2. Review of the Real Number System
Chapter 1
Review of the Real Number System

3. Operations on Real Numbers
1.2
Operations on Real Numbers

4. 1.2 Operations on Real Numbers
Objectives
Add real numbers.
Subtract real numbers.
Multiply real numbers.
Find the reciprocal of a number.
Divide real numbers.
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5. 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 –3 –1 5 –5 –4 –3 –2 –1 1 2 3 4
Sum
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6. 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 6 –4 5 –5 –4 –3 –2 –1 1 2 3 4
Sum
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7. 1.2 Operations on Real Numbers Rules for Adding Real Numbers
Same signs
To add two numbers with same signs, 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 lesser absolution value from the greater. The sum has the same sign as the number with the greater absolute value.
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8. 1.2 Operations on Real Numbers Adding Two Negative Real Numbers
–25 + (–6) =
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
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9. 1.2 Operations on Real Numbers
Adding Two Negative Real Numbers
–2.4 + (–4.8) =
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) = –( ) = –(7.2) = –7.2
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10. 1.2 Operations on Real Numbers Adding Two Negative Real Numbers
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11. 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.
– = –13
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12. 1.2 Operations on Real Numbers
Adding Real Numbers with Different Signs
19 + (–11) =
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) = 8
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13. 1.2 Operations on Real Numbers
Adding Real Numbers with Different Signs
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14. 1.2 Operations on Real Numbers
Adding Real Numbers with Different Signs
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15. 1.2 Operations on Real Numbers Subtracting Real Numbers
Subtraction
For 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.
Subtraction changes to addition
Change the sign
7 – 5 = 7 + (–5) = 2
Difference
Subtrahend = 5
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16. 1.2 Operations on Real Numbers Subtracting Real Numbers
Change to addition
Sign changed
5 – (–19) = 5 + [–(–19)] =
= 24
Change to addition
Sign changed
– 17 – (–6) = –17 + [–(–6)]
= –17 + 6
= –11
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17. 1.2 Operations on Real Numbers Subtracting Real Numbers
Find the differences:
–6.3 – 3.2 =
–9.5
(a)
(b)
5.1 – ( 4.7) = –
9.8
(c)
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18. 1.2 Operations on Real Numbers Adding and Subtracting Real Numbers
For problems involving both addition and subtraction:
Add and subtract in order from left to right.
Work inside brackets or parentheses first
–2 + 7 –12 = (–2 + 7) –12
= 5 – 12
= 5 + (–12)
= –7
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19. 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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20. 1.2 Operations on Real Numbers Adding and Subtracting Real Numbers
7 – [(– 12) – (–5)] – 2 = 7 – [(– 12) + 5] – 2
= 7 – [–7] –2
= – 2
= 14 – 2
= 14 + (–2)
= 12
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21. 1.2 Operations on Real Numbers Multiplying Real Numbers
Same signs
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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22. 1.2 Operations on Real Numbers Multiplying Real Numbers
Same Sign (+)
Different Sign (–)
–9(11) = –99
–7(–4) = 28
1.5(–6) = –9
–0.2(–0.6) = 0.12
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23. 1.2 Operations on Real Numbers Finding the Reciprocal of a Number
The reciprocal of a nonzero number a is .
Number
Reciprocal –9
0.02 50 –4
None
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24. 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.
Opposite signs
–8 and 8 are additive inverses of each other
Same signs
are reciprocals of each other
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25. 1.2 Operations on Real Numbers
Dividing Real Numbers
Division
For all real numbers a and b (where b ≠ 0),
a ÷ b =
In words, multiplying the first number by the reciprocal of the
second number.
CAUTION
Dividing 0 by a nonzero number gives 0.
Division by 0 is undefined.
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26. 1.2 Operations on Real Numbers
Dividing Real Numbers
Same signs
The quotient of two nonzero real numbers with the same sign is positive.
Different signs
The quotient of two nonzero real numbers with different signs is negative.
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27. 1.2 Operations on Real Numbers
Dividing Real Numbers
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28. 1.2 Operations on Real Numbers Equivalent (Equal) Fraction
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