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Review of the Real Number System

Chapter 1

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1.2

Operations on Real Numbers

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1.2 Operations on Real Numbers

Objectives

1. Add real numbers.

2. Subtract real numbers.

3. Multiply real numbers.

4. Find the reciprocal of a number.

5. Divide real numbers.

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1.2 Operations on Real Numbers

Adding Real Numbers

Number lines can be used to illustrate addition and subtraction of real

numbers. Move left (the negative direction) to add negative numbers.

–1 + (–3) = –4

0 5–5 –4 –3 –2 –1 1 2 3 4

–1–3

Sum

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1.2 Operations on Real Numbers

Adding Real Numbers

Number lines can be used to illustrate addition and subtraction of real

numbers. Move right (the positive direction) to add positive numbers

or left (the negative direction) to add negative numbers.

–4 + 6 = 2

0 5–5 –4 –3 –2 –1 1 2 3 4

6

–4

Sum

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1.2 Operations on Real Numbers

Rules for Adding Real Numbers

Same signs

Different signs

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

To add two numbers with different signs, find the absolute values of the numbers, and subtract the lesser absolution value from the greater. The sum has the same sign as the number with the greater absolute value.

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1.2 Operations on Real Numbers

Adding Two Negative Real Numbers

First find the absolute values.

| –25| = 25 and | –6| = 6

Because –25 and –6 have the same sign, add their absolute values.

Both numbers are negative, so the answer is negative.

–25 + (–6) = –(25 + 6) = –(31) = –31

–25 + (–6) =

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1.2 Operations on Real Numbers

Adding Two Negative Real Numbers

First find the absolute values.

| –2.4| = 2.4 and | –4.8| = 4.8

Because –2.4 and –4.8 have the same sign, add their absolute values; the answer will be negative.

–2.4 + (–4.8) = –(2.4 + 4.8) = –(7.2) = –7.2

–2.4 + (–4.8) =

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1.2 Operations on Real Numbers

Adding Two Negative Real Numbers

Write each number with a common denominator:

12 12Both numbers are negative, so the answer is nega

3 2

4 3

3 9 2 8

ti

and 4 3

3 2 9 8 9 8 17 17

4 3 12 12 1

3 4

2 12 12

3 4

12

ve.

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1.2 Operations on Real Numbers

Adding Real Numbers with Different Signs

First find the absolute values.

| –23| = 23 and | 9| = 9

Because –23 and 9 have different signs, subtract their absolute values.

23 – 9 = 14

The number –23 has the larger absolute value, so the answer is negative.

–23 + 9 = –13

–23 + 9 =

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1.2 Operations on Real Numbers

Adding Real Numbers with Different Signs

First find the absolute values.

| 19| = 19 and | – 11| = 11

Because 19 and –11 have different signs, subtract their absolute values.

19 – 11 = 8

The number 19 has the larger absolute value, so the answer is positive.

19 + (–11) =

19 + (– 11) = 8

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1.2 Operations on Real Numbers

Adding Real Numbers with Different Signs

9 5 9 5 4

15 15 15 15 15

3 1 3 1

5 3 5

3 5

3 53

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1.2 Operations on Real Numbers

Adding Real Numbers with Different Signs

7.3 9.8 9.8 7.3

2.5

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1.2 Operations on Real Numbers

Subtracting Real Numbers

SubtractionFor all real numbers a and b,

a – b = a + (– b).In words, to subtract b from a, add the additive inverse (or opposite) of b to a.

Difference7 – 5 = 7 + (–5) = 2

Subtrahend = 5

Change the sign

Subtraction changes to addition

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1.2 Operations on Real Numbers

Subtracting Real Numbers

5 – (–19) = 5 + [–(–19)]

= 5 + 19

= 24

Change to addition

Sign changed

– 17 – (–6) = –17 + [–(–6)]

= –17 + 6

= –11

Change to additionSign changed

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1.2 Operations on Real Numbers

Find the differences:

Subtracting Real Numbers

–6.3 – 3.2 =

5.1 – ( 4.7) =

–9.5

9.8–

(a)

(b)

(c)9 2

7 7

11

7

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1.2 Operations on Real Numbers

Adding and Subtracting Real Numbers

For problems involving both addition and subtraction:

1. Add and subtract in order from left to right.

2. Work inside brackets or parentheses first

–2 + 7 –12 = (–2 + 7) –12

= 5 – 12

= 5 + (–12)

= –7

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1.2 Operations on Real Numbers

Adding and Subtracting Real Numbers

19 – 11 – (– 8) = (19 – 11) – (– 8)

= (19 + (–11)) – (– 8)

= 8 – (–8)

= 8 + 8

= 16

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1.2 Operations on Real Numbers

Adding and Subtracting Real Numbers

7 – [(– 12) – (–5)] – 2 = 7 – [(– 12) + 5] – 2

= 7 – [–7] –2

= 7 + 7 – 2

= 14 – 2

= 14 + (–2)

= 12

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1.2 Operations on Real Numbers

Multiplying Real Numbers

Same signs

Different signs

The product of two numbers with the same sign is positive.

The product of two numbers with different signs is negative.

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1.2 Operations on Real Numbers

Multiplying Real Numbers

Same Sign (+) Different Sign (–)

–7(–4) = 28

112 3

4

–0.2(–0.6) = 0.12

–9(11) = –99

4 3 3

7 4 7

1.5(–6) = –9

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1.2 Operations on Real Numbers

Finding the Reciprocal of a Number

Reciprocal

The reciprocal of a nonzero number a is . 1

a

Number Reciprocal

–9

0.02 50

–4

0 None

1

9-

13

5

5

131

4-

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1.2 Operations on Real Numbers

Finding the Reciprocal of a Number

CAUTION

A number and its additive inverse have opposite signs; however, a number and its reciprocal have the same sign.

–8 and 8 are additive inverses of each other

Same signs1

and 66

are reciprocals of each other

Opposite signs

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1.2 Operations on Real Numbers

Dividing Real Numbers

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

a ÷ b = .

In words, multiplying the first number by the reciprocal of the

second number.

1

aa

b b

CAUTION

Division by 0 is undefined. Dividing 0 by a nonzero number gives 0.

6 is un

0defined 0 since 5

00 = 0

5

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1.2 Operations on Real Numbers

Dividing Real Numbers

Same signs

Different signs

The quotient of two nonzero real numbers with the same sign is positive.

The quotient of two nonzero real numbers with different signs is negative.

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1.2 Operations on Real Numbers

Dividing Real Numbers

24(a) 24 4

15(b) 15

6

1

3 35

1

6

54(c) 54 6

22 105(d)

1

9 9

253 325

5 3

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1.2 Operations on Real Numbers

Equivalent (Equal) Fraction

equals equals x x x

y y y

--

-

2 2 2

9 9 9

equal sx x

y y

--

11 11

3 3