---
tag: 00S7C68D51
kind: definition
name: Absolute value
msc: [26A03]
defines: 0FFXKTKKQC
uses: [0QSPNASE50]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value
    locator: Real numbers
---

## Definition

Let $x$ be a real number.
The absolute value of $x$ is the real number
$$|x| = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}$$

## Remarks

For each real $x$: $|x| \ge 0$; $|x| = 0$ only for $x = 0$; $|-x| = |x|$; and $|x| = \max(x, -x)$.
The distance between two real numbers $x$ and $y$ is $|x - y|$.
For $r \ge 0$, the inequality $|x| \le r$ holds exactly when $-r \le x \le r$.

## Explanations

### L1 human

The absolute value removes the minus sign from a number.
On the number line, $|x|$ is the distance from $x$ to $0$, and $|x - y|$ is the distance from $x$ to $y$.