Absolute Value Calculator

Calculate |x| for any number, or compute |a - b| for the distance between two numbers.

Select Mode & Enter Values

Can be positive, negative, or zero

Result

|x|
7.5
Input x = -7.5
Absolute Value 7.5
|-7.5| = -(-7.5) = 7.5
Properties -

Understanding Absolute Value

The absolute value of a real number x, written |x|, is its non-negative value regardless of sign.

Definition

Formally: |x| = x if x ≥ 0, and |x| = -x if x < 0. The absolute value strips away the sign and returns only the magnitude.

Geometric Interpretation

|x| represents the distance of x from zero on the number line. Similarly, |a - b| is the distance between points a and b, regardless of which is larger.

Key Properties

Non-Negativity

|x| ≥ 0

|x| = 0 if and only if x = 0.

Symmetry

|x| = |-x|

A number and its negative have equal absolute values.

Multiplicativity

|xy| = |x| · |y|

Absolute value distributes over multiplication.

Triangle Inequality

|x + y| ≤ |x| + |y|

The most important inequality in analysis.

Reverse Triangle

||x| - |y|| ≤ |x - y|

A useful consequence of the triangle inequality.

Division

|x/y| = |x| / |y|

For y ≠ 0, absolute value distributes over division.

Applications

  • Error measurement: |measured - actual| quantifies measurement error.
  • Engineering: Parts within |x - target| ≤ tolerance pass quality control.
  • Statistics: Mean absolute deviation uses |x_i - mean| to measure spread.
  • Physics: Distance, speed, and field strengths are absolute-value quantities.
  • Computer science: Sorting, distance computation, and difference calculations.