Chapter 1Chapter 1Section 2Section 2

Copyright © 2011 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc.

1

2

3

Operations on Real Numbers

Add real numbers. Subtract real numbers.Find the distance between two points on a number line.Multiply real numbers.Divide real numbers.

4

5

1.21.2

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Objective 1

Add real numbers.

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Adding Real Numbers

Same Sign To add two numbers with the same sign, add their absolute values. The sum has the same sign as the given numbers.Different Signs To add two numbers with different signs, find the absolute values of the numbers, and subtract the smaller absolute value from the larger. The sum has the same sign as the number with the larger absolute value.

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EXAMPLE 1

Find each sum. a. 6 + (15)

Find the absolute values.| 6| = 6 | 15| = 15

Because they have the same sign, add their absolute values.6 + (15) = (6 + 15) Add the absolute values.

= (21) = 21

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Both numbers are negative, so the answer will be

negative.

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continued

b. 1.1 + (1.2)= (1.1 +1.2)= 2.3

c.

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3 54 8

58

34

58

68

118

Add the absolute values. Both numbers are negative, so the answer will be negative.

The least common denominator is 8.

Add numerators; keep the same denominator.

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EXAMPLE 2

Find each sum. a. 3 + (7)

Find the absolute values.| 3| = 3 | 7| = 7

Because they have different signs, subtract their absolute values. The number 7 has the larger absolute value, so the answer is negative. 3 + (7) = 4

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The sum is negative because

|7| > |3|

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continued

b. 3 + 7 = = 4

d.

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3 18 4

3 28 8

18

3 28 8

c. 3.8 + 4.6 =

= 0.8

Subtract the absolute values. -3/8 has the larger absolute value, so the answer will be negative.

Subtract numerators; keep the same denominator.

7 – 3 4.6 – 3.8

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Objective 2

Subtract real numbers.

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Subtraction

For all real numbers a and b, a – b = a + (–b). That is, to subtract b from a, add the additive inverse (or opposite) of b to a.

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EXAMPLE 3

Find each difference. a. 12 (–5)

= 12 + 5 = 17

b. 11.5 – (6.3) = 11.5 + 6.3 = 5.2

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Change to addition.The additive inverse of 5 is 5.

Change to addition.

The additive inverse of 6.3 is 6.3.

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continued

c.

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3 24 3

3 24 3

1712

3 43 4

3 24 3

Write each fraction with the least common denominator, 12.

Add numerators; keep the same denominator.

9 812 12

To subtract, add the additive inverse (opposite).

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EXAMPLE 4

Perform the indicated operations. a. 6 – (–2) – 8 – 1 Work from left to right.

= (6 + 2) – 8 – 1 = 4 – 8 – 1 = 12 – 1 = 13

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continued

b. 3 – [(7) + 15] – 6 Work inside brackets.

= 3 – [8] – 6 = 11 – 6 = 17

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Objective 3

Find the distance between two points on a number line.

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To find the distance between the points 2 and 8, we subtract 8 – 2 = 6. Since distance is always positive, we must be careful to subtract in such a way that the answer is positive.

Or, to avoid this problem altogether, we can find the absolute value of the difference. Then the distance is either |8 – 2| = 6 or |2 – 8| = 6.

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Distance

The distance between two points on a number line is the absolute value of the difference between their coordinates.

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EXAMPLE 5

Find the distance between the points 12 and 1.

Find the absolute value of the difference of the numbers, taken in either order. | 12 – (1)| = 11

or

| 1 – (12)| = 11

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Objective 4

Multiply real numbers.

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Multiplying Real Numbers

Same Sign The product of two numbers with the same sign is positive. Different Signs The product of two numbers with different signs is negative.

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EXAMPLE 6

Find each product. a. 7(–2)

b. –0.9(–15)

c.

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5 (16)8

Different signs; product is negative.

Same signs; product is positive.

Different signs; product is negative.

= 14

= 13.5

= 10

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Objective 5

Divide real numbers.

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CAUTION Division by 0 is undefined. However, dividing 0 by a nonzero number gives the quotient 0.

For example,

Be careful when 0 is involved in a division problem.

undefi6 is ned0

, 0 but = 06

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Recall that Thus, dividing by b is the same a multiplying by 1/b. If b ≠0, then 1/b is the reciprocal, or multiplicative inverse, of b.

When multiplied, reciprocals have a product of 1.

1 .a ab b

Number Reciprocal

6

0.05 200 None

25

711

52

16

117

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Division

For all real numbers a and b (where b ≠ 0),

a b =

That is, multiply the first number by the reciprocal of the second number.

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Dividing Real Numbers

Same Sign The quotient of two nonzero real numbers with the same sign is positive. Different Signs The product of two nonzero real numbers with different signs is negative.

1 . a ab b

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EXAMPLE 7

Find each quotient. a.

b.

c.

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315

Same signs; quotient is positive.

The reciprocal of 11/16 is 16/11.

3115

5

6

38

111

1611

38

611

716

34

3 16

4 7

4828

4 2 64 7

127

Multiply by the reciprocal.

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Equivalent Forms of a Fraction

The fractions

Example:

The fractions

Example:

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, , 0).

and are equivalent (x x x y

y y y

4 4 47 7 7

, , 0).

and are equivalent (x x y

y y

4 47 7

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EXAMPLE

Identify which fractions equal

a. b.

c. d.

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35

3.5

35

35

35

Equivalent fraction

Equivalent fraction

Not Equal

Not Equal