Absolute value

In mathematics, the absolute value of a real number ${\displaystyle x}$, written as |x| or ${\displaystyle {\text{abs}}(x)}$, is the value of ${\displaystyle x}$ when the sign is dropped (or ${\displaystyle x}$ without its sign). That is, |x| ${\displaystyle =x}$ for a positive ${\displaystyle x}$, |x| ${\displaystyle =-x}$ for a negative ${\displaystyle x}$, and |0| ${\displaystyle =0}$

For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3 (${\displaystyle \left\vert x\right\vert =\left\vert -x\right\vert }$). The absolute value of a real number may be thought of as its distance from zero. It can be defined as follows:

${\displaystyle {\begin{cases}\ \ \ x&{\mbox{if}}\ \ x\geq 0\\-x&{\mbox{if}}\ \ x<0\\\end{cases}}}$

Because a distance is always positive, the absolute value of a number is always positive.

Similarly, the absolute value of a complex number may be thought of as its distance from the origin.

It is defined by the equation