Absolute value

January 22, 2026

Left vertical box line a right vertical box line.

The absolute value or modulus of a real number x, denoted by |x|, is defined as

$\left| x \right| = \begin{cases} x, &\text{if } x \geq 0 \\ -x, &\text{if } x < 0 \end{cases}$

The absolute value has the following four fundamental properties (a, b are real numbers):

$\left( i \right) \left| a \right| \geq 0 \qquad \qquad \qquad \quad \text{Non-negativity}\\\\ \left( ii \right) \left| a \right| = 0 \iff a=0 \quad \text{Positive-definiteness}\\\\ \left( iii \right) \left| ab \right| = \left| a \right| \left| b \right| \qquad \;\;\;\;\;\, \text{Multiplicativity}\\\\ \left( iv \right) \left| a+b \right| \leq \left| a \right| + \left| b \right| \qquad \text{Subadditivity (triangle inequality)}$