MTH070Elementary Algebra

Chapter 1Review of Real Numbers

and Problem Solving

Copyright © 2010 by Ron Wallace, all rights reserved.

What is Algebra?

What is Algebra?

• A generalization of arithmetic.

Mathematics Dictionary, 4th Ed.

by James/James

Therefore, you need to know

and understand arithmetic!

What is Algebra?

• A generalization of arithmetic in which symbols, usually letters of the alphabet, represent numbers or members of a specified set of numbers and are related by operations that that obey specified laws.

The American Heritage Dictionary, 2nd College Edition

What is Algebra?

• A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.

www.dictionary.com

Terminology & Notation

• Mathematical Vocabulary• Precise Communication

• Notation• Symbols instead of words

• Semantics & Syntax• Multiplication & Division in Algebra?

Yes, you are expected to be able to read, understand, and use correct mathematical terminology and notation.

Terminology & NotationExample …

Terminology

• Variable• A symbol used to represent a value.

• Usually a letter (Latin or Greek)

• The value is …– unknown– to be determined– assigned different quantities

x

Terminology

• Constant• A specific value.

– Integers– Fractions– Decimals– Special

17

7.53

21

57.53

Terminology

• Algebraic Expression• A legal combination of variables,

constants, operators, and grouping symbols.

• AKA: Expression

3( 7)

2

x

5 3x

Terminology

• Equivalent Expressions• Expressions that are equal for all

values of their variables.

• Simplifying: Changing expressions into simpler equivalent expressions.

2( 3)x 2 6x

Terminology

• Term• A constant, variable, or product of a

constant and one or more variables.

• When grouping symbols are removed, all expressions are a sum of terms.

• Coefficient – The constant part of a term

• Linear Term – A constant times a variable.

x = 1x -x = -1x

3x

Terminology

• Factor• The constants or variables that are

multiplied together in a term.

• Common Factor – A constant or variable that is a factor of each term of an expression.

3 12x

7 5x x

3 3(4)x

Terminology

• Identities• additive identity: 0

• multiplicative identity: 1

0x x 1x x

Terminology

• Inverses• Opposite – The negative of the number.

– aka: Additive inverse

• Reciprocal - One divided by the number.– aka: Multiplicative inverse

( ) 0x x 1

1xx

Terminology

• Commutative• Order of addition and multiplication

may be reversed.

5 5x x (3) 3x x

5 5x x 5 ( )x

Terminology

• Associative• Sums and products may be grouped

in any way.

( 5) 7 (5 7) 5 7x x x

2(7 ) (2 7) 2 7x x x

What about subtraction and division?

Terminology

• Distributive (involves both operations)

3( 5) 3 3 5 3 15x x x

4 7 (4 7) 11x x x x

“3 is distributed over the sum x+5.”

What about subtraction and division?

Terminology

• Equation• A mathematical statement that two

expressions are equal.

• Three possibilities …– always true “identity”– always false “fallacy”– neither “conditional”

2 3 5x x x

5x x

2 5 7x

Terminology

• Inequality• A mathematical statement that one

expression is ...><≥≤

… a second expression.

2 5 7x

Terminology

• Solution• A value for the variable that makes an

equation or inequality a true statement.

• Solution Set: All such values.

The solution of 2 5 7

is 1.

x

x

Terminology

• Absolute Value• How far a number is from zero.

• AKA: Magnitude

x

if 0

if 0

x xx

x x

Terminology

• Real Numbers• The set of all numbers that

correspond to the points on a number line.

– Can be thought of as directed distances from zero.

0 1 xy

R

Terminology

• Integers • The set of positive and negative

counting numbers and zero.

... 3, 2, 1, 0, 1, 2, 3, ...

Z

Terminology

• Positive Integers • The set of counting numbers.

• AKA: Natural Numbers

1, 2, 3, ...

Z

Terminology

• Negative Integers• The set of opposites of the positive

integers.

... 3, 2, 1

Z

Terminology

• Non-Negative Integers• The set of counting numbers and zero.

• AKA: Whole Numbers

0, 1, 2, 3, ...

*Z

Terminology

• Rational Numbers• Set of numbers that can be written as a

quotient of two integers.

• Decimal forms will either be terminating decimals or repeating decimals.

where , and 0a

a b Z bb

Q

Terminology

• Irrational Numbers• All Real numbers that are not Rational.

2 0.1011011101111...

Properties of Zero

is "undefined"0

x

00 if 0x

x

0 0 or 0ab a b

0x x

0 0x

0x x

Addition

Signed Number Arithmetic

Think Position on # Line• Start @ 1st #

– Negative to left of 0– Positive to right of 0

• Add 2nd #– Negative move left– Positive move right

• Are you moving towards or away from zero?

Think Money in Your Pocket• Start w/ 1st #

– Negative in debt (owe)– Positive cash on hand

• Add 2nd #– Negative spending– Positive receiving

• Are you …– increasing your debt?– increasing your savings?– reversing your state?

“Rules” on Page 28.

Subtraction

Signed Number Arithmetic

Think …– Add the opposite!

“Rule” on Page 32.

a -b = a + (-b)

7 12 7 ( 12)

( 7) 12 ( 7) ( 12)

7 ( 12) 7 12

( 7) ( 12) ( 7 2 ) 1

SEE THINK

Multiplication

Signed Number Arithmetic

Multiplying by -1

“Rules” on Page 37.

( 1)x x 1 if is even

( 1)1 if is odd

n n

n

So …– Multiply numbers

• ignore signs

– Even # of negatives • Positive result

– Odd # of negatives• Negative result

Division

Signed Number Arithmetic

Treat as a Fraction

Therefore … reduce!

“Rules” on Page 38.

aa b

b

Division is …– multiplication by the

reciprocal– sign of the result is

just like multiplication.

a a a

b b b

a a

b b

Division

Signed Number Arithmetic

Warning: Be careful w/ Cancelling

“Rules” on Page 38.

4 5?

4

5 2?

3

x

x

“Value of an Expression”

The Question…

Find the value of 4(2 3) when 5.

--- OR ---

Evaluate 4(2 3) for 5.

x x x

x x x

Solution

Replace x with its value and do the arithmetic!(be careful when x is negative)

Order of Operations

• Grouping symbols { }, [ ], ( ), —• may be nested

• Exponents

• Multiplication & Division• left to right

• Addition & Subtraction• left to right

Simplify!

• Remove Grouping Symbols• Distributive & Associative Laws

( )a b c d a bc bd

( )a b c d a bc bd

( )a c d a c d

( )a c d a c d

( )a b c d a bc bd

( )a b c d a bc bd

( )a c d a c d

( )a c d a c d

Simplify!

• Remove Grouping Symbols• Distributive & Associative Laws

• Group Like Terms • Commutative & Associative Laws

• Combine Like Terms• Distributive Law (i.e. add coefficients)

• Complete All Arithmetic

Translate: Words Algebra

• Addition– added to; sum of;

plus; more than; increased by

• Subtraction– subtracted from;

difference of; minus; less than; decreased by

• Multiplication– multiplied by;

product of; times; twice; of

• Division– divided by; quotient

of; divided into; ratio of; per

Table of Terms w/ Examples on Page 50.

… and combinations of these (e.g. twice the sum of …)

Problem Solving: The 4P’s

Problem Solving?

• Analyzing a situation• Organizing information• Choosing a strategy• Implementing the strategy

The 4P’s

• Prepare– Read (not a novel, it’s technical)

– What do you know?

– What are you trying to find?

• Plan (i.e. strategy)

– Guess & check?

– Patterns?

– Algebraic?

• Process (i.e. implementation)

• Ponder– Did you answer the question?

– Does the solution make sense?

– Units?