Positive real number that is equal to

*x*if*x*is positive and that is equal to –*x*if*x*is strictly negative.### Symbol

The symbol is “| |” which is read as: “the absolute value of”.

The absolute value of a real number corresponds to the distance that separates this number from the origin on a number line. The distance between 0 and –10 is the same as between 0 and 10.

The absolute value of

*x*and –

*x*is

*x*and we can write: | –

*x*| = |

*x*| =

*x*.

### Properties

There are four laws governing the concept of absolute value:

- | 0 | = 0
- If
*x*≠ 0, | x | > 0 - |
*x*×*y*| = |*x*| × |*y*| - |
*x*+*y*| ≤ |*x*| + |*y*|

### Examples

- | –12 | = 12
- | 12 | = 12

### Historical Note

The first person to use the absolute value symbol (like |24|) was Karl Weierstrass (1815-1897) in 1841 to represent the absolute value of a complex number *a* + *bi* like: |a + bi|.