---
title: "Definition of Absolute Value"
slug: definition-of-absolute-value
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 7
tags: [algebra, absolute-value]
---

The absolute value of a number represents its distance from 0 on the number line, regardless of direction.

It is defined piecewise as:

$$
|x| =
\begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
$$

This means:

- If $x$ is positive or zero, $|x| = x$
- If $x$ is negative, $|x| = -x$ (which makes it positive)

Absolute value is often used in contexts involving:

- Distance between numbers
- Inequalities (e.g., $|x - a| < \delta$ in limits)
- Piecewise definitions
- Symmetry of graphs

## Related

- [Absolute Value as Piecewise Function](absolute-value-as-piecewise-function.md)
- [Limit with Absolute Value - Piecewise Behavior](limit-with-absolute-value-piecewise-behavior.md)
- [Limit with Absolute Value - Jump Discontinuity](limit-with-absolute-value-jump-discontinuity.md)
- [Function Notation](function-notation.md)
- [Differential Calculus](differential-calculus.md)
- [Notation of a Limit](notation-of-a-limit.md)
- [When a Limit Exists](when-a-limit-exists.md)
- [Computing Basic Limits - Direct Substitution](computing-basic-limits-direct-substitution.md)
- [Computing Limits with Algebraic Simplification](computing-limits-with-algebraic-simplification.md)
- [Limit with Difference of Cubes](limit-with-difference-of-cubes.md)