Absolute Value Calculator

Explore magnitude, compare expressions, and see distance on the number line.

Enter your values

Real numbers

Single-number magnitude

Number · x

ABSOLUTE VALUE|−15| = 15

Negative input

Two-number analysis

First number · aSecond number · b

Result display

Calculate inside the bars first.
|a + b| means add a and b, then take the magnitude. It is not generally equal to |a| + |b|.

Finite decimals and scientific notation supported. Division is undefined when b = 0.

Precision & input limits

Finite decimal entries are interpreted exactly. Maximum 60 significant digits per entry; scientific exponents −100 to 100. Input fractions such as 1/3 are not supported; exact fractional outputs are supported. Diagram coordinates are approximate and do not determine the numerical result.

DISTANCE BETWEEN a AND b · |a − b|8 units

|−5 − 3| = |−8| = 8 · distance is never negative.

Magnitude · |a|5

Magnitude · |b|3

Relative positiona < b

Your numbers, visualized

Equal-unit scale

Signed differencea − b = −8→Absolute distance|a − b| = 8

The minus sign indicates direction. Absolute value keeps the distance.

Expression breakdown

Evaluate → magnitude

Expression	Inside the bars	Absolute value
\|a + b\|	−5 + 3 = −2	2
\|a − b\|	−5 − 3 = −8	8
\|a × b\|	−5 × 3 = −15	15
\|a / b\|	−5 / 3	53≈ 1.666667

The division result is shown as a reduced fraction with an approximate decimal.

Triangle inequality

Verified

|a + b| ≤ |a| + |b|2 ≤ 8

|a + b|

|a| + |b|

Opposite directions partially cancel.

−5 + 3 = −2, so the magnitude of the sum is 2. Adding the individual magnitudes gives 5 + 3 = 8.

Equality holds for real a and b with the same sign, or when either is zero.

Magnitude is distance from zero

SAME DISTANCE · DIFFERENT SIGNS

|−15| = |15| = 15

The sign describes which side of zero a number lies on. Absolute value describes how far away it is.

|x| = x if x ≥ 0; |x| = −x if x < 0

Product rule|a × b| = |a| × |b|15 = 5 × 3

Quotient rule · b ≠ 0|a / b| = |a| / |b|5/3 = 5 ÷ 3

Symmetric distance|a − b| = |b − a|8 = 8

Read the notation correctly

1. Substitute: |a + b| = |−5 + 3|
2. Simplify: |−5 + 3| = |−2|
3. Take the magnitude: |−2| = 2

Absolute value is not a sign label.
A negative input can have a positive magnitude. Zero has magnitude zero and is neither positive nor negative.

Frequently asked questions

Can absolute value be negative?

No. For a real number x, |x| is nonnegative. Only zero has absolute value zero.

Is |a + b| always |a| + |b|?

No. For −5 and 3, |−5+3| = 2 but |−5|+|3| = 8. The triangle inequality gives the correct relationship.

What happens if b is zero?

The quotient a/b is undefined, including 0/0. Other expressions and the distance remain valid.

Can I enter decimals and negative numbers?

Yes. Both are supported. A reduced fraction can preserve an exact quotient; a displayed decimal may be approximate.

Sources & Reference Guides

OpenStaxPrecalculus · Absolute Value FunctionsMagnitude, distance and the piecewise definition.NISTDLMF · Triangle InequalityAbsolute-value bounds for sums.

Independent educational references · No institutional endorsement implied.

↗ Math