~ of "3x + 7 = s + j-5"

is & solution of "x = 4." The solution of the l a t t e r equa t ion is

obv ious ly 9. But sre we s u r e that is a solution of

"3x -I- 7 = x + 15"? We could eas i ly check that it is, but let us

use t h i s example to suggest a general procedure.

The problem is this : We showed t h a t , if x is a s o l u t i o n

of

3 x + 7 = x + 1 5 ,

then x is a solution of

'What we must now show is t h a t , if x is a solution o f

then x i3 n solution o f

3 e s e two statements a r e usually written together as

x is a solution of ' I & + 7 = x i- 15" if and only -- if x i s a s o l a t i o n of " x = 4 . " -

One way t o show t h a t the second of these statements is true is to reirerse thc atcpo In the proof' of l;he first. Thus, if x = 4,

we multiply by 2 t o obtain

1 ( ~ o t i c e t h a t 2 is the reciprocal o f 1. ) Then we add (x + 7 ) t o cb ta in

( ~ o t i c e tha t (x + 7 ) I s the o p p o s i t e of ((-x) + ( - 7 ) ) . Hence,

every s o l u t i o n of "x= 4" is a solution of "3x - 7 = x + 15." mat is, the one and only s o l u t l o r ~ is 4.

We say that "x = 4" and "3x + 7 = x + 15" a1.e

equivalent; sentences in the sense that h e i r -

t r u t h s e t s are the same.

What have we learned? If t o bo th rnernbers oi' an eql;atior, we

add a real nurnber or nultiply b y a non-ze ro rea l n;int,er. t h ~ i i r w

sentence o b t a l n e d i s --- equivalent to t h e or ip , i t -~al scnt;er,.:c. This is

t r u e because these o p e r a t i o n s are Itrevers i b l e . l1 Thcr; if wt' succeed

Ln o b t a i n i n g an equiva len t scntcrlce whose so lu t i o r . is o b v i o u s , we

are s u r e that we have the required t r u t h s e t without checking. Or"

course, a check may te des i rab le t o ca tch mistskes i:i a r : ~ th :ne t ic .

As another example, solve the e q u a t i o n

T h i s equa t ion is equivalent to

( :Y + ti) + ((-zy) + ( -8) ) = (8y i (-10)) + ( ( -%J} +- ( - 6 ) ) ,

t ! ~ a t is, to

( ~ J T + (-py)) + ( 8 + (-8)) = (?J + (-3)) + ((-10) + ( - ~ ) \ J'

and to

3y = -18.

In o t h e r words, y is a S o l u t i o n of' "5y -k 8 = 2y -k (-1G)" if 2nd -- t f o n l j ~ if y 1s a s o l u t i o r ~ o f 3y = -18." he latter sentence is

equ iva len t t o

( + 1 ( 3 ~ ) = (+)(-16), t h a t is, to

y = -6.

Thus y is a solution of "3y = -18'' i f and o n l y if y i~ a solu- ---- t f o n of "y = -6." hence, all th ree s e n t e n c e s are equivalent, and

the i r t r u t h s e t is {-61. Here, we were cer ta in t ha t oach s t e p was

r e v e r s i b l e without a c t u a l l y doing it. When we s o l v e an equation

we ask ourse lves at each s t e p , "1s t h i s s t e p r eve r s ib l e? " If tt

is, we o b t a i n an e q u i v a l e n t equa t i-on.

Later , we s h a l l learn how to solve o t h e r types of sentences

by means o f app ly ing proper t ies of numbers. To do t h i s we shall

learn more a b o u t o p e r a t i o n s which y i e l d e q u i v a l e n t sentences and

o the r s which do n o t .

Problem S e t 7-7 -- 1. Tor each of the pairs o f sentences glven below, show how the

second is obtained from the f i r s t . Show ff poss ib l e how the

first can b e obta ined from the second. Which p a i r s of sen- tences a r e equivalent?

2 . Find t h e truth s e t of each of the following equations.

Solve : ( - 3 ) + qx = (-2x1 + ( - I ) . This sen tence is e q u i v a l e n t t o

((-3) + 4 4 + (ex + 3 ) = ( ( - 2 4 + (-I$ + ( 2 ~ + 31, 6x = 2 ;

and t h l s is equ iva l en t t o 1 1 6 ( 6 ~ ) = ;;-2,

1 Hence, the t r u t h s e t is

( a ) 2 a + 5 = 1 7

(b) 4~ + 3 = 3y -t 5 -+ Y + (-2) ( c ) 1 2 x + (-6) = 7x s 24

8x + (-3x) + 2 = 7x + 8 ( c o l l e c t t e r m s f i r s t . ) 62 + g t (-42) - 18 + 22

Translate t h e following I n t o open sentences and f i n d their

t r u t h s e t s , then answer t h e question in each prohlen.

(a ) The per imeter of a triangle is 44 inches. The second

s ide Is t h ree inches more than twice the length of t h e

third s i d e , and the f I r a t s i d e is f i v e inches longer than

the t h i r d s ide. Find t h e l eng ths o f the th ree s ides of

t h i s t r i a n g l e ,

( b ) If an in teger and i t s successor are added, the r e s u l t is

one more than t w i c e t h a t in teger . What I s the i n t ege r?

( c ) The sum of t w o consecutive odd integers is 11. What are

the in tegers ?

( d ) !<IF. Johnson bought 3 0 ft. of wire and l a t e r Lought 55 J o r e

f e e t of the same kind of wire. He found that he p a i d

Bi:.20 more than h i s neighbor pa-id f o r 25 ft . of the same k i n d of w:re at the sane p r i c e . h%at w3.s t h e c o s t p e r

f o o t cf the w i r e "

( e ) Four t imes a11 i n t ege r is t e n more than twice the successor

of that i n t e g e r . t h a t is the i n t e g e r ?

*(f) In an autclmotjile race, one d r i v e r , s t a r t i n g w i t h t h ~ f i r s t group of cars , d rove f o r 3 hou r s at s c e r t a l n s p e e d and

was then 120 miles frorn the F i n i s h line. Another d r i ' ~ e r ,

who s e t o u t w i t h a l a t e r hea t , had traveled a t the same

r2te as the f i r s t d r i v e r f cr 3 hours and w a s 250 m i l e s

horn t h e f i n i sh . How f a s t were these men driving?

( g ) P l a n t A grows two inches each week, and it is now 20 inches t a l l . Plant B grows three inches each week, and it is now 12 inches t a l l . How many weeks from now will they be equal ly t a l l ?

(h) A number is increased by 17 and the sum is multiplied by

3 . If the resulting product is 192, what is the number?

(1) One number is 5 times another and t h e i r sum is 15 more than 4 times the smaller. What I s the smaller number?

( 3 ) Tho quar t s of a l c o h o l a r e added t o the water in a radiator and the mixture then contains 20 percent alcohol . How many quarts of water are in the radia tor?

7-8 . Reciprocals

We s h a l l find it convenient to use the s h o r t e r name for the multiplicative inverse, and we represent the

r e c i p ~ o c a l o f a by the symbol 'I$". Thus f o r every a except 0 , 1 a,- = 1, a

You probably noticed that for p o s i t i v e integers the symbol we chose for "reciprocal" is the familiar symbol of a fraction.

Thus, the rec iproca l of 5 is This certainly agrees w i t h your 5- former experience. 1 1 1 The r ec ip roca l of however, is of -9 is -q; of 2.5 is =. 3'

f 3 1 - 2 2 3 Since 2 is the rec iprocal of 5, and 3 x = 1, it fol lows that

I 3 3 1 7 and g m u s t name the sene number; s ince - is the r ec ip roca l of

3 -9 1 1 -9 and since -9 x ( - V) = 1,-9 and must name the same number. -9

must name the same number. We Since (2 .5) (0 .4) = 1, 0.4 and

shall be in a better pos i t i on to continue t h i s discussion after we cons ider d i v i s i o n of real numbers in a l a te r chapter .

P r o b l e m -- S e t 7-8a

1. I f t h e reciprocal of any non-zero number a is represented by

t h e symbol represent t h e r e c i p r o c a l of: a '

( a ) 15 5 ~ ( 4 7

2. Write t h e common name f o r the reciprocal of each nurnber

of Problem 1. In which cases is it the same as the

r e c i p r o c a l w r i t t e n i n Problem lr:

3 . P r o v e the theorem: For each non-zero rea l number a t he re is

o n l y one multiplicative inverse o f a. ( H i n t : We know there 1 I

is a mu1tLplicative i nve r se of a, namely a. - Assume t h e r e 1s

another, say x. Then ax = 1.)

Nhy did we exclude 0 from o u r d e f i n i t i o n of reciprocals?

Suppose 0 d i d have a r e c i p r o c a l . What c o u l d it be? If t h e r e

were a number b which is the reciprocal of 0 , then 0.b - 1 . - What is the t r u t h s e t o f the sen tenee 0 - b = l? You should conclude

t h a t U s imply cannot have a reciprocal. Here we have an oppor-

tun i ty to demonstrate, us ing a rather s imple example, 3 v e r y power-

f u l type o f p r o o f . This proof depends on t he f a c t t h a t a g i v e n

sentence e i t h e r true or it is fa l se , b u t n o t b o t h . An asser- ----- --- tion t h a t a g i v e n sentence is b o t h t r u e and f a l s e is called a

c o n t r a d i c t i o n . If we reach a contradiction in a c h a i n a ? c o r r e c t

r ea son ing , then we are fo rced t o admit that t h e reasonlng : s based

on a false statement . This is the idea behind the proof of t h e

next theorem.

Theorem 7-6a. The number 0 has no rec iproca l .

Proor : The sentence " 0 has no reciprocal " is either true or f a l s e , b u t n o t both . Assuming t h a t it is f a l s e , t h a t is, assuming that 0 does have a reciprocal, we have t h e following chain of reasoning:

1. There is a real number a Assurription.

such t h a t 0-a - 1.

2 . O m a = O The product of 0

with any real

number is 0 .

3 . Therefore 0 = 1. I

Thus we are led to t h e assertion that " 0 = 1" is a t rue sentence.

But " 0 = 1" is obviously a f a l se sentence, and so we have a con-

t r a d i c t i o n . Conclusion: Step 1 of the argument cannot be true. Therefore it is fa l se t h a t 0 has a r ec ip roca l ; that is, 0 has

no reciprocal. -

A proof of the above type is c a l l e d ind i rec t o r a proof b~ cont;radiction o r a reducti.0 ad absurdum. - We s h o u l d l i k e now t o see what we can d i s cove r and what we

can prove about t h e way reciprocals behave.

In each of the following s e t s of numbers, find t h e reciprocals.

h k a t conclusion do you draw about reciprocals on examining the two

s e t s ?

Observat ion of rec iproca ls on the number line strengthens our

belief t ha t the following theorem is true.

Theorem 7-8b- The rec iprocal of a positive number is p o s i t f v a , and t h e reciprocal of a negative number is negative.

1 Proof ' : We know t h a t a-; = 1; t h a t is, the product of a non-

zero nmber and its reciprocal is the positive number 1. L e t us

first assume that a is positive. Then exactly one of three p o s s i b i l i t i e s must be true:

1 1 We see that if - is negative, then a . ~ is negatlve, a contradiction a 1 1 -of the :act that am- is p o s i t i v e . Also, If = 0 when a is

a I ;positive, then ama = 0, agaln a contradictf on. This leaves but one

i remaining possibility: a is p o s i t i v e . 1 I n the same w a y , we may show t ha t if a is negative, then

, is negative.

For each of the following numbers, find the reciprocal o f t h e I

m b e r ; then f i n d the reciprocal of t h a t reclprocal. What conclu- [$ion is suggested?

I Theorem 7-8c. The r e c i p r o c a l o f the r e c i p r o c a l o f a non-zero kreal number s l a a i t s e l f .

1

I Proof: The reciprocal of the reciprocal of a I s T, - 1 a

i - 1 [Since 1 and a arc both reciprocals of a (why?), and since

a 1 I "

ithere I s exactly one r ec ip roca l of i;, it follows that T - and a

1 "a must be names of the s m e number. Renc e ,

Problem -- Set 7-8b

1. Find the reciprocals of the following numbers:

2. For what real values of a do the following numbers have no

reciprocals? a 2 1 a + ( - I ) , a + 1, a* + (-11, a(a + I ) , T i , a + 1, -

a + l

3. Consider t h e sentence

(a t (-3)) (a + 1) = a + (-31,

which has the t r u t h s e t {O, 3 ) . (ver i fy this f a c t . ) If both

members of the sentence are multiplied by the rec iprocal of a -k (-31, that is by , + , and some properties of real

(-3) numbers are used (which p r o p e r t i e s ? ) , we obta in the sentence

For a = 3, we have 3 + 1 = 1, and t h i s is clearly a false

senterice. Why doesn't the new sentence have the same t r u t h I

s e t as the or ig ina l sentence?

4. Write a property of opposites which corresponds to Theorem 7-8b, Write a proper ty of opposites which corresponds to Theorem 7-8c.

5. Consider three pairs o f numbers: (1) a = 2, b = 3;

(2 ) a = 4, b = -5; ( 3 ) a = -4, b = -7. Does the sentence ' hold t rue in a11 three cases? a 'b = z5

1 1 6. Is the sentence - > 6 true in a l l t h r e e cases of Problem 5? a Show the numbers and t h e i r reciprocals on the number line.

1 1 7 . Is it true that if a > b and a, b are positive, then 6 > z?

Try this f o r some p a r t i c u l a r values of a and b .

1 1 8 . Is it t r u e that if a > b and a, b are negative, then > Z? S u b s t i t u t e some particular values of a and b.

3. Could you t e l l immediately whizh reciprocal 1s greater than

another if one of the numbers is positive and t h e o the r is

r l e g a t l v e ? lllus t ra te on t h e n u m b e r l-l ne .

*lo. F f n i s ~ t h e proof of Theorem 7-8b for the case when a is

! negative . I

In P r o b l e m 5 above you showed that f o r three p a r t i c u l e r pairs 1 1 1 of values of a and b , ;;;-b = . In other words, t h e product

o f the r ec ip roca l s of these t w o nwnbers is t h e recip-ocal of their

product. How many times would we need t o t e s t t h i s sentence f u r

particular numbers in order t o be s u r e it is true f o r a l l r e a l numberls except zero? Would 1,000,000 tests be enoug?? How would

we know that the sentence would n o t be fa l se f o r the 1000,001st

t e s t ?

We can o f t e n reach probable conclusions by observing hat

happens i n a nmber of p a r t i c u l a r cases. We call this i n d u c t i v e reasoning. No matter how mny cases we observe, inductive r eason-

i n g a lone cann~t assure us t h a t a s t a t e n e n t is always true. 1 1 1 Thus we cannot use i ~ d u c t i v e reasoning to prove t h a t a.5 = - ab

is always true. ~e can prove it f o r a l l non-zero r ea l numbers by - deductfve reasoning as follows. e em ember t h a t In the proof we nay

use only proper t ies which have already been s t a t ed . )

Theorem 7-8d. For any non-zero real nwnbers a and b,

1 1.1 =, a b ab'

1 1 Discussion: Since our object 1s t o prove that a . T5- is t h e

reciprocal o f a b , w e reca l l t h e d e f i n i t i o n of rec iprocal . Then we 1 1 concentrate on the product ab.(-.-) and try to show t h a t it Is 1. a b

Proof: 1 1 ab . (- -- I 1 b) = a(z).b(E) by commutative

and associative p r o p e r t i e s .I

1 s i n c e x b - = 1 X

1 1 Hence, -*- is the rec iprocal of ab. In other wonis, a b

Notice how c lose ly the proof of Theorem 7-8d parallels the

proof t h a t the sum of the opposites of two numbers is the opposite of the i r sum. Remember how t h i a result was proved: a + b) + ( - a + - 1 ) = (a + (-a)) + (b + (-b)) = 0; hence, ( - a ) + (-b) = - (a + b).

Problem S e t 7 - 8 c

1. Do the fol lowing multiplications. (1n these and in fu tu re problem s e t s we assume that the values of the variables are such that the fractions have meanin&)

2. What is the value of 87 x ( -9 ) x 0 x 5 x 642?

3. IS 8.17 = 0 a true sentence? Why?

4. If 11-50 = 0, what can you say about n?

5 . If p.0 = 0, what can you say about p?

6 . If p * q = 0, what can you say about p or q?

7. If p . q = 0, and we h o w that p > 10, what can we say about q?

The idea suggested by the above exercises will be a very use-

ful one, especial ly in finding t r u t h sets of cer ta in equations. We are able to prove the following theorem now by using the proper- ties of reciprocals .

I 179

Theorem 7-8e. For real numbers a and b, ab = 0 if and o n l y i f a = O or b = 0 .

Because of the "if and only if," we really m u s t prove two theorems: (1) If a - 0 or b = 0, then ab = 0 ; (2) If ab = 0, then a = O or b = 0 .

Proof: If a = 0 or b = 0, then ab = 0 by the multiplica-

tion property of 0. Thus, we have proved one p a r t of t h e theorem. To prove t h e aecond part of the theorem, note t h a t e i t h e r

a = 0 or a # 0, but not both. IS a = 0, the requirement that a = 0 or b = 0 is s a t i s f i e d . Why?

If ab 3 0 and a f 0, then there is a reciprocal of a and

Thus, in this case alao the requirement tha t a = 0 or b = 0 is satisfied; hence, we have pmved the second par t of the theorem.

Problem -- Set 7-8d

07 = 0, what m u s t be true about 7 or (x + (-5$ ?

Can 7 be equal to O? What about x + ( - 5 ) then?

2. Explain how we know t h a t the only value of y which will make

9 x y x 17 x 3 = 0 a true sentence is 0, 1 1 1 6.3. If a is between p and q, is between - and -? Explain.

P 9

4 . Theorem 7-8e enables us to dctcrmlne t h e t r u t h set cf an equa-

tion such as

(X c (-3)) (x + (-8)) = 0

w i t h o u t guesswork. With a = (x + (-3)) and b = (x + (-8)) , the theorem t e l l s us that t h i s sentence is equivalent to t h e

sentence x + ( - 3 ) = 0 or x + (-8) = 0 .

From t h i s sentence we read off the t ru th set a s 13, 81. Find

the t r u t h s e t of each of the following equations:

5 . Prove; If a, b, c are real numbers, and if ac = bc and c & 0, then a = b.

7 -9 . The Two Basic Operations and the I n v e r s e o f a Number Under -- -- These O p e r a t i o n s

In the f a s t two chapters we have focused our a t t e n t i o n on addi t ion and nultiplication and on the inverses under these two

operations. These four concepts zre basic to t h e rea l number sys-

t e m . Addition and multiplication have a number of properties by themselves, and one p r o p e r t y connects addition w i t h multiplication,

namely,.the distributive property . All our work in algebraic

simplification rests on these propert ies and on t h e various conse- quences of them whlch relate additlon, multiplication, opposite, and reciprocal.

We have pointed out that the distributive property comeeta

addition and multiplication. St is instructive to see whether some relationship occurs which connects every combination of addi-

tion, multiplication, oppos i t e , and reciprocal In pairs . Let us w r i t e down a l l possible combinations.

1. Addition and multipllcation: The d i s t r i b u t i v e property,

a(b + c) = ab + ac. 2. Addition and opposite: We have proved that

- (a + b) = ( - a ) + (-b). 3 . Addition and reclprocal: We f i n d t h a t there is no atmple

1 1 1 --

relatlowhip connecting a t 6 and 7. In f ac t , there a + are no real numbera at a l l f o r which these two phrases represent the same number. Thls unfortunate l ack of relationship is considerable cause of trouble in algebra f o r atudents who unthinkingly assume tha t these express- ions represent the same nmber.

4 . Multiplication and opposite: We have proved that -(ab) = ( - a ) (b) = ( a ) (-b) .

5. Mult lp lFcat ion and reciprocal: We have proved that 1 1 1 x = z.5

1 1 6. oppost te and reciprocal: = -(;).

This las t re la t ion is a new one and should be proved. !/!he proof may be obtained from (5 ) above by replacing b by -1. The proof is left to the students. (Hint:

kkat is the rec iproca l of -I?)

s ta te ( I ) , (2 ) , ( 4 ) , (51, ( 6 ) In words. Do you aee any

s i m i l a r i t y between addition and opposite , on the one hand, and multi- : plicatlon and rec lprocal , on the other , in these propertfes?

Explain.

Review Problems

1. Write a sumnary of the important ideas in this chapter, similar

t o t h a t written at the end of Chapter 6 .

2 . Change to indicated s m :

( a ) 3a (a + ( - 2 9

3 . Write each of the following as an indicated product: (a) pax -t- 2ay

(bj ac + ( - b c ) + c

(c) c ( a + S)X t (a t b)y ( d ) 1cx2 + (-15x1 + (-5) ( e ) 9x3 -I- 6x2 + (-3x)

4. Find t h e truth s e t of each of the following equatfons:

(a) 4a + 7 = 2a + 1I

( b ) 8x + (-18) = 3x + 17 f c ) 7x 4- 2 + ( - 5 4 = 3 + 2x + (-1) (a) 1-21 r 2x - ( - 3 ) 4 3x + 5 ( e ) 3x2 + I-2)x = x2 + 2 + 2x2

5. Co l l ec t terns in the following phrases:

( a ) 3a + b + a t (-?b) + 4b (b) 7x 4 b + ( - 3 ~ ) + ( - 3 b ) ( c ) 6a c ( - 7 a ) -:, 13.2 -+ ( - 5 ) a + ( -8 .6 )

( d ) 1x1 + 31-xl + (-2]-xl

*6. Given tke s e t S = [ - 2 , -1, 0 , 21

( a ) Find t h e s e t P o f a11 products of the elements o f S taken 3 a t a time. (~lnt: F i r s t f ind the s e t of all products of pa i r s u f elements. Then f ind the s e t of the products

of each element of this new s e t with each element of S*)

(b) Flnd the s e t R o: a l l sums of elements of S taken 3 a t a

t i m e .

For each of the following problems, write an open sentence, f lnd it,s truth s e t , and answer the question asked in t h e prob- lem,

(a) Jim and I plan to buy a basketbal l . J i m I s working, so

he agrees t o pay a2 more than I pay. If t h e baske tba l l

c o s t s 811, how much does Jim pay?

(b) The sum of two consecutive odd i n t ege r s is 41. What are the Integers?

(c) The length of a rectangle is 27 yards more than the width. The perineter is 395 yards. FEnd Lht! length and the

width.

( d ) Mary and J i m added thetr grades on a t e s t and f'ound the

sum to be 170. They subtracted the grades and Mary's

grade was 14 pafnts higher thzn Jim's. What were t h e i r

grades?

(el A man worked 4 dzys on a job and h i s son worked half as

long. The son's daily wage was that of his fa thcr . If 5 they earned a t o t a l of h96, what were the i r daily wages?

(f) In a rarmerl s yard were some p i g s and chickens, and no other creatures except the famncr himself. There were, in fact , sixteen more chickens than pigs. Observing this

fac t , and f u r t h e r observing t h a t there were 74 f ee t in t h e yad , not counting hls own, t he farmer exclaimed

hap?ily to himself--for he was a mthematician a3 w e l l as

a farmer, and was given to talkbig to himself--"Now I can tell how many of each kind of creature the re are in my yard." How many were there? (~int: Pigs have 4 Eeet, chickens 2 fee t . )

( g) A t the target shootlng booth at a fair, Montmorency was

paid 106 f o r each time he hit the target, and was charged 5 $ each time he missed. If he lost 25# at t h e booth and

made ten more misses than h i t s , how many h i t s dld he mke?

Chapter 8

PROPERTIES OF ORDER

8-1. The Order Relation for Real Numbers -- -- In Chapter 5 we extended the concept of order from the nwn-

bers of arithmetic t o a l l real numbers. This was done by using

the number line, and we agreed that:

"Is leas than, " f o r real numbers, means "to the left of 'I on the real number l ine .

If a and b are real numbers, then "a is less than b" is written "a < b. "

He speak of the relation "is less thw" for real numbera as an - o r d e r r e l a t i o n . It Is a binary r e l a t i o n s ince it expresses a relation between - two numbers. What are aome of the facts which we already know about the order relation? What are some of i ts general properties? 'I

Two basic properties of the order r e l a t i on f o r real numbera

were obtalned in Section 5-2.

Comparison propepty: If a is a real nmber, then exactly one of the follow- In@; ie true:

a < b , a = b , b(a.

1 Transitive property: If a, b, c are I!:

1 real numbers and if a < b and b < c , !

then a ( c.

Another property of order which was obtained in Sect ion 5-3 ,

connects the order r e l a t i o n with the operation of taking opposites:

If a and b are real numbers and if a < b, then -b < -a.

You may wonder a t this point why we are so careful t o avoid talking about "greater than". As a matter of fact , the r e l a t i o n "18 greater than", f o r which we use the symbol ">", I s also an

order re la t ion . Does this order relation have the comparison

property and the transitive property? Since it does, we actually

have two different (though very closely connected!) order relations

f o r the real numbers, and we have chosen to concentrate our atten- t ion on "leas thantt. We could have decided to concentrate on I1 greater than"; but if we are going to study an order relatlon and its properties, we must not confuse the issue by shifting

from one order r e l a t i on to anather in the middle of the discussion. Thus we state the last property mentfoned above in terms of

"<", but in applying the property we feel free to say, "1f a < b, then -a > -b ",

In the next two sections we obtain some properties of the

order r e l a t i o n "<" which involve the operations of addition and multiplication. Such properties are essential if we are to make

much use of the order re la t ion in algebra.

Problem Set 8-1 -- 1 . For each pair of numbers, determine t h e i r order.

( c ) c, 1 ( ~ ~ n s l d e r the comparisonproperty.)

2. Continuing Problem l(c), what can you say about the order of c and 1 If it is h o w n tha t c ) 4 ? What property

of order did you use here?

3. Decide in each case whether the sentence is true.

(a) - 3 + (-2) < 2 + (-2) (b) ( -3 ) + (0) < 2 + 0

( d ) ( - 3 ) + a ( 2 + a 4. Decide in each case whether the sentence is true.

( a ) ( -3 ) (5 ) < W ( 5 ) (4 (-31(-21 < (2)(-21 (b) (-3)(0) < (2)(0) (d) ( -3 ) ( a ) < (2) (a) (What h she

truth s e t of t h i s sentence?)

Decide i n each case whether the sentence is true.

A given aet may be described In many ways. Describe in three ways the truth se t of

(a) 3 < 3 + x

(b) 3 + x < 3

Determine the truth s e t of

(a) 3 = 2 + x

(b) 3 = (-2) + x

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

( d ) r = f i+ c

Determine the truth s e t of

( a ) Y < 3 (4 -lul < 3 I b ) lul < 3 ( d l -Y < 3

: 8-2. Addition Property of Order -- !

What is the connection between order of numbers and addition ' of numbers? We shall f i n d a basic property, and from it prove

other properties which relate order and addition. As before,

we concentrate on the order r e l a t i o n "<"; similar properties can

be stated f o r the order relation ")",

It is helpful to vlew addition and order on the nmber l i n e . We remember that adding a poei t ive number means moving to the right; adding a negative number means moving to the left. Let us fix two points a and b on the number line, with a ( b. If we add the same number c to a and t o

b , we move to the r igh t of a and of b If c is

p o s i t i v e , to the left if c is negative. We could think

of two men walking on the number l i n e carrying a ladder between them. A t the start the man at a ia to t h e left of

t h e man at b. If they walk c units in ei ther direction,

the fixed length of the ladder will insure t h a t the man t o the left w i l l stay to t h e left. In the i r new positions the man a t a + c will s t i l l be to the l g f t of the man at b + c, Thus,

Here we have found a fundamental property of order which we sha l l assume for a l l r e a l numbers.

Addition Property of Order. If a, b, c -- are real numbers and if a < b, then

a + c < b + c .

Illustrate this p rope r ty f ~ r a = -3 and b = - p with 1 c having, successively, the values -3 , 0, -7. Here

1 is " ( -3 ) + ( -3) < (- + ( - 3 ) " a true sentence? -3 < - 2. Continue with the other values of c . Phrase t h e addition

property of order in words. Is there a corresponding property

of equality?

Problem -- S e t 8-2a

I. By applying the add i t i on propertles of order, determine which

of the following sentences are t r u e . 6

4 8 (4 (-5.3) + (-2)(- < ( - 0 . 4 ) +

( d l (;I(- $1 -+ 2 2 ( - + 2

2. Formulate an addition property of order f o r each o f the

relations "<I1, ">", "2".

I 3. An extension of the order property states that: If' a, b, c, d are real numbers such that

I a < b and c < d, then a + c ( b + d. This can be proved In three s t eps . Give the reason f o r

If a < b, then a + c < b + c ; / each st9i If c < d, then b + c < b + d; i I

hence,

I, I

a + c ( b + d .

1 4. Find the truth s e t of each of the following sentences.

1 3 ! Example: If (- $) + x < (-5) + g is true f o r some x,

then 3 3 x ( ( - 5 ) + H + 2 is true r o r the same x.

x < -2 is true for the same x.

! Thus, if x is a number which makes the or iginal sentence i true, then x < 2 If "x < -2" is true for some x, then t

3 3 ( - g) + x < (- F} + (-2) is true for the same x,

3 3 ( - F) + x < ( -5) + p is true for the same x.

Hence, the truth set is the set of all real numbers less ; '

! than -2.

(a) 3 + x < ( -4 ) ( f l {-x) + 4 < ( -3) + 1-31

I I 5. Graph the truth eeta of parts (a), ( c ) , and (h) of Problem 4.

6. In Chapter 5 the following property of order was stated: "If a < b, then -b < - a * " Prove this property, using the addltion property of order. (Hint: Add ((-a) + ( - b )) to

both members of the inequality a < 31; then use property of addi t ive inverses. )

"7 . Show that the property: I t If 0 < y, then x < x + y. t t

is a special case of t he addi t ion property of order .

(Hint: In t h e statement of the addftion property of order, l e t a - o, b = y, c = x.)

Many results about order can be proved as consequences of the addition property of order. Two of these are of special Interest t o us, because they give direct translations back and forth between statements about order and statements about equality.

The f irst of these results w i l l be a specLal case of t h e

property. Let us consider a few numerical examples of the pro-

perty with a = 0. If a = 0 , then ' 'a < b f t becomen "0 < b"; tha t is, b Is a p o s i t i v e number. Thus, we may write: If 0 < b, then c + 0 < c + b.

wt a = ~ , b = 3 and c = 4 ; t h e n Y - t O < Q t - 3 ;

that is, since 7 = 4 - t 3 , then 4 < 7 .

L e t a = 0, b = 5 md c = -4; then (-4) + 0 < ( - 4 ) + 5 ; 1 that is, s ince 1 = (-4) + 5, then - 4 < 1. I

These two examples can be thought at as saying: Since 7 = !I + 3 and 3 is a positive number, then

4 < 7 4

Since 1 = (-4) +, 5 and 5 is a positive number, then

- 4 < 1. This result we state as

Theorem 8-2a. If z = x + y and y is

a p o s i t l v e number, then x < z .

191

Proof. We may change the add i t i on property of order to read:

If a < b, then c + a ( c + b. (my?)

Since the property is true for - a l l real numbers a , b, c, we

may let a = 0, b = y , c = X. Thus,

if 0 ( y, then x + 0 ( x + y,

z = x + y, then "x + 0 < x + y'' means "x < z", Hence we

have proved t h a t if z = x -+ y and 0 ( y (y is p o s i t l v e )

then x < z. meorem 8-2a now gives us a translation from a statement

about equality, such as

-4 = ( - 6 ) + 2,

t o a statement about order , in t h i s case

-6 < - 4 ,

Notice that adding 2, a p o s i t i v e number, to ( - 6 ) ylelds a number

t o the r i gh t of -6. 1 Changethesentence i

4 = ( -2) + 6

t o a sentence Involving order.

The second result of the addLtion property is a theorem ; whfch translates from order to equality, instead of from equality

t o order, as Theorem 8-2a does. You have seen that if y is

p o s i t i v e and x is any number, then x is always less than

x + y. If x < z , then does there exist a p o s i t i v e nmber y

such that = x + y? Consider, for example, the numbers 5 and

7 and note tha t 5 < 7 . What i a the number y such that

How did you determine y? Did you find y to be p o s f t i v e ?

Consider the nmbers -3 and -6, noting that -6 < -3. What is

the truth se t of

Is y again p o a f t i v e ?

What kind of number make8 each o f the above equations tme? In each caae you added a p o s i t i v e number to the smaller number

t o get the greater. By this time you see that the theorem we have in mind is

Theorem 8-2b. ff x and z m e two real numbera such that x < z, then there is a positive real. number y such that

z s x + y .

*Proof. - There are really two things to be proved. First, we m a t find a value of y such that z = x + y; aecond, we must prove that the y we found is pos i t ive , if x < z .

It l a no t hard to find a value of y such that z = x + y. Your experience with solvlng equations probably suggests adding

(-x) to both members of "2 = x + yrt to obtain y = z + (-x). L e t us try this value of y. L e t

Then

P ~ u s we have found a y, namely z + (-x), such that: z = x + y. Tt remaina to show that if x < z, then t h i s y l a posit ive.

We know there is exactly one true sentence among these: y is

negative, y is zero, y is p o s i t i v e , (by?) If we can show that two of these poss ib l l l t l e s are false, the t h l r d must be true, Try t h e f i rst posslhllity: If I t were true that y is negative

and z = x + y, then the addition property of order would assert that z < x, (Let a = y, b = 0, c = x.) But t h i s contradicts

fact that x < z ; cannot be true that y is negative. Try the second poss ibi l i ty: If it were true that y is zero

and z = x + y, then it would be true that z = x. This again

contradicts the fac t that z < x; so it cannot be true that y

is zero. Hence, we are left with only one poss ibi l i ty , y is

positive, which must be true, This completes the proof.

Theorem 8-2b allows us to translate from a sentence involv- ing order to one involving equality. Tfius,

-5 < -2 can be replaced by

-2 = ( - 5 ) -t 3,

which gives the same "order information" about -5 and - 2 . That is, there is a p o s i t l v e number, 3 , which when added to the lesser, -5, yields the greater , -2.

I Problem Set 8-2b -- 1. For each pair of numbers, determine t he i r order and f i n d

the p o s i t i v e number b which when added to the smaller

gives the larger.

(a) -15 and -24 ( e ) -254 and -345

$ and & 1 - T and 3

33 - and -

1.47 and -0.21

Show that the following is a true statement: If a and c

are real numbers and if c < a , then there is a negative real number b such that c = a + b. (Hint: Fol low the similar dlacussion for b pos i t i ve . ) Whfch of the following sentences are true for a11 real values of the variables?

(a) If a + 1 = b, then a < b. (b) If a + (-1) = b, then a < b. ( c ) If (a + c ) + 2 = (b + c ) , then a + c < b + c .

(d) If (a + c ) + ( -2) = (b + c ) , then b + c < a + c .

( e ) If a < -2, then there is a pos i t ive number d such

that -2 = a + d .

(f) If -2 < a , then there is a p o s i t i v e number d such

that a = (-2) + d.

4. (a) Use 5 + 8 = 13 to suggest two true sentences involving 11 11 < re la t ing pairs of the numbers 5 , 8, 13.

(b) Slnce ( -3 ) + 2 = (-11, how many t m e sentences involvfng "<" can you write using pairs of these three numbers?

( c ) If 5 < 7, write two true sentences Involving "=" relating the numbers 5 , 7,

5 . Show on the nmber l ine that if a and c are real nmbers and If b is a negative number such that c = a + b, then

c < a.

6 . Which of the following sentences are true for all values of the variables? (a) If b < 0, then 3 + b < b. (b) If b < 0, then 3 + b < 3. ( c ) If x ( 2, then 2x ( 4.

7. Verify that each of the following is true.

(a> 13 4- 41 5 131 + 141

(b) I ( - ~ I + 4 1 s 1-31 + 141

( c ) 1 ~ 3 1 + ( -411 1-31 + 1-41

( d ) Sta te a general property relating 1 a + b 1 , 1 a 1 and Ib 1 for any real numbers a and b.

8. What general property can be stated for multiplication

slmilar to the property for addition in Problem 7?

9. Translate the following into open sentences and f ind the i r truth sets .

(a) The sum of a number and 5 is lesa than twlce the number. What is the number?

[set. 8-21

( b ) When Joe and Moe were planning to buy a sailboat, they asked a salesman about the c o s t of a new type of a

boat t h a t was being designed. The salesman repl ied,

lllt wontt cost more than $380." If Joe and Moe had agreed that Joe was to cont r ibu te $130 more than Moe when the boat was purchased, how much would Moe have

t o pay?

( c ) Three more than six times a number Is greater than

seven increased by f i v e times t h e number, What I s the number?

( d ) A teacher says, "1f I had twice as many students in

my class as I do have, I would have at l eas t 26 more than I now have." How many students does he have in h i s class?

+(el A student has test grades of 82 and 91. What must he

score on a third t e s t to have an average of 90 or higher?

*(f) 3111 is 5 years older than Norman, and the sum of t h e i r

ages is less than 23. How o l d is Norman?

I ; 8-3. Multiplication Property -- of Order

In the preceding section we s ta ted a basic property giving the order of a -+ c and b + c when a < b. L e t us now ask about the order of the products ac and bc when a < b.

Consider the true sentence 5 < 8.

If each of these numbers is multiplied by 2 , the products are involved in the true sentence

What is your conclusion about a multiplication property of order?

Before making a decision, let; us try more examples. Just as

above, where we took the two numbers 5 and 8 In the true sentence " < 8" and inserted them in ' I ( ) (2 ) < ( j ( 2 ) " to make a true sentence, do the same in the following.

2, -9 -< 6 and ( ) (5) < ( ) ( 5 )

3. 2 < 3 and ( I ( -4) < ( )(-4)

5. - < 8 and ( )(-3) < ( ) [ - 3 )

We are concerned here with the order r e l a t i o n "<", observing the p a t t e r n when each o f the numbers in the otatement "a < b" is m u l t l p l l e d b y the same number. Dld you notice that it makes a difference whether we m u l t i p l y by a p o s i t i v e number or a

negative number ?

The above experience swgeuts that l r a < b , then ac < bc, provided c is a p o s i t i v e number; - - bc < ac, provided c - - is a negative number.

Thus we have found another important set a f propert i e s of

order. How can you u s e these properties to t e l l quickly whether the

fol lowing sentences are true? 1 2 5 10

Since g < 7 then < - 7 '

Since - 14 5 l4 then - g < - V J 51 < 78. 5 Slncc g < 4, then - & < 5 - iF.

These properties of order turn out t o be consequences of

t h e other properties of order, and we state them together as

Theorem 8-3a. Mult lp l ica t lon Property of

Order. Tf a , b , and c are real numbers and if a ( b, then

ac < bc, lf c is p o s i t i v e , bc < ac, If c is negat ive.

P roo f . There are two cases. Let us consider t h e case of

positive c. Here we must prove that if a < b, then ac < bc.

You fill in the reason fo r each s tep of the proof. 1. There I s a positive number d such that b = a + d .

2. Therefore, bc = (a + d ) c .

3. bc = ac + dc 4. The number dc is p o s i t i v e . 5. Hence, ac < bc.

The proof of the case f a r negative c is left to the student I n the problems.

We could equally well have discussed the multiplication property of the order r e l a t i o n "is greater than'' instead of "is less than".

When we are comparing numbers, the two statements "a < b" and "b > a" say the same thing about a and b. Thus, when

we are concerned primarily with numbers rather than a particular order r e l a t i o n , it may be convenient t o shift from one order

/ relation to another and mlte such sentences as:

I Since 3 ( 5, then 3 ( - 2 ) > 5(-2). I !

Since -2 > -5, then (-2)(8) > ( - 5 ) ( 8 ) . I Since 3 > 2, then (3)(-7) < (2 ) ( -7 ) .

Verify that these sentences are true. When we are focusing on the nwnbers involved instead of on

an order relation, we can say that

ac < bc if c is posi t ive , if a < b, then

ac > bc if c is negative. State these propertles of orders in your o m words.

In our study we shall also need some results such as

Theorem 8-3b. then

Proof. If x 0, then e i the r x is negative or x is

positive, but not both. If x is positive, then

x > 0,

If x is negative, t h e n

In e i t h e r case, the r e s u l t is t he desired one.

Tneorem3-3bstates tha t khe square of a non-zer~ number

is positive. What can be said about x2 f o r any x? Tqe p r o p e r t i e s of order can be used t o advantage in f i n d i n g

t r u t h s e t s of inequalities. For example, let us find t h e truth

s e t of

By the a d d i t i o n property of order we may add ( - 2 ) + (-5x) t o both members of t h i s i n e q u a l i t y t o o b t a i n

which when simplified is . 1

-8x < -8.

Since ( ( - 2 ) + (-5-r) is a real number f o r every value of x , ) the new sentence has the same t r u t h s e t as the o r i g i n a l .

(mat must w e add to the members of "-8x ( -8" t o ob ta in the

o r i g i n a l sentence, t h a t is, t o reverse the s tep?) Then, by t he multiplication p r o p e r t y of order.

1 ( - 8 - 6) < (-8x) ( - 8)

Here we mult ip l ied by a non-zero r ea l number. Tfius, t h i s sen-

t ence is equivalent t o the former sentence. (What must we multiply t h e members of "1 ( xu by t o obtain the former sentence?)

Obviously, t h e truth s e t of "1 < x" is t h e s e t of all numbers greater t h a n 1, and this is the truth s e t of the o r i g i n a l in-

equality .

Problem Set 8-3 -- [ 1. Salve each of the following inequalities, using the form

of the following example. ( R e c a l l that to "solve" a sen- tence is to find its truth set. ) &ample: (.-3x1 + 4 ( -5.

This sentence is equivalent to

3 < ( - 5 + - 4 , (add ( -4) to both members)

which is equivalent t o

1 ( - $(-91 < ( - 31(-3~19 (multiply both members by ( - +I)

3 < x. Thus, the tmth s e t consists

( a ) (-2x1 + 3 < -5

(b) (-2) + (-4x1 > -6

(4 (-4) + 7 < I-2x1 + (-5)

(d) 5 + (-2x1 < 4x + (-3)

(4 ($1 + ( -81 < (-;I + (

of all numbers greater than 3 .

(g) zx < 3 + 1-21 I - $ 1 (h) (-2) + 5 -t (-3x1 < 4~ + 7 t (-2x1

(1) -(2 + x) < 3 + ( - 7 )

2. Graph the truth sets of parts ( a ) and (b) of Problem I.

3. Wanslate the following into open sentences and solve.

(a ) Sue has 16 more books than Sally. Together they have

more than 28 books. How many books does Sally have?

(b) If a cer ta in variety of bulbs are planted, less than

of them will grow into plants. If, however, the B 3 bulbs are given proper care more than 8 of them w i l l

grow. If a careful gardener has 15 plants, how many

bulbs d i d he p l a n t ?

Isec. 8-31

4, Prove that if a < b and c is a negative number, then bc ( ac. Hint: There Is a negative number e such that a = b + e, Therefore, ac = bc t ec. Mat kind of number is ec? Hence, what I s the order of ac and bc?

45. If c is a negative number, then c ( 0. I3y t&king ogposftes,

O < ( - c ) , Since (-c) is a positive number, we may prove the theorem of Problem 4 by noting that if a < b, then a(-c) ( b ( - c ) ; i .e., -(ac) < -(bc). Why does the canclu- s l o n then f o l l o w ?

6 . If a < b, and a and b are both p o s i t i v e real numbers, 1 1 prove t h a t < . Hint: Nultiply the inequality a < b

3. i by (h.6). Ikmonstrate t h e theorem on the number l i n e .

7 . Does the re la t ion i~.( 5 hold if a < b and both a and

b are negative? Prove it or disprove it,

' hold if a < b and a < 0 and 8. Does the r e l a t i o n < b > O? Prove o r disprove,

9 . State the add i t ion and multiplication properties of the o rde r ">".

2 10. prove: Tf O < a ( b, then a < b2. H i n t : Use properties

2 2 of order t o obtain a ( ab and ab < b .

8-4. The Fundamental Proper t ies of Real Numbers -- --- - - -- I n th1s and thc preceding three chapters, we have been deal-

ing w i t h two main problems. The first problem was to extend t h e o rde r r e la t ion and the operations of addltlon and multipli-

cation fros the numbers of arithmetic to a l l real numbers. Until

t h i s was done we really d i d not have the real number system t o

work w i t h . The second problem was to discover and a t a t e care-

f u l l y the fundamental proper t ies o r the real number system.

!The t w o problems, as we have been forced to deal with them, are c lose ly intertwined. In t h l s sectlon we shall separate out the moat important problem, the second one, by summarizing the funda- mental properties which have been obtained.

Before continuing, we should admit that the decision as to

i what is a fundamental property I s n o t made because o f s t r i c t - i mathematical reasons but is to a large extent a matter of con-

i venience and common agreement. We tend t o think of the real

i number system and i t s many p roper t i e s a s a "structure" b u i l t i upon a foundation consisting of fundamental proper t ies . This : is the way you should begin to th ink of the real number system.

A good question, which can now be answered more precisely than

: before, is: What is the real number system?

The real number system is a s e t ,of elements f o r which

binary operations of addition, "tl', and multiplication, I t . " ,

along with an orde r relat ion, "<", are given with the following Properties.

1. For any r ea l numbers a and b,

a + b is a real number. ( Closure ) 2 . For any real numbers a and b,

a + b = b + a . (~ornmutat ivi ty) 3. For any real numbers a, b, and c,

( a + b) + c = a + (b + c ) , (~ssociativf ty ) 4. There is a apecial real number 0

such that, for any real number a,

a + O = a . ( ~ d e n t i t y element) 5. For every real number a there is

a real number -a such t h a t

a + ( - a ) = 0. (~nverses)

6. For any real numbers a and b,

a . b is a real number. (Closure)

7 . For any real numbers a and b , a-b = boa. (Commutativity)

8 . For any real numbers a, b, and c ,

(a-b). c = a. ( b . c ) . (~ssociativity)

9 . There is a special real number 1

such t h a t , f o r any real number a , a.1 = a. ( ~ d e n t i t y element)

10. For any real number a different 1 from 0, there is a real nmber g

such that I a - (-1 = 1. a (Inverses)

11. For any real numbers a, b, and c,

ad(b + c ) = a ~ b + a - c . (Distributivity )

12. fir any real numbers a and b,

exactly one of t he following is true: a ( b, a = b, b < a.

13. For any real numbers a, b, and if a ( b and b < c , then a <

14. For any real numbers a, b, and if a ( b , then a + c < b -k c .

1 For any real numbers a, b, and

if a < b and 0 < c, then

a - c < b m c ,

if a < b and c < 0, then

b . c < a h c .

( comparison)

C ,

c. (~ransitivity)

C 9

(Addition property)

c ,

(Multiplication property

You have probably noticed that there are several familiar and useful proper t i e s whlch we have failed to mention. This

is not an oversight. The reasons for omi t t ing them is that they can be proved from the properties listed here. In fact, by adding just one - new property, we could obtaln a list of proper t lea from which everything about the real numbers could be proved. We shall not consider this additional property since that would take us beyond the limits of t h i s course . You w i l l see i t i n a later course.

Pract ica l ly all of the algebra in this course can be based on the above l i s t of p rope r t i e s . It is by means of proofs that we bridge the gap between these basic properties and a l l of the many ideas and theorems which grow out of them. The chains of reasoning Involved i n proofs are what hold together the whole structure of mathematics -- or of any logical system.

Thus, If we are going to appreciate fully what mathematics

is l i k e , we should begin t o examine how ideas are linked in these chains of reasoning -- we should do some proving and not always be satisfied with a plausible explanation. It is true that some of the statements we have proved seem very obvious, and you might wonder, quite Justifiably, why we should bother

to prove them. As we progress further in mathematics, there will be more ideas which are not at all obvious and which are established only through proofs. During the more elementary stages of our t r a in ing we need the experience of seeing some simple proofs and developing gradually some feeling for the chain of reasoning on which the whole structure of mathematics depends. This is our reason for looking closely at proofs of some rather obv2ous statements.

The abi l i ty to dlscover a method f o r proving a theorem is something which does not develop overnight. It comes with seeing a variety of different proofs, by learning to look f o r connecting links between something you know and something you

want to prove, by thinking about the suggestions which are given to lead you i n t o a proof. On the other hand, the kind of thfnk-

ing required is not used only in mathematics but i s involved In a l l logical reasoning.

Let us now return to the fundamental properties of real num- bers and summarize a few of the other proper t ies which can be proved from those given above, Some of these were proved in

the text and some were included i n exerc i s e s .

16. Any real number x has just one a d d i t i v e Inverse, namely -x,

17. For any real numbera a and b,

- ( a + b) = (-a) + (-b). 18. For real numbers a , b, and c, If a + c = b + c,

then a E b.

19. For any real number a, a. 0 = 0.

20. For any real number a, (-1)a = -a.

For any real numbers a and b, (-a)b = -(ab)

and (-a)(-b) = ab. The opposite of the opposi te of a real number

a is a. Any real number x d i f f e r e n t from 0 has

1 jus t one multiplicative inverse, namely - x ' The number 0 has no reciprocal . The rec iproca l of a p o s i t i v e number i s p o s i t i v e , and the r e c i p r o c a l of a negative number is negat ive . The r ec ip roca l o f the r e c i p r o c a l of a non-zero

real number a is a .

For any non-zero real numbers a and b ,

For real numbers a and b, ab = 0 if and only

if a = O o r b = 0 .

For real numbers a, b, and c with c # 0, if

ac = be, then a = b.

For any real numbers a and b, if a < b, then

-b < -a.

If a and b are real numbers such that a < b, then there I s a positive number c such that

b = a + c .

If x # 0, then x2 > 0. 1 1 If 0 ( a < b, then 6 < a.

2 P If 0 ( a ( b, then a < b .

You may have noticed that we gave a proof of t h e multiplication

proper ty of order in Sect ion 8-3. In fact, this property (No. 15

i n the list) fol lows from the other 14 fundamental properties. Therefore it could have been omitted from the l i s t without limit-

ing i n any way its scope. Eowever, we have included the proper ty

i n o rde r t o emphasize the parallel between t he p rope r t i e s of add-

i t i o n and the p r o p e r t i e s of multiplication.

[ s e c . 8-41

You may have noticed a l s o that nowhere in the above discussion , of fundamental properties is there any mention of absolute values.

This important concept can be brought into t h e framework of the

basic prope r t i e s by the d e f i n i t i o n :

If 0 < a , then la1 = a.

If a ( 0, then la1 = -a .

We close t h i s summary with a mention of some properties of

a rather d i f f e r e n t kind, namely the properties - of equality.

These are properties of t he language of algebra rather than

properties of real numbers. Recall that the sentence "a = b",

where "a" and "b" are numerals, asserts that "a" and "b" name

the same number, The first two properties of equality which

we l i s t have not been stated before but have actually been used

many times. In the following, a, b, and c are any real numbers.

35, If a = b, then b = a. (symmetry)

36. If a = b and b = c , then a = c. (Transitivity)

37. If a = b, then a + c = b + c. ( ~ d d f t i o n p r o p e r t y )

38. If a = b, then ac = bc.

3 9 . If a = b y then -a = -b.

40. ~f a = b, then 131 = lbl.

(Multiplication property 1

Review Problems 1. For each pair of numbers, determine their order .

{a) -100, -99 6 5 (d l 7 (b ) 0 .2 , -0 .1 ( e ) 3.4+ (-4),

( c > 1-31, 1-71 2 (f) x + 1, 0 2. If p ) 0 and n < 0, determine which sentences are true

and which are false. (a) If 5 > 3, then 5n < 3n. (b) If a > 0, then ap < 0. ( c ) If 3x > x, then 3px > px.

(d) If ( i ) x > 1, then x > n. 1 1

( e ) If p > n, then - < E. P

1 1 1 ( f ) If ~f > > 0, then p < x and x > 0.

3. Which of the following pairs of sentences are equivalent? (a) 3a > 2, ( - 3 ) a > (-2)

(b) 3x > 2 + x, 2x ) 2

( c ) 37 + 5 = y + (-I), 2y = ( - 6 ) ( d ) -X < 3 , x > (-3) ( e ) -P + 5 < P + (-11, 6 > 2p

4. If p > 0 and n < 0, determine which represent positive numbers, which represent negative numbers.

(a) -n ( d l Pn

5. Solve each of the following inequalities.

(a) -x > 5 ( d ) (-4) + (-x) > 3x + 8 ( b ) (-1) + 2~ < 3~ ( e ) b + b + 5 + 2b =+12<381

6. Find the truth s e t of each of the following sentences.

1 1 ( a ) 2 < and x ) 0

1 2 ( d ) (x) > 0 and x # 0

1 1 ( b ) = - 2 ( e ) o 2x < 180 1 1 ( c ) ;; < and x < 0

i 7 . If the domain of the variable is the set of integers f i n d

the t r u t h sets of t he following sentences.

(a) 3x +- 2x = 10 ( d l ~ ( x + ( - 3 3 = 5

( b ) x + (-1) = 3x t 1 (e) 3x + 5 < 2x -I- 3

( c > 2X f 1 = -3x -t ( -9) 1 ( f ) ($1 + (-XI > ( - + (-2x1

8. Solve the fol lowing equations.

( a ) 3x = 5 (4 7~ + 3 = y + (-3) ( b ) 3 + x = 5 ( e l 3x = 7x + ( - 2 ) x

( c ) 2n + n + (-2) = o ( f ) 3q + (-q) + 5 + q = ( - 2 )

9. Solve the fo l lowing equations.

( a ) 3(x + 5 ) = (x + 3 ) + x (b) x(x + 3 ) = (X + (-4) ( X t 3 )

1 1 1 1 (4 2Y T"+ ( - = ( - + ( -

( d ) a2 = .(a 4. 1)

( e ) ( X t 2) (x -t 3 ) = x(x t 5) -+ 6

10. The length of a rectangle is known to be greater than or equal t o 6 units and less than 7 units. The width is known t o be 4 units. Find the area of the rectangle.

11. The l e n g t h of a rectangle is known t o be greater than or

equal to 6 units and less than 7 units. The width is kcnown to be greater than or equal to, 4 units and less than 5 unlts. Find the area of the rectangle.

"12. The l eng th of a rectangle is known to be greater than or

equal to 6.15 inches and less than 6.25 inches, The width

I s known to be greater than or equal to 4.15 inches and less than 9.25 inches. Find the area of the rectangle.

13. ( a ) A c e r t a i n variety of corn plant y i e l d s 240 seeds per plant . N o t a l l the seeds w i l l g r o w i n t o new plants

3 when planted. Between and of the seeds will pro- duce new plants. Each new p l a n t w i l l also yield 240

seeds. From a single corn p l a n t whose seeds are har-

vested i n 1960 how many seeds can be expected i n 1961?

(b) Suppose instead that a corn plant d i d not y i e l d exactly 240 seeds, but between 230 and 250 seeds. Under this

condition how many seeds can be expected in 1961 from

the 240 seeds planted at the beginning of the season?

14. Write open sentences and f ind the so lu t ion t o each of the questions which follow..

(a ) A square and an equi la tera l triangle have equal per i - meters. A side of the trfangle l s 3.5 inches longer than a side of the square. What is the length of the

s ide of the square?

(b) A boat traveling downstream goes 10 miles per hour faster than the ra te of the current. Its velocity

downstream is not more than 25 miles per hour. What is t h e rate of the current?

( c ) Mary has typing to do which will take her at least 3 hours. If she starts at 1 P.M. and must finish by

6 P,M., how much time can she expect to spend on the

job ?

( d ) Jim rece ives $1.50 per hour for work which he does in

his spare time, and is saving his money t o buy a car. 1f the car will cost him at least $75, how many hours

must he work?

Chapter g

SUBTRACTION AND D I V I S I O I J FOR REAL NUMBERS

9-1. D e f i n f t i o n o_f Subtraction

Suppose you make a purchase which amounts t o 83 c e n t s , and

give the cashier one d o l l a r . What does she do? She p u t s down two

c e n t s and says "85" , one n i c k e l and says "go", and one dime and

says "one dollar". What has she been doing? She has been sub-

t ract ine 83 from 100. How does she do it? - S y f i n d i n g what she

has t o add to 8 3 to obtain 100. The ques tLon "100 - 83 = w h a t ? "

means t h e same as "83 + what = 100?", And how have we solved the

equation 83 t x = 100

so fa r In t h i s course? We add the opposi te of 83, and f i n d

x = 100 + ( - 8 3 ) .

Thus "100 - 83" and "100 + (-83)" are names for the sane number.

T r y a few more examples:

From these examples you will agree t h a t subtracting a p o s i t i v e

number b from a larger p o s i t i v e number a, g ives the saxe r e s u l t

as adding the opposite of b to a. Since subtractLon f o r p o s i t i v e nunbers 1 s already familiar

t o you, you probably wonder what we have a c c o m ~ l i s h e d . Our prob-

lem I s to d e c i d e how Lo d e f i n e subtraction f o r - all rea l num5ers.

We have now d e s c r i b e d subtraction in the f a l l l i a r case of t h e pos-

itive numbers in terms of o p e r a t i o ~ l s we h o w how t o do f o r a l l

real numbers , namely adding and t ak ing opposi tes , And so w e define

subtraction for a l l real numbers as adding the opposite. I n this

way, we extend sll 'otraction t o real numbers so t h a t it still h a s the

prope r t i e s we know from arithmetic; and our definition has used

o n l y ideas with which we have previously become familiar.

To subt rac t the real number b from t h e real number a,

add the o p p o s i t e of b to a . Thus, f o r real numbers a

and b,

a - b = a + ( -b ) .

Examples : 2 - 5 = 2 + (-5) = -3

( - 5 ) - 2 = ?

5 - (-2) = ?

(-2) - (-5) = ?

(-5) - ( -2 ) = ?

Read the express ion "5 - ( -2) ", Is the symbol "-I1 used In two different ways? What is the meaning of the f i r s t " - I 1 ? What i s t h e meaning of t h e second 'I-"?

To he lp keep these uses of the symbol clear, we make t he following para l le l statements about them.

1 1 - l t stands between t w o numerals 11 - l t is part of one numeral and ind ica tes , t h e operation of and ind ica tes the opposi te

s u b t r a c t i o n , We read t h e - of. We read the above as above as "a minus b". "a plus t he opposi te of

b" . We see t h a t t h e operation of s u b t r a c t i o n is c lose ly related

to t h a t o f a d d i t i o n . We may s t a t e this as

Theorem 9-1. F o r any real numbers a, b, c,

a = b + c if and only if a - b = c . Proof: Remember that in order t o prove a theorem invo lv ing 'tif and o n l y i f " w e really must prove two theorems.

Let us first prove: if a = b + c, then a - b = c .

Next we prove: if a - b = c, then a = b + c. To do t h i s ,

note that "a - b = c" means "a + (-b) = c". The student

may now complete the proof.

Problem Set 9-1

(-5000) - (-2000)

Subtract -8 from 15.

From -25, subtract -4 -

What number is 6 less than -9?

-12 is how much greater than -17?

How much greater is 8 than -5?

Let R be the set of a l l real numbers, and S the s e t of

a l l numbers obtalned by performing t h e opera t ion of sub- traction on pairs of numbers of R. Is S a subset of R?

Are the rea l numbers closed under sub t r ac t ton? Are the num-

bers of arithmetic closed under sub t r ac t i on?

15. Show why "a - a - 0" is true for a l l real numbers.

16. Find the t r u t h set of each of t h e following equations:

(a ) y - 7 2 5 = 2 5 ( a ) 3y - 2 = -14

(b) z - 34 = 76 (e ) x + 23.6 = 7.2

( c ) 2~ + 8 = -16 3 1 ( f ) z + ( - T ) = - ?

17. From a temperat~re of 3' below zero, the temperature dropped 10'. What was the new temperature? Show how t h i s question

is related t o subtraction of real numbers.

18. Mrs. J. had a credit of $7.23 in her account at a department

store. She bought a dress f o r $15.50 and charged it, What was t h e balance in her account?

19. Billy owed his brother 80 cents. He repaid 50 cen t s of the debt. How can t h i s t ransac t fon be written as a s u b t r a c t i o n of real numbers? ( ~ e p r e s e n t the debt of 80 cents by (-80.))

20. The bottom of Death Valley is 282 f e e t below sea Level, The

top of Mt. Whitney, which is v i s i b l e from Death Valley, has

an altitude of 14,495 feet above sea level. How high above

Death Valley is Mt. Whitney?

9-2. Properties of Subtraction - What are some of the p r o p e r t i e s of subt rac t ion? Is

5 - 2 = 2 - 5 ?

What do you conclude about the commutativity of subtraction? Next, is

8 - (7 - 2) = ( 8 - 7 ) - 2? Do you t h i n k subtraction is associa t ive?

If subtraction does not have some of the prope r t i e s t o which

we have become accustomed, we shall have t o learn to subtract by going back to the d e f i n i t i o n in terms o f adding t h e opposite. Ad-

dition, after all, does have the familiar properties.

For example, s ince subtraction I s not associative, the ex- pression

3 - 2 - 4

r e a l l y l a not a numeral because it does not name a specific num- ber. Recall that subtraction is a binary operation, that is, involves two numbers. Then does "3 - 2 - 4" mean "3 - ( 2 - 4 ) " or does it mean " ( 3 - 2 ) - 4"? To make a decis ion, we convert subtraction to addition of opposite. Then

On t he other hand,

The second of these I s the meaning we decide upon. We shall

agree that

a - b - c meana (a - b) - c , that is

a - b - c = a + (-b) + (-c) . &ample 1, Find a common name for -

6 1 We can think of this as (5 + 2) + ( ) and then we b o w t h a t we

can write

~ x m p l e - 2. U s e the properties of addition to write

-3x + 5x - 8x in simpler form. -3x + 5x - 8x = ( - 3 ) ~ + 5x + (-8)x,

where we have used the theorem, -ab = (-a) b, f a r the f l r s t term,

and the d e f i n i t i o n of subtraction the same theorem f o r the last term, That is, we think of -3x + 5x -8x as the o f (-3)x, 5x, and (-8)x. Then

-3x + 5x - 8x s ((-3) + 5 + (-8)) x by the d i s t r i b u t i v e property

= - 6 ~ .

While it is not as preclse, we use the commonly accepted word "simplify" f o r d i r ec t ions such as "find a common name for"

and "use the proper t ies of addition to write the fo l lowing in simpler form", When there is no posslbLlity of confusion, this

term will appear henceforth.

mample - 3. Simplify (5y - 3) - (6y - 8).

= j y + (-3) + (- (6y) ),+ (- (-8)) , since the o po ite of the sum is the sum of the opposites.

Instead of t he f ac t t h a t the opposlte of a sum is the sum of the opposf tes , we could a l so have used Theorem 7-2a which states

tha t -a = (-l)a, and then t he distributfve property. Then our example would have proceeded as follows:

When you understand the steps involved, you can abbre-

viate the atepa t o :

(5y - 3) - (6y - 8) = 5~ - 3 - 6~ -I- 8

You may be impressed by the way we are now doing a number

of steps mentally, T h i s a b i l i t y to comprehend several steps without writing them a l l down is a sign of our mathematical growth. We must be careful , however, to be able at any time to

p i c k out a l l t h e detailed steps and explain each one. For instance, give the reason f o r each of the fo l lowing

steps:

(6a - 8b + c) - (4a - 2b + 7c)

Problem Set 9-28 -- Simplify (In Problems 7 and 20 show and explain each step aa in

the first two parts of Example 3. In the remaining problems, uae the abbreviated form of the t h i r d part of Example 3. )

1. (a) 3x - 4x 7. ( 3 ~ + 9 ) - ( 5 ~ - 9 )

( b j -5a - 3a 8. ( 2 6 + 8) + ( 2 -fi)

( c ) 4x2 - (-7x2) 9. -4y + 6y ( d ) -4xz - xz I 10. -3c + 5c + p

1 (b) -3c + 5c - TC 12. ( 3 x - 6 ) + ( 6 - 3x)

( c ) 7x2 - 4x2 - llx 2 13. (5a - 17b) - (ha - 6b)

3. ( a ) - (3x - 4y) 15. (3a + 2b - 4 ) - (5a - 3b + c )

(b ) - I-5a - 3y) 16. (7x2 - 3x1 + ( 4 2 - TX - 8)

( c ) - (4a - 6a) 17. (7x2 - 3x) - (4x2 - 7x - 8)

21. From l l a + 13b - 7c subtract Ra - 5S , bc.

2 22. What is the r e s . ~ l t 3T subtracting -3x + 5x - 7 from -3x + 12?

23. What must be added to 3s - 4t + 7u to obtain -9s - 3u? 24. Prove: If a ) b, then a - b is p o s i t i v e .

25. If (a - b) I s a positive number, which of the statements, a < b, a = b, a ) b, I s t rue? What if (a - b) is a negat ive

number? What i f (a. - b) I s zero?

26. If a, b, and c are real numbers and a ) b, what can we say about the order of a - c and b - c? Prove your statement.

The d e f i n i t i o n of subt rac t ion in terms of addition permits us t o extend further our app l i ca t ions of the di s tr ibut ive pro- per ty , and t o describe i n d i f f e r e n t language some of ou r steps

in finding t r u t h s e t s .

Example Simplify

By applying the d e f i n i t i o n of aubtraction, we have

(-3)(2x - 5 ) = (-3) (2X + ( - 5 ) )

= (-3)(2x) + (-3) ( -5) (my?)

= ((-3)(2)) x + 15 What properties of muXtiplication have we used here?

= ( - 6 ) x + 15

You would perhaps have done some of these steps mental ly ,

and would have written directly:

(-3)(2~ - 5) = - 6 ~ + 15, t h l n k i n g " ( - 3 ) times (2x1 is (-6x)"

"(-3) times (-5) is 15."

Problem Set 9-2b

1. Perform indicated operations and simplify where possible:

(a> 4 ( 3 - 5) (f) 5 (3 - 2x1

( b ) 2 - - (-a)) ( 9 ) (-2)(3x - (-3))

(4 (-3)(4 - ( - 5 ) ) (h) 2(-3x - 3)

( d l (-x)(-2 + 7) (1) a ( b - 2)

( 4 ( - 4 ) ( 3 - x) (3 ) ( -y)( -x - 4 )

2. Perform Indicated operations and simplify where poss ib l e :

(a) ( -3 ) ( -a + 2b - c )

(b) (-3x i- 2y) + 2 ( - 2 x - y)

( d ) 4u (2u + 3) - 3(2u + 3 )

(el X ( X + Y) - Y ( X + Y)

( s ) 2a(a - b ) + b ( a - b) ( g ) 3(a - b + c ) -(2a - b - 2c)

( h ) (-X) ( 4 ~ - y) + 2x(-x - y )

(i) a(b + c + 1) - 2a(2b -I- c - 1) ( J ) a(a + b + 3) + b ( a + b + 3) + 3(a + b + 3)

3. Solve :

(a ) 3 x - 4 = 5 (f) 0 . 7 ~ + 1.3 = 3.2 -I- 1.4~ - 0.3 (b) 2 a m 1 = 4 a - 3 (g) - x - 1 < 4 - x

( c ) -3y = 2 - y - 6 (h) 3a + 3 = 7a .+ 4 - ba - 1 (d) -2 - 2y < - 1 (i) 1.2 - 2 . 5 ~ < - 3 . 3 - c

( e ) 4u + 3 > -5u - 3 4 (a) The width of a rectangle is 5 inches less than i t s

l eng th . What is I t s l ength if its perimeter is 38 inches'!

(b) If 17 I s subtracted from a number, and the result I s

multiplied by 3, the product is 102. What is the number?

(c ) A teacher says, "If I had 3 times as many students in my class as I do have, I would have less than 46 more than I now have." How many students does he have in his

class?

9-3, Subtraction --- in Terms of Distance Suppose we ask: On the number l i ne , how far is it from 5

to 8? If A represents the number of units in th i s distance, then

The solut ion of t h i s equation, as we have seen, can be w r i t t e n as x = 8 - 5. Thus, 8 - 5 can be interpreted as the distance from

5 to 8 on the number l i n e .

L e t us now ask how far I t is from 3 t o (-2). If y repre-

sents the number of u n i t s in t h i s distance, then

Thus (-2) - 3 can be interpreted as the distance from 3 to (-2).

The quant i ty 8 - 5 = 3 is p o s i t i v e , while ( 2 - 3 is

negative. What does this d i s t i n c t i o n tell us? It t e l l s us that the distance from 5 to 8 Is to the right, while from

3 to (-2 is to the left. Therefore, a - b really gives us the distance f r o m b to a, that is, both the length and its - - di rec t ion ,

Suppose we are not Interested I n the d i rec t ion , b u t only in

the distance between a and B . Then a - b is the distance from b to a, and b - a is the dis tance from a to b, and

the distance between a and b is the positive of these two,

From our earlier work, we know that this is la - bl. For example, the d i s t ance - from 3 - t o (-2) is (-2) -3, t h a t is,

-5; the distance between 3 (-2) is 11-21 - 31, that is,

5. In the same way, the dlatance from 2 x is x - 2; the

distance between 2 -- and x I s Ix - 21.

Problem S e t 9-3 -- 1. What is the distance

(a) from -3 t o 5? If) between 5 and I?

(b) between -3 and 5? ( g ) from -8 to -2?

( c ) from 6 to -2? (h) between -8 and -2?

( d ) between 6 and -2? ( 5 ) from 7 to O?

( e ) from 5 to l? (j) between 7 and O?

2. What I s the distance

(a) from x t o 5? ( e ) from -1 t o -x?

(b) between x and 5? (f) between -x and -1

( c ) from -2 to x? (g) from o to x?

( d ) between -2 and x? (h) between 0 and x?

3. For each of the fol lowing pairs of expressions, fill in the symbols "<" , "=", or l a > " , which will make a true sentence.

(a) 19 - 21 ? 191 - 121

(b) 12 - 91 ? 121 - 191

( C 1 19 - (-211 ? 191 - 1-21

Id) 11-21 - 91 ? 1-21 - 191

4. Write a symbol between 1 a - b 1 and 1 a \ - lb 1 which will

make a true sentence for a11 real numbers a and b. Do the same f o r )a - bl and IbI - l a l . For la - b ) and

llal - l b l I 5. Describe the resu l t ing sentences in Problem 4 in terms of dis-

tance on the number l ine .

6. What are the two numbers x on the number l i n e such that

I x - 41 = I?

7. What is the t r u t h set of the sentence

I X - 4 1 < 1, Draw the graph of t h i s set on the number l i ne ,

8. What is the truth set of the sentence

I x - 4 1 > I ?

9, Graph the t r u t h set of

x > 3 and x < 5

on the number l ine . Is this s e t the same as the t r u t h s e t of

/ x - 4 I ( l? (we usual ly write "3 ( x < 5' for the sentence "x ) 3 and x ( 5".)

10. Find the t r u t h set of each of the following equations; graph each of these sets:

(a) I x - 6 1 = 8

(b) y + ( -6 ) = 10

(d l 1x1 < 3

(4 I v l > -3

(f) i y l + 12 = 13

( g ) l y - 81 ( 4 ( ~ e a d this: The distance between y and 8 is less than 4.)

(h) 121 + 1 2 = 6

(1) IX - (-1911 = 3

I j ) I Y + 51 = 9

11. For each sentence in the left column pick the sentence in the r i g h t column which has the same truth set :

1x1 = 3 x = - 3 a d x = 3

1x1 1 3 x > - 3 and x < 3

1x1 > 3 x > - 3 or x < 3

1x1 2 3 x < -3 and x > 3

1x1 6 3 x < - 3 or x > 3

1x1 $ 3 x i - 3 or x 2 3

*12. From a po in t marked 0 on a straight road, John and Rudy ride

bicycles. John rides 10 miles per hour and Rudy rides 12

miles per hour. Find the distance between them after

(a) They start from the 0 mark at the same time and John goes east and Rudy goes we st.

(b) John is 5 miles east and Rudy is 6 miles west of the 0

mark when they start and they both go east.

( c ) John starts from the O mark and goes east. Rudy

starts from the 0 mark 15 minutes later and goes west.

( d ) Both s tart at the same time. John starts from the 0

mark and goes weat and Rudy starts 6 miles west of the

0 mark and a l s o goes west.

You will recall that we defined subtraction of a number as addition of the opposlte of the number:

In other words, we defined subtraction In terms of addition and the additive inverse.

Since d i v i e i o n is related to multiplfcation in much the same way as subtraction I s related to addit ion, we m i g h t expect to def ine d i v i s i o n in terms of multiplication and the multiplicative inverse, This is exactly what we do.

For any real numbers a and b (b # 0), "a divided by bl1 means "a multiplied by

the reciprocal of b" ,

'la'' T h i s We shall indicate "a divided by btl by the symbol 6 . symbol is n o t new, You have used it as a fraction indicat ing division. Then the definition of division I s :

As in arlthmetie, we shall call "an the numerator and "b" H I1

the denominator of the fraction . When there is no posai- b i l i t y of confusion, we ehall a l s o cal l the number named by "a1'

the numerator, and the number named by "b" the denominator, Here are some examples of our def in i t ion . By LQ , we mean

Does t h i s definition of d i v i s i o n agree w i t h the ideas about

division which we already have in arithmetic? An elementary way

to talk about is t o ask "what times 2 gives lo?" Since 2 10 5'2 = 10, then 7 = 5.

Why in t h e definition of d i v i s i o n d i d we make the restriction

"b f O"? Be on your guard agalnst being forced into an impossible situation by i n a d v e r t e n t l y t r y i n g to divide by zero.

Problem Set 9-4a -- Write common names f o r the fol lowing:

Mentally complete the following, and compare

10- 2 - 5 and 10 = 5-2,

- - -I8 - 6 -3 and -18 = 6( ),

and 8y = 2y(4).

What do these suggest about the r e l a t i o n between multiplication and division? Is the following theorem c o n s i s t e n t with your experience in arithmetic?

Theorem 9-4. For b # 0, a = cb

a if and only if = c.

This amounts to saying tha t a divided by b is the number which

multiplied by b gives a. Compare t h i s with Theorem 9-1 which says that b subtracted from a I s the number which added to b

gfves a.

Again, in order to prove a theorem involving "if and on ly if l1 we must prove two things. First, we must show tha t if $ = c

(b # 0), then a = cb. The fact t h a t we want to obtain cb on the

r ight suggests s t a r t i n g the proof by multiplying both members of

1 Proof: If = o (b d 0), then a-b = C,

1 a(%.b) = cb,

aml = cb,

a = cb.

a Second, we must show that i f a = cb ( b 0), then -E;= c. This

time, the f a c t that we do not want b on the r ight suggests

1 starting the proof by multiplying both members of ''a = cb" by - b ' This is possible, since b # 0.

Esec. 9-41

1 1 Proof: If a = cb (b # 0), then am5 = ( ~ b ) ~ ,

a-E 1 = c(b*t),

Supply the reason for each step of the above proofs. The second part of this theorem agrees with our customary

method of checking division by mult iplying the quotient by the divisor.

The multfplication property of 1 states that a = a(1) f o r any real number a. If we apply Theorem 9-4 to this, we

obtain two familiar specia l cases of division. For any real

number a,

and for any non-zero real number a,

8 - = 1. a

Problem Set 9-4b -- 1. Prove that for any real number a,

2. Prove that f o r any non-zero real number a,

[sec . 9-41

In the following problems perfom the indicated divisions

and check by multiplying the quot ien t by the divisor.

28 4. Comment on . 5 . When d i v i d i n g a p o s i t i v e number by a negatfve number, I s the

quot ien t positive or ia it negative? What if we d i v i d e a negat ive number by a p o s l t f v e number? What if we divide a negative number by a negative number?

6. Find the t r u t h set of each of the f o l l o w i n g equations :

( a ) 6y = 42 1 (h) ?X = 20

(b) -6y = 42

( c ) 6y = -42

( e ) 42y = 6

( f ) 42y = 42

( g ) 4~ = 43

Find t he truth set of each of t h e following equations.

(b) & + 13 = 25

( c ) x + .30x = 6.50

1 ( e ) +a = ga + 4

If six times a number is decreased by 5, the r e su l t is -37.

Find the number.

If two-thi rds of a number is added to 32, the result is 38. What is the number?

2 If it takes 7 of a pound of sugar to make one cake, how many

pounds of sugar are needed f o r 35 cakes f o r a banquet which

320 people will attend.

A rectangle is 7 times as long as it Is wide. I t s perimeter 1s 144 inches. How wide is the rectangle?

John is three times a s o l d as Dick. Three years ago the sum

of their ages was 22 years. How o l d is e-ach now?

Find two consecutive even integers whose sum is 46.

Find two consecut ive odd p o s i t i v e integers whose sum is less

than o r equal t c 83.

On a 20%dlscount sale, a chalr cost $30. What was the price

of the cha i r before the sale?

Two t ra ins leave Chicago at the same time: one travels north

at 60 m.p.h. and the other south at 40 m.p.h. After how

many hours will they be 125 miles apar t?

One-half of a number is 3 more than one-sixth of t h e same

number. What is the number?

Mary bought 15 three-cent stamps and some four-cent stamps.

If she paid $1.80 f o r all the stamps, was she charged the

correct amount?

John has 50 coins which are nickels, pennies, and dimes. He

has f o u r more d i m e s than pennies, and six more n i c k e l s than

dimes. How many of each kind of coin has he? How much money does he have?

John, who is saving hi3 money f o r a bicycle , said, "In f i v e weeks I shall have one dollar more than three times the amount I now have. I shall then have enough money f o r my bicycle." If the bicycle c o s t s $76, how much money does John have now?

A p l a n e which f l i e s at an average speed of 200 m.p.h. (when 1 no wind is blowing) is held back by a head wind and takes 32

hours to complete a flight of 630 miles. What is t h e average speed of the wind?

The sum of three successive pos i t ive integers is 108. Find

the integers ,

The sum of two successive posi t ive in t ege r s is less than 25.

Find the in tegers .

A syrup manufacturer made 160 ga l lons of symp worth $608 by mixing maple syrup worth $2 per quart with corn syrup worth

60 c e n t s per quart. How many g a l l o n s of each kind did he use?

Show Chat if the quot ien t of two real numbers is posit ive, the product of the numbers also is p o s i t i v e , and If the quo-

t i e n t is negative, the product is negative.

9-5. Common Names In Chapter 2 we noted some special names f o r rat ional nwnbers

which are in some sense the s i m p l e s t names f o r these numbers, and

which we cal led "common names1'. Two particular items of Interest

20 'l about i n d i c a t e d quotients were the fol lowing: We do n o t cal l - 5

"14" a common name f o r "fourtt, because "4" is simpler; similarly,

"21' is simpler. We is n o t a common name f o r "two-thirds" because - 3

ob ta in these common names by uslng the property of 1 and the theo- a rem - = 1. a

and

On the other hand, we cannot slmplify "4 " and 1 f 2 " 7 any fu r the r . In the above example, what permitted us to write

z(T)? This familiar practice from arithmetic is one which can be 3 7 proved f o r a l l real numbers.

Theorem 9-5. For any real numbers a, b, c , d,

If b # 0 and d & 0, then

a c 1 1 Proof: x*, = (aab)(cea)

1 1 = (ac ) (---I b d

(why?

(why?)

Theorem 7-8d

(why? 1

3azb Example 1. Simplify 5aby

3a2b - = * l ab{ , by assoc ia t ive and commutative 5aby 5y ab p r o p e r t i e s ,

3a ab = ) , by Theorem 9-5, 5~ ab

3a - - - 5~

, by t h e p roper ty of 1.

Example - 2. Simplify . ( ~ o t e : When we write thls phrase, we assume autoinat ical ly that the domain of y excludes 1, Why?)

.* = 3[* , by t h e d i s t r i b u t i v e p roper ty , 2 Y 2 Y

(%I , by Theorem 9-5, = F y

= $(I) , s ince a a = 1, (here a = y - 1) 3

= 2 , by the multiplication property of 1.

After further experience, your mental agility will undoubtedly permit you to skip some of these steps.

2x + 5 ) - Example 3. Simplify ' (5 - 2x1 - -

By the d e f i n l t i o n of subtraction,

I2x + 5 ) - ( 5 - 2x1 - 2x + 5 - - + 2x - - 4x =x 9

- - - " by the multlplf ca t ion proper ty -2 of 1.

X -X The numerals -2 and - 2 and -? a l l name the aame number,

and a l l look equally simple; the accepted common name Is the laat of these. Therefore,

2x 4- 5 - (5 - 2x1 X - = - T o

Problem Set 9-5 1. We have used the property of real numbers a and b that

if b # 0 , then

-a a a 't5 "-b= '6

Prove this theorem.

In each of the following problems, simplify:

5. (a ) w. (b) -*, ( c ) X - x y + y (dl yyx"Y1j Y

6. (a) &+ (4 ++-

6a2b 10. (a) - 5a2b 6a2b 3ab(2a + a) a (b) (4 7 2b a -15abc

11. ( a ) (X + l)Cx - l) ( 1 (X - 1 ) ( 2 x - 3 + x ) x + 1 4x - 4

9-6 F r a c t i o n s

A t the b e g i n n i n g of S e c t i o n 9-5, we recalled two c o n v e n t i o n s on common names which we have been us ing ever s i n c e Chapter 2:

A common name con t a in s no Indicated d i v i s i o n which can be per- formed, and if it contains an ind ica ted d i v i s i o n , the resulting

f r a c t i o n should be " i n lowest terms1'. Then during Section 9-5, we

s ta ted a n o t h e r conven t ion , this one about opposites: We prefer

writing - - " t o any of the other simple names f o r the same number, b

Le t us return t o the convent ions about f rac t ions . In this course a " f r ac t ion" i s a symbol which indicates t h e quotient of t w o numbers. Thus a fraction i n v o l v e s t w o numerals , a numera to r

and denominator, When there is n o possibility o f confusion, we

shall use the word " f r a c t i o n " t o r e f e r also to the number i t s e l f

which is represented by t h e fraction. When there a possibility

of confus ion we must go back t o our s t r i c t meaning of f r a c t i o n as

a numeral, a I n some app l i ca t ions of mathematics the number g iven by

is called the ra t io of a t o b. Again we shall sometimes speak

of the ratlo when we mean the symbol indicating the quo t i en t .

In the preceding sec t i on we used Theorem 9-5 t o write a fraction as the lndlcated product of two fractions. F o r Ins tance , we wrote

Let ua now apply Theorem 9-5 In the "other d i r e c t i o n " to write an indicated product of fractions as a s ing le f r a c t i o n .

x 5 Example 1: Simplify 5.6. -

x 5 BY Theorem 9-5, 3m6 =

Sometimes we use Theorem 9- "both ways" in one problem. 3 1

- I

Example 2: Simplify (T)(3).

3 14 3.14 (s)(T) = (why? 1

14 = 2.7 a d - 3.w. because = 3 0 3 - 2 - 3 . 3

(why?)

= ( by Theorem 9-5.

7 = 3 by the property of 1.

Problem Set 9-6a -- In Problems 1-10 simplify:

h 21 2' ( 8 ) y.5 (b) E*T ( c ) 4 . 7 5+2 7 + 1 4 ( d ) -*- 1 4-21

7 l o

1 1 4. ( a ) ,*E 1 1 1 I (b) " '6 (4 i * ~ ,ii

10.3 3 6- ( 8 ) 3 2 (b) 2 + ( c } (-?)(- $) ( d ) (-%) + (- $)

11. C a n every ra t iona l number be represented by a fraction? Does

every fraction represent a ra t ional number? 2 12. The ratio of faculty to students in a col lege is 5 . If

there are 1197 students, how many faculty members are there?

13. The profits from a student show are to be given t o two

echolsrship funds in the r a t io 5 . If the fund rece iv ing the larger amount was given $387, how much was given to the o t h e r fund?

We can state what we have done so far in another way. A

product of two indicated quotients can always be written as one indicated quotient. Thus, in certain kinds of phrases, which involve the product of s e v e r a l fractions, we can always simplify the phrase t o just one f r ac t ion . If a phrase conta ins several fractions however, these f r a c t i o n s might be added or subtracted, or divlded. We shall see in thls sectlon tha t in U these cases, we may a lways f i n d another phrase for the same number which in-

volves only one indicated division. We are thus able t o state

one more convent ion about i nd i ca t ed quot ien ts : No common name for a number s h a l l contain more t h a n one i nd i ca t ed d iv i s i on . Thus

the instruction "simplify" will always include the idea "use t he properties of t h e real numbers to find another name which conta ins at most one indicated d iv i s i on . "

The key to simplifying the sum of two fractions is us ing the

p r o p e r t y of one to make the denominators alike.

Example 3: Sirnpllfy 5 + . X + Y - X - (1) + * (I), by the proper ty of 1,

a = " (5) + (31, since = 1 , 3 5 5 3

by Theorem 9-5,

1 1 = 5x(=) + 3y (-), by the def . of

l5 division,

I = (5x + 3 ~ ) ~ , by t h e distributive

property ,

= , by t he deflnltion o f d i v i s i o n ,

Once again, you will soon learn t o t e lescope these steps.

Problem S e t 9-6b In Problems 1-5 simplify:

4 5 3. (a) a +a

7. Prove that - a + = 8 + b - for real numbers a, b, and c C C

( c d 0). +8. Prove that a + = f o ~ r e a l n u m b e r s a, b, c , and

C

9. Find the t r u t h s e t of each of the following open sentences:

~ x a m p l e : 4 - 2 -5. Two di f fe ren t procedures are possible.

X - = 2 I 9 * We multiplied by 9 because

we could see that the re- x = 18 s u l t l n g equation would

contain no f r a c t i o n s . 1 1

( a ) ~ Y + ~ = T Y a (f) F - 3 = - - 3 a

(b) : + % = I ( P ) 3 ( ~ 1 + 8 = 1 l w l + T 41

(4 2- = 3 3 22 (h) - -7 + 1x-31 <

7 1 ( d ) 3~ = JX + 8

(e ) & = 3 5 - x

3 10. The sum of two numbers I s 240, and one number is - times 5 t h e other. F i n d the t w o numbers.

4 11. The numerator of the fraction - is increased by an amount 7 x. The v a l u e of the r e s u l t i n g fraction is 3 . By what

amount was the numerato~ increased?

13 1 2 of a number is 13 more than of the number. What

is the number?

13. Joe is as o l d as h i s father. In 12 years he will be 3 1 - as old as his father then 1s. How old is Joe? His father? 2

14. The sum of two p o s i t i v e integers is 7 and t h e i r difference

is 3. What are the numbers? What is the sum of the rec ip- roca l s of these numbers? What is t h e difference of the

r ec ip roca l s?

of the radios were defec- 15. In a shipment of 800 rad ios , 20 tive. What is the ra t io of defec t ive radios to non-defective

radios in the shipment?

16. (a) If I t takes Joe 7 days to pa in t his house, what part

of the job will he do in one day? How much in d days?

(b) If it t akes Bob 8 days to p a i n t Joets house, what p a r t of t h e job would he do in one day? In d days?

( c ) If Bob and Joe work together what por teon of t h e j o b

would they do in one day? What por t i on in d days?

( d ) Refer r ing t o parts (a), (b), ( c ) , translate the fo l lowing into an mglish sentence:

Solve t h i s open sentence f o r d. What does d r ep re sen t ?

( e ) What p o r t i o n of the pa in t ing will Joe and Bob,

working together, do in one day?

"17. A b a l l team on August 1 had won 48 games and lost 52. They had 54 games left on t h e i r schedule. L e t us suppose

that t o win the pennant they must f i n i s h with a s tanding

of at l eas t .600. How many of their remaining games must

they win? What is t h e highes t s t a n d i n g they can get? The

lowest?

For s i m p l i f y i n g the indicated product of two f r a c t i o n s , a key property was Theorem 9-5; f o r simplifying the ind i ca t ed sum

of two f r ac t ions , a key property was the property of 1. When

handling t he indicated quotient of two fractions, we have several

alternative procedures invo lv ing these properties. Let us con- sider an example.

Example 4 : Simpl i fy - 5 z- 1 - 2

Method 1. L e t us apply the property of 1, where we shall think 6 6 of 1 as 5 . (YOU w i l l see why we chose g as the

work proceeds. )

10 = 3 9

by ou r p rev ious work on multiplication.

Method 2. Let us use the property of 1, where we shall t h i n k of

* We choose 2 because it is the reciprocal 1 as 3 .

5 5 then

1 = 1 - - 2 2

10

numerator by previous work 1 denominator by choice of reciprocal of 2

because 1 f o r any a.

Method 3: L e t us apply the d e f i n i t i o n of d i v i s i o n

1 = $ (2) Since = a - a

by previous work on multiplication. = 3

You may apply any one of these methods which appeals to you, pro-

vided t h a t (1) you always understand what you are doing, and (2)

you receive no i n s t r u c t i o n s to the contrary.

Problem Set 9-6c In each of the following, s impli fy , using the most appropriate method:

a - b 9 2

a - b Ti -

10. a - b a - b 2 -4

9-7. Summary

Definition - of subtraction: To subtract the real number b

from the real number a, add the opposite of b to a.

Theorem 9-1. For any real numbers a, b, c, a = b + c if

and only if a - b = c.

On the number l i n e ,

a - b is the dlstance from b to a b - a is the distance from a to b

la-bl is the distance between a and b

D e f i n i t i o n - of d i v i s i o n : To divide the real number a by

the non-zero real number b, multiply a by the rec iproca l

of b.

Theorem 9-4. For any real numbers a, b, c, where b # 0 ,

a a = cb if and only if i;-= c ,

Theorem 9-5. For any real numbers a, b, c, d, if b f O - and d # 0, then

The simplest name for a number:

(1) Should have no indicated operations which can be performed. (2) Should In any indicated dlvislon have no common factors in

the numerator and denominator. -& ( 3 ) Should have the form - g In preference to or -:

(4) Should have at most one indicated d i v i s i o n .

Review Problems 1. Which of the following name real numbers? For each part write

either the common name f o r the number or the reason why it is not a number.

(b) 3 4

( c ) 3.0 (fj 5*0 (i) 8 - 2 - 3

2. Insert parentheses in each of the following expressions so that the r e su l t ing sentence is t r u e

(a) 5 - 5-7 = 0 ( d ) 7 - 6 - 2 = - 1

( b ) 5 - 5.7 - -30 ( e ) 3-2 - 2.5 = o

( ~ 1 7 - 6 - 2 = 3 (f) 3.2 - 2 -5 = 20

3. Find the value of the phrase, b2 - k c , f o r each of the fol lowing:

(a) a = 2, b = (-I), c = 5 (d ) a = 1,654, b = 2, = 0

(bj a = 5, b = 6, c - (-3) ( e ) a = 5 , b = 0, -5

( c ) a = 1, b = (-31, c = ( - a ) (r) 1 1 a = ~ , b = T , 1 = = -5

4. Given the f r a c t i o n 3x + ; what is the only value of x f o r rn which t h i s is not a real number?

5. Use the distributive property t o write the following as

ind ica ted sums.

(b) ab2 ( a - b) ( f ) (x - 3) 7x2

( c ) rn2 (m + 1) (g) (2x - 3yHx + 4 ~ )

(d ) -(3x - 2 ~ ) (h) ( 2 8 - 3b12 6. Solve the fo l lowing sentences

(a) 2 a - 3 < a + 4 3 ( d l $ - y < $

(b) 7x + 4 + ( -X) = 3x - 8 1 ( e ) ,z + 1 = g z - 2

( c ) 6 m 2 1 3 5 (f) -31x1 -6

1 1 7. If of a number increased by g of the number is less than

the number diminished by 25, what is the number?

8. Find t h e average of t h e numbers

x - ! - 3 X - 3 x + k X - k X ' X ' X ' - X

, where x # 0.

3 and 2 < & are true sentences. Then t e l l 9, show that 8 < 20 20 - 2 i s t m e . why you h o w Immediately t h a t 3 < 15

*10, A haberdasher sold two sh i r t s for $3.75 each, On the first he lost 25% of the cost and on the second he gained 25% of the

cost. How much did he gain or lose, or did he break even on t h e two sales?

11. If y = ax + b and a d 0, f i n d an equivalent sentence f o r x in terms of a, b, and y.

12. Last year t e n n i s ba l l s cost d dollars a dozen. This year the price is c cen t s per dozen higher t h an l a s t year, What w i l l half a dozen b a l l s cost at the present rate?

13. Find the t r u t h s e t of each of the following equations.

(a) (X - l)(x + 2 ) = 0 ( d ) ( 2 m - l)(m - 2) = o

( b ) (Y + 5 ) ( ~ + 7 ) = 0 ( e ) (x2 + 1) 3 = o

( c ) 0 = z ( 2 - 2) (f) ( x - 3) + (x - 2) = 0

19. If g a = a where a, b, c, and d are real numbers with

(a) Prove that ad = bc

a b ( b ) Prove that, if c # 0 , then = a d ( c ) Prove that, if a 0 , c d 0, then =

a + b c + d ( d ) Prove that - d b =-

15. Prove the theorem: ac a If b 4 0 and c # 0 , then = i; .

16. Prove the theorem: b If a d 0 and b 4 0, t h e n l - -

a - - a ' b

*17. Given the set {I, -1, A , -j] and the fo l lowing r n u l t l p l i c a t i o n

t a b l e . Second Number

(a) Is the set closed under multiplication?

(b) Verify that this multiplication is commutative f o r the

cases (-1, J ) , (j, -j) and (-1, -JL

( c ) Verify that this multiplication is associa t ive f o r the

cases (-1, j, -j) and (1, -1, j).

(d ) Is it true that a x 1 = a, where a is any element of

[I, -1, J , - J l *

( e ) Find the rec iprocal of each element in this set.

If x is an unspecified member of the set, find the

t r u t h sets of the following (make use of questlan (e)).

(f) j x x = 1. (h) j2 x x = -1.

W - J x x = 3 . (I) j3 . . = -1.

INDEX

The reference is to the page on which t h e t e r m occurs .

abscissa, 406 absolute value, 113, 115, 118 addition

fractions, 258 of r a t i o n a l expressions, 355 of real numbers, 121, 125, 127, 141 on the number line, 1 4

addition method, 476 addition property, 202, 205

associa t ive , 201 commutative, 201 i d e n t i t y element, 162, 201 of equality, 133, 141 of opposites, 131, 141 of order, 187, 188, 190 of z e r o , 57, 71, 131, 141

additive inverse , 13 5, 201 approximation, 299

o f 6, 306 a s s o c i a t i v e p rope r ty

of addi t ion , 24, 71, 130, 1 4 1 , 201 of multiplication, 27, 71, 153, 201

ax is , 500 base, 267 bas ic operat ions, 180 blnary opera t ion , 23, 110, 201 binary r e l a t i o n , 185 c losu re property

of addition, 61, 71, 201 of multiplication, 17, 27, 71, 201

c o e f f i c i e n t , 317 collecting terms, 1% combining terms, 158 common names, 19, 229 commutative proper ty

of addition, 25, 71, 130, 141, 201 of multiplication, 28, 71, 151, 146, 201

comparison property, 105, 118, 185, 202 completing the square, 333, 368, 503 compound sentence , 56

w i t h t h e connective and, 52 with the connective or, 53

cons t an t , 317 constant of variation, 438 contains, 42 j contradiction, 173

coordinate , 9 , 406 coordinate a x l a , 407 correspondence, 8 counting numbers, 2 cube, 267 cube root, 285 degree of a polynomial, 317 denominator, 223

rationalizing, 296 difference of squares, 325, 367 dlstance between, 114, 220 distance from b t o a, 219, 220 d i s t r i b u t i v e property, 31, 66, 67, 71, 146, 155, dividend, 3 59 d i v i s i b l e by, 248 division, 223, 241

checking, 226 of polynomials, 358-359

d i v i s o r , 359 domain, 38, 102 domain o f d e f i n i t i o n , 516 element o f a s e t , 1 empty set, 2 equal s ign, 19 equality

addition property of, 133, 141 properties of, 205

equation, 50 equations involving fac tored expressions, 388 equiva len t inequa l i t i e s , 385 equivalent open sentences, 377 equivalent sentences, 169, 198 existence of multiplicative inverse, 164 exponents, 266, 267

negat ive exponents, 273 z e r o exponents, 273

f a c t o r i n g , 347 f ac to r tng a polynomial, 316 fac tors and d i v i s i b i l i t y , 247 f a c t o r of the integer , 248 facforlng over the positive i n t ege r s , 252 f ac to r s of sums, 263 f i n i t e s e t , 5 f o u r t h r o o t s , 286 fraction, 9, 233

adding, 236, 258 subtracting, 258

fractional equations, 391 function, 516 function notation, 519 fundamental p r o p e r t i e s , 201 fundamental theorem of arithmetic, 257

graph, 13, 416 of a sentence, 55, 56 of' functions, 524 of open sentences fnvolvlng absolute value, 448 of open sentences invo lv ing Integers only, 440 of open sentences with two variables, 411 of quadratic polynomials, 493 o f the polynomial, 437 of the t r u t h s e t o f an open sentence, 48

greater than , 49, 185 g rea te r than or equal to , 54 grouping terms, 323 h o r i z o n t a l change, 428 identity element

for addition, 57, 162, 201 for multiplication, 57, 162, 201

if and only if, 169 indicated d i v i s i o n , 236 ind ica ted product, 19 indicated quotient, 235, 236 indicated sum, 19 indirect proof , 174 i n f i n i t e s e t , 5 i n t ege r s , 98, 252 inverses , 180

addition, 135, 201 multiplication, 162, 163, 172, 202

irrational numbers, 98, 118, 299 JF is i r r a t i o n a l , 287

is approximately equal to , 302 l e a s t common denominator, 259 least common multiple, 59, 258 less than, 49, 102, 118, 185 less than or equal t o , 54 l i n e a r function, 529 linear in x, 437 nember of a s e t , 1 monomia 1, 3 17 multiple, 5

least common, 59, 258 multiplication

fractions, 230 of a real number by -1, 155 on the nurriDer llne, 1 4 real numbers, 145

multiplication property , 202, 205 associative, 27, 71, 153 commutative, 25, 71, 151, 201 identity element, 57, 162, 201 of equa l i t 167 of one, 58' 71, 151, 236 of o r d e r , i 9 5 , 196 o f zero, 58, 72, 151, 179 of zero - converse, 179 use of, 156

multiplicative inverse, 162, 163, 172, 202 natural numbers , 2 negative exponents, 273 negative number, 118 negat ive real number, 98 n th r o o t , 286 null set, 2 number,

i r r a t t ona l , 98, 118, 287, 299 line, 7 , 9 , 97, 101 natural, 2 negative, 118 negative real , 98 of arithmetic, 11 positive, 113 positive real, 98 r a t i o n a l 9 , 98 real, 96, 118 whole, 2

number line, 7, 9, 97, 101, 155 nurnber plane, 405 numbers of arithmetic, 11 numerals, 19 numerical phrase, 21 numerical sentence, 22 numerator, 223 one, multiplicatfon property of, 58, 71, 151 open phrase, 42, 77 open sentence, 42, 56, 82 oppos i t e s , 108, 110, 118, 209

addition property of, 131, 141 ordered pairs, 406 order relation, 185 ordinate, 406 parabolas, 500, 534 pattern, 196 perfec t square, 286, 330, 367 phrase, 21 polynomial

inequalities, 398 over the integers, 314 over the rational number, 347 over the real numbers, 347

p o s i t i v e number, 113 positive real number, 98 pos i t fve square root , 283 power, 267 prime f a c t o r i z a t i o n , 255 prime numbers, 252, 254 product of real numbers (definition), 147 p r o o f , 137

by contradiction, 174, 288 proper f ac to r , 248

properties of addition (see addition property) of comparison, 105, 118, 185, 202 of equality, 205 of exponents, 2'71, 273 of multiplication (see multiplication properly) transitive property, 106, 118, 185, 202, 205

proper subset, 14 property, 23 proportional, 480 quadratic equation, 505 , 540 quadratic formula, 545 quadratic function, 531 quadratic polynomial, 317, 334, 368, 493 quotient, 359 radicals, 286, 290, 294 range, 516 ratio, 233, 480 rational expression, 351 ra t l onallzing the denominator, 296 rational number 9, 98 real number, 96, 118 reciprocals, 172, 175 reduct io ad abaurdum, 174 re f l ex iv i ty , 205 remainder 3 59 roots , 263 aatisfy a sentence, 412 sentence, 22, 41

compound, 56 numerical, 22 open, 42, 56, 82

s e t , 1 empty, 2 element of , I f i n i t e , 5 infinite, 5 member of, 1 null, 2 solution, 134 truth, 45, 56

Sieve of Eratosthenes , 254, 255 simplest name f o r a number, 242 s lope , 427 134 solutions,

of equations, 167 solution of a sentence, 412 solution set , 134 solve, 134 square, 267

square of a non-zero number is pos i t i ve , 197, 198 square of a number, 44

square root, 283, 284, 300 trial and er ror method, 300

is irrational, 287 to a p p r o x i m a t e n , 306

squaring in equationa, 394 standard form, 300, 503, 537 s t ructure , 201 subset, 3 subs ti t u t i o n method, 482 subtrac tlon

d e f i n i t i o n o f , 209, 241 fractions, 258 in terms of distance, 219 is not associative, 213 on the number line 241

substitution method, 462 successor, 7 symbol "Z" , 302 systems of equations, 465 sys terns of inequalities, 485 terms of a phrase, 158 theorem, 138 transitive property, 106, 118, 185, 202, 205 trans la tion

from equality to order, 191 from order to equality, 191, 193

t r u t h set , 45, 56, 167 of a system, 466

unique, 137, 164 value of a function, 519 variable, 37, 77, 102

value o f , 37 v a r y d i r e c t l y , 438 vary inversely, 439 vertical change, 428 vertex, 500 whole number, 2 y-form, 414 y-intercepts, 426 y-Intercept numbers, 426 zero

addition property of, 57, 71, 131, 141 has no reciprocal , 174 multiplication property of, 58, 72, 146 zero exponents, 273