Chapter 2 Real Numbers and algebraic expressions ©2002 by R. Villar All Rights Reserved...

Chapter 2Real Numbers and algebraic expressions

©2002 by R. VillarAll Rights Reserved

Re-engineered byMistah Flynn 2015

Real Number SystemRational Numbers

i.e. -4, 0, 5/8, ,

-2.475, , 72

Irrational Numbers

i.e. pi ≈ 3.14159…

-2,010010001…

≈ 3.4641…Any decimal that terminates or repeats is rational.Any number that can be written as a fraction.

Any decimal that does NOT terminate or repeat

Rational Numbers

Natural or Counting Numbers

Whole Numbers

Integers

Rational Numbers

Ordering real numbers› If the numbers are natural

or whole, then numbers on the right of the number line are greater than those on left.

› If the numbers are integers, then positives are greater than negatives. Numbers on right of number line are greater than the left.

› If the numbers are rational, fractions with same denominator, greater numerator is larger. If the numbers are rational, fractions with same numerator, smaller denominator is larger.

› 3/5 > 2/5 but 5/3 < 5/2

› If the numbers are irrational, then use a calculator to compare their place values starting on left and moving right.

Real Numbers

The Set of Rational Numbers

united with the Set of Irrational Numbers Equals the Set of Real

Numbers.

Natural (counting) numbers

Whole numbers

Integers

Rational numbers

Irrational numbers

Absolute Value

› Absolute value of a number is the DISTANCE to ZERO.

› Distance cannot be negative, so the absolute value cannot be negative.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

5 5

5 5

0 0

Absolute Value

Evaluate the absolute value:

Ask yourself, how far is the number from zero?

1) | -4 | =

2) | 3 | =

3) | -9 | =

4) | 8 - 3 | =

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Absolute Value

Evaluate the absolute value:

Ask yourself, how far is the number from zero?

1) | 12 ÷ -4 | =

2) | 3 ● 15 | =

3) | -9 + 1 | - │1 + 2│ =

4) | 8 - 3 | + │20 - 20│=

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Opposites

› Two numbers that have the same ABSOLUTE VALUE, but different signs are called opposites.

Example -6 and 6 are opposites because both are 6 units away from zero.

| -6 | = 6 and | 6 | = 6

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Opposites

What is the opposite?

1) -10

2) -35

3) 12

4) 100

5) 1

6) X

Adding and Subtracting Real NumbersAdd two numbers with the same sign.Add numbers with same denominators.Add positive and negative numbers.Add numbers with different denominators.

Use the definition of subtraction.Subtraction is the opposite of addition.Use the rules for order of operations with real numbers.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Objective 1

›Add two numbers with the same sign.


›To add two numbers with the same sign, add the absolute values of the numbers. The sum has the same sign as the given numbers.

›Example:

Add two numbers with the same sign.

The sum of two negative numbers is a negative number whose distance from 0 is the sum of the distance of each number from 0.

That is, the sum of two negative numbers is the negative sum of the sum of their absolute values.

4 3 7

To avoid confusion, two operation symbols should not be written successively without a parenthesis between them.


EXAMPLE 1

› Use a number line to find each sum.

Adding Numbers on a Number Line

Solution:

1 4

2 5

5

−7


EXAMPLE 2

› Find the sum.

Solution:

Adding Two Negative Numbers

15 4 15 4

15 4

19


Objective 2

›Add positive and negative numbers.


Add two numbers with the different signs.

›For instance, to add −12 and 5, find their absolute values:

›and

›Then find the difference between these absolute values:

›

›The sum will be negative, since ,

›so the final answer is .

To add two numbers with different signs, find the absolute values of the numbers and subtract the smaller absolute value from the larger. Give the answer the sign of the number having the larger absolute value.

12 12 5 5

12 5 7 12 5

12 5 7

COPYRIGHT © 2008 PEARSON EDUCATION, INC. PUBLISHING AS PEARSON ADDISON-WESLEY

EXAMPLE 3

Solution:

Adding Numbers with Different Signs

›Use a number line to find the sum.

6 3 3


EXAMPLE 4

Solution:

Adding Mentally

Correct3 11 5

4 8 8

3.8 9.5 5.7

›Check each answer.

Correct

Properties of Real Numbers

ClosureCommutativeAssociativeDistributiveSubstitutionZero ProductIdentity + ×Inverse + ×

Closure Properties

› The result or answer to an operation is also a member of that set of numbers.

› Closure Property of Addition states: two real number added together are still a real number

› Closure Property of Multiplication states: two real numbers multiplied together will still be a real number

› Closure Property of Subtraction states: two real number subtracted together are still a real number

› Closure Property of Division states: two real numbers divided by each other will still be a real number

Commutative Properties

›Changing the order of the numbers in addition or multiplication will not change the result.

› Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a.

› Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

Associative Properties›Changing the grouping of the numbers in addition or multiplication will not change the result.

›Associative Property of Addition states:

3 + (4 + 5)= (3 + 4)+ 5 or

a + (b + c)= (a + b)+ c ›Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

Distributive Property

›Multiplication distributes over addition.

a ca bcba

5323523

acabcba

5323523 SECTION 1.1

Distributive Property

›Multiplication and Division also distributes over addition or even subtraction.

a

c

a

b

a

cb

5323523

Substitution Property

Substitution property allows replacing a number with an equivalent value.

Teaching Concept #9

Since 2/3 = 4/6, then2/3 + 1/6 = 4/6 + 1/6 = 5/6If a = 5, then -3a = -3(5) = -15

Think: The substitute teacher takes the place of the regular teacher.

Zero Product Property

Zero product property is the product of any number, and zero will always be zero.

Teaching Concept #9

-½ (0) = 0( -½) = 07.4(0) = 0(7.4) = 0

a(0) = 0(a) = 0

Think: No matter how large or small your allowance is, if it isn’t paid, its value is zero.

Identity Properties

0 + -19 = -19

2/3 + 0 = 2/3

0 + 1.97 = 1.97

0 + a = a + 0 = a

1 (-5) = -5

¾ (1) = ¾

1 (0.35) = 0.35

a(1) = 1(a) = a

Teaching Concept #10

Think: Identity Property is like a student ID card, it

identifies the student.

Think: The Identity element shows you the original number

as your answer.

For MultiplicationMultiplicative Identity

element is 1

For AdditionAdditive Identity

element is 0

Inverse Properties

3 + -3 = 0

-3/5 + 3/5 = 0

2 (1/2) = 1

9/4 (4/9) = 1

Teaching Concept #11

For Addition The sum of a number and its

inverse is 0.

Think: The answer will be the additive identity.

For MultiplicationThe product of a number

and its inverse is 1.

Think: The answer will be the multiplicative identity.


Objective 2

›Subtracting two real numbers.


Subtracting Integers or Rationals.

›For instance, to subtract −12 and 5, add the opposite of 5 to negative 12:

- 12 – 5 = (-12) + (-5) = -17

›For instance, to subtract −12 and -5, add the opposite of negative 5 to negative 12:

- 12 – -5 = (-12) + (5) = -7

›For instance, to subtract 12 and -5, add the opposite of negative 5 to positive 12:

12 – -5 = (12) + (5) = 17

To subtract two numbers, take the opposite of the second number and follow the rules for addition. Recall the opposite of negative number is a positive number. When using a number line, count left when subtracting positive, right when subtracting negative.


EXAMPLE 4

𝟑𝟒

−−𝟏𝟏𝟖

=𝟏𝟕𝟖

Adding Mentally


−𝟑 .𝟖−𝟗 .𝟓=−𝟏𝟑 .𝟑


EXAMPLE 4

|−𝟓|−|𝟑|+𝟏=𝟑

Absolute Value differences


−|𝟑 .𝟖|−𝟗 .𝟓=−𝟏𝟑 .𝟑

›Multiplication and Division

› Find the product of a positive number and a negative number.

› Find the product of two negative numbers.› Identify factors of integers.› Use the reciprocal of a number to apply the

definition of division.› Use the rules for order of operations when

multiplying and dividing signed numbers.› Evaluate expressions involving variables.

COPYRIGHT © 2008 PEARSON EDUCATION, INC. PUBLISHING AS PEARSON ADDISON-WESLEY

Multiplying and Dividing Real Numbers

Slide 1.6- 35

The result of multiplication is called the product. We already know how to multiply positive numbers, and we know that the product of two positive numbers is positive.

We also know that the product of 0 and any positive number is 0, so we extend that property to all real numbers.

Multiplication by Zero says,for any real number x, .0 0x

Multiplicative Identity

› The number 1 is special for multiplication.

› It is called the multiplicative identity.

– This is because a . 1 = a for any real number a.


EXAMPLE 1

Solution:

Multiplying a Positive and a Negative Number

›Find the product.

516

32

4.56 2

80

32

8 5 2

8 2 2

5

2

9.12

Slide 1.6- 37


Find the product of two negative numbers.

›The rule for Multiplying Two Negative Numbers states that:

Slide 1.6- 38

For any positive real numbers x and y,

That is, the product of two negative numbers is positive.

Example:

( )x y xy

5( 4) 20


EXAMPLE 2

Solution:

Multiplying Two Negative Numbers

›Find the product.

3 2

4 5

6

20 3 2

10 2

3

10

Slide 1.6- 39


Identify factors of integers.›In Section 1.1, the definition of a factor was

given for whole numbers. The definition can be extended to integers. If the product of two integers is a third integer, then each of the two integers is a factor of the third.

The table below show several examples of integers and factors of those integers.

Slide 1.6- 40

Use the reciprocal of a number to apply the definition of division.

›The quotient of two numbers is found by multiplying by the reciprocal, or multiplicative inverse. By definition, since

› and ,

›the reciprocal or multiplicative inverse of 8 is and of is .

Pairs of numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other.

1 88 1

8 8

5 4 201

4 5 20

1

85

44

5

Suppose that k is to be the multiplicative inverse of 0. Then k · 0 should equal 1. But, k · 0 = 0 for any real number. Since there is no value of k that is a solution of the equation k · 0 = 1, the following statement can be made:

0 has no multiplicative inverseSlide 1.6- 41


Use the reciprocal of a number to apply the definition of division. (cont’d)

›The Definition of Division says that,

›for any real numbers x and y, with y ≠ 0,

That is, to divide two numbers, multiply the first by the reciprocal, or multiplicative inverse, of the second.

1xx

y y

If a division problem involves division by 0, write “undefined.”

In the expression , x cannot have the value of 2 because then

the denominator would equal 0 and the fraction would be undefined.

1

2x

Slide 1.6- 42

Division›We refer to a/b as:

The quotient of a and b or as the fraction a over b.

– a is the numerator. – b is the denominator (or divisor).


EXAMPLE 3

Solution:

Using the Definition of Division

Find each quotient, using the definition of division.

36

6

12.56

0.4

10 24

7 5

6

1012.56

4

136

6

31.4

10 5

7 24

50

168

2 25

2 84

25

84

Slide 1.6- 44


Dividing Signed Numbers›When dividing fractions, multiplying by the

reciprocal works well. However, using the definition of division directly with integers is awkward.

›It is easier to divide in the usual way and then determine the sign of the answer.

The quotient of two numbers having the same sign is positive.The quotient of two numbers having different signs is negative.

Examples: , , and153

5

15

35

153

5

Slide 1.6- 45


EXAMPLE 4

› Find each quotient.

Solution:

Dividing Signed Numbers

16

2

16.4

2.05

1 2

4 3

116

2

8

116.4

2.05

8

1 3

4 2

3

8

Slide 1.6- 46

Division

› Every nonzero real number a has an inverse, 1/a, that satisfies a . (1/a).

› Division undoes multiplication.– To divide by a number, we multiply by

the inverse of that number. – If b ≠ 0, then, by definition,

a ÷ b = a . 1/b– We write a . (1/b) as simply a/b.


EXAMPLE 6Evaluating Expressions for Numerical Values

Solution:

›Evaluate if and .

2 22 4x y 2x 3y

2 22( )2 34( )

2(4) 4(9)

8 36

28

Slide 1.6- 48

Division › To combine real numbers using division, we use these properties.

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