## absolute value

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## Absolute Value

# Absolute Value

Absolute value is an operation in mathematics, written as bars on either side of the expression. For example, the absolute value of −1 is written as |−1|.

Absolute value can be thought of in three ways. First, the absolute value of any number is defined as the positive of that number. For example, |8| = 8 and |−8| = 8. Second, one absolute value equation can yield two solutions. For example, if we solve the equation |*x* | = 2, not only does *x* = 2 but also *x* = −2 because |2| = 2 and |−2| = 2.

Third, absolute value is defined as the distance, without regard to direction, that any number is from 0 on the **real number** line. Consider a formula for the distance on the real number line as |*k* − 0|, in which *k* is any real number. Then, for example, the distance that 11 is from 0 would be 11 (because |11 − 0| = 11). Likewise, the absolute value of 11 is equal to 11. The distance for −11 will also equal 11 (because |−11 − 0| = |−11| = 11), and the absolute value of −11 is 11.

Thus, the absolute value of any real number is equal to the absolute value of its distance from 0 on the number line. Furthermore, if the absolute value is not used in the above formula |*k* − 0|, the result for any negative number will be a negative distance. Absolute value helps improve formulas in order to obtain realistic solutions.

see also Number Line; Numbers, Real.

*Michael Ota*

## absolute value

**absolute value** The magnitude of a number, regardless of its sign (positive or negative). For example, 25 is the absolute value of 25 and –25. Most spreadsheet programs include a function that returns the absolute value of a number.

## absolute value

absolute value, magnitude of a number or other mathematical expression disregarding its sign; thus, the absolute value is positive, whether the original expression is positive or negative. In symbols, if |*a*| denotes the absolute value of a number *a,* then |*a*| = *a* for *a* > 0 and |*a*| = -*a* for *a* < 0. For example, |7|= 7 since 7 > 0 and |-7| = -(-7), or |-7| = 7, since -7 < 0.