Absolute value

From AoPSWiki

The absolute value of a real number $x$ , denoted $|x|$ , is the unsigned portion of $x$ . Geometrically, $|x|$ is the distance between $x$ and zero on the real number line.

The absolute value function exists among other contexts as well, including complex numbers.

Real numbers

When $x$ is real, $|x|$ is defined as $|x| = \begin{cases} x & \text{for } x \ge 0,\\ -x & \text{for } x \le 0.\end{cases}$ For all real numbers $x$ and $y$ , we have the following properties:

(Alternative definition) $|x| = \sqrt{x^2}$
(Non-negativity) $|x| \ge 0$
(Positive-definiteness) $|x| = 0 \iff x=0$
(Multiplicativeness) $|xy| = |x| |y|$
(Triangle Inequality) $|x+y| \le |x|+|y|$
(Symmetry) $|x| = |-x|$

Note that

$|x| \le y \iff -y \le x \le y$

and

$|x| \ge y \iff x \ge y \text{ or } x \le -y.$

Complex numbers

For complex numbers $z$ , the absolute value is defined as $|z| = \sqrt{x^2+y^2}$ , where $x$ and $y$ are the real and imaginary parts of $z$ , respectively. It is equivalent to the distance between $z$ and the origin, and is usually called the complex modulus.

Note that $|z| = |\overline{z}| = \sqrt{z\overline{z}}$ , where $\overline{z}$ is the complex conjugate of $z$ .

Examples

If $|x|=k$ , for some real number $k$ , then $x=k$ or $x=-k$ .
If $|ax| = k$ , for some real numbers $a$ , $k$ , then $ax = k$ or $ax = -k$ , and therefore $x = \frac{k}{a}$ or $x = -\frac{k}{a}$ .