We indicate the additive inverse of a number by writing the symbol in front of
the number. For example, the additive inverse of 7 is written We could write the
additive inverse of as , but we know that 3 is the additive inverse of.
Since a number can have only one additive inverse, 3 and must represent the
same number, so
This idea can be generalized.
- 1 - 32 = 3.
- 1 - 32
- 3 - 1 - 32 - 3
- 7.
-
32 CHAPTER 1 The Real Number System
Double Negative RuleFor any real number x, 1 x 2 x.
The table in the margin shows several numbers and their additive inverses.
OBJECTIVE 4 Find the absolute value of a real number. Because additive
inverses are the same distance from 0 on a number line, a number and its additive in-
verse have the same absolute value. The absolute valueof a real number x, written
and read“the absolute value of x,”can be defined as the distance between 0 and
the number on a number line. For example,
The distance between 2 and 0 on a number line is 2 units.
The distance between and 0 on a number line is also 2 units.Distance is a physical measurement, which is never negative. Therefore, the absolute
value of a number is never negative.
In symbols, the absolute value of xis defined as follows.
|- 2 |= 2. - 2
| 2 |= 2,
x
By this definition, if x is a positive number or 0, then its absolute value is x itself.
For example, since 8 is a positive number,
If x is a negative number, then its absolute value is the additive inverse of x.
|- 8 |=- 1 - 82 = 8 The additive inverse of - 8 is 8.
| 8 |= 8.
Absolute ValueFor any real number x,
x e
x if x» 0
x if x<0.
The additive inverse of a nonzero
number is found by changing the
sign of the number.Number Additive Inverse
7
3
00
19 190.52 -0.522
(^3)
2
3
- 1 - 32 , or 3
7
Additive InverseThe additive inverseof a number xis the number that is the same distance from
0 on the number line as x, but on the oppositeside of 0.
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