In mathematics, the absolute value (sometimes called the modulus) of a real number $a$ is denoted $\left | a \right |$. It refers to the distance of $a$ from zero. Therefore, $\left | a \right |>0$ for all numbers. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0. 

Absolute value

The absolute values of 5 and -5 shown on a number line.

When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.

History

Absolute value is closely related to the mathematical and physical concepts of magnitude, distance, and norm. The term "absolute value" has been used in this sense since at least 1806 in French and 1857 in English. The notation $\left | a \right |$ was introduced by Karl Weierstrass in 1841. Other names for absolute value include "numerical value," "modulus," and "magnitude."

Examples

The following are some examples of equations involving absolute value:

• $\left | 7 \right |=7$
• $\left | -2 \right |=2$
• $-\left | 4 \right |=-4$
• $-\left | -3 \right |=-3$