Absolute Value — Mathematics Knowledge System

Overview

The absolute value of a real number is its distance from zero on the number line, regardless of direction. Denoted |x|, it strips away the sign of a number: |3| = 3 and |-3| = 3. It's the simplest example of a function that measures magnitude without regard to direction — an idea that generalizes to distance in the plane, norms of vectors, and moduli of complex numbers.

Intuition

Think of a number line as a road with zero at your house. |x| answers 'how far did I walk?' regardless of whether you went left or right. Walking 3 steps right or 3 steps left both cover the same distance: 3.

Interactive Graph

Formal Definition

Definition

The absolute value is defined piecewise by the sign of x:

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

Definition

|x| = \sqrt{x^2}

Equivalent definition via the principal square root

Notation

Notation	Meaning
$\|x\|$	The absolute value (or modulus) of x

Properties

Non-negativity

|x| \geq 0 \text{ for all } x, \text{ and } |x| = 0 \iff x = 0

Evenness

|-x| = |x|

Multiplicativity

|xy| = |x||y|

Triangle inequality

|x + y| \leq |x| + |y|

Applications

Absolute value expresses tolerances and error bounds, e.g. |measured − true| < ε.

Worked Examples

Both measure distance from 0, regardless of sign.
$|-7| = 7, \quad |7| = 7$

Answer: Both equal 7

Practice Problems

Difficulty 2/10

Solve |2x + 1| = 9.

Common Mistakes

Common Mistake

Assuming |x - a| always equals x - a.

|x - a| = x - a only when x ≥ a; otherwise it equals a - x. Always check the sign of the inside expression.

Summary

|x| measures distance from 0 on the number line, always ≥ 0.
Defined piecewise: |x| = x if x ≥ 0, else |x| = -x.
Equivalently |x| = √(x²).
Satisfies the triangle inequality |x+y| ≤ |x|+|y|, which generalizes to vector norms and complex moduli.

References

WebsiteWikipedia — Absolute value