Absolute value

In mathematics, the absolute value of a real number ${\displaystyle x}$, written as ${\displaystyle |x|}$ or ${\displaystyle {\text{abs}}(x)}$, is the value of ${\displaystyle x}$ when the sign is dropped (or ${\displaystyle x}$ without its sign).^[1][2] That is, ${\displaystyle |x|=x}$ for a positive ${\displaystyle x}$, ${\displaystyle |x|=-x}$ for a negative ${\displaystyle x}$, and ${\displaystyle |0|=0}$

For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3 (${\displaystyle |x|=|-x|}$). 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^[2]

${\displaystyle |a+bi|={\sqrt {a^{2}+b^{2}}}}$

1. ↑ "Absolute Value". www.mathsisfun.com. Retrieved 2020-08-28.
2. ↑ ^{2.0 2.1} Weisstein, Eric W. "Absolute Value". mathworld.wolfram.com. Retrieved 2020-08-28.