Definition

The absolute value of a number is how far it is from zero, regardless of direction, always positive or zero, written with vertical bars: $|5| = 5$ and $|-5| = 5$. For a real number $x$, $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$, with key properties $|x| \ge 0$, $|xy| = |x||y|$, and the triangle inequality $|x + y| \le |x| + |y|$; $|x - y|$ measures the distance between $x$ and $y$ on the number line. More abstractly, $|x|$ defines a norm on $\mathbb{R}$ ($|x| \ge 0$, $|x| = 0$ iff $x = 0$, $|cx| = |c||x|$, $|x+y| \le |x|+|y|$), generating the metric $d(x,y) = |x-y|$ that makes $\mathbb{R}$ a complete metric space; on $\mathbb{C}$, $|z| = \sqrt{a^2 + b^2}$ for $z = a+bi$ is the Euclidean norm, while on $\mathbb{Q}$ the $p$-adic absolute value $|x|_p = p^{-v_p(x)}$ gives a completely different metric.