Absolute Value Calculator

Definition

The absolute value (modulus) of a real number x is the function |·| : ℝ → [0, ∞) given piecewise by |x| = x when x ≥ 0, and |x| = −x when x < 0. Equivalently |x| = √(x²) with the principal (non-negative) square root. Geometrically |x| is the distance of the point x from zero on the number line, that is d(0, x) in the Euclidean metric on ℝ. The calculator implements |x| in the browser’s floating-point arithmetic — the result should match this definition for finite inputs.

The notation |x| with vertical bars is standard in analysis and in schoolwork. The same symbol is sometimes used for set cardinality or for a determinant; meaning follows from context. Here it means only the modulus of a single real variable. The definition appears in Rudin (Principles of Mathematical Analysis), in school algebra texts, and in MathWorld (Absolute Value) and NIST elementary-function notes.

In teaching practice it helps to separate the mathematical formula from floating-point limits: this calculator shows the browser’s floating-point computation (about 15–16 significant digits), not a paper proof of symbolic identities. When inputs are extreme, check the domain first, then the floating-point range.

Historical and conceptual context

The idea of “magnitude without sign” is older than the modern |x| notation. In Greek geometry a segment length was non-negative by definition; algebra with negative numbers needed an explicit “drop the sign” rule. The modern piecewise definition and the link to √(x²) are part of the standard real-analysis curriculum of the nineteenth and twentieth centuries. In metric topology |x − y| is the canonical distance on the line; in functional analysis the generalization is a norm ‖·‖, and on the complex plane the modulus |a + bi| = √(a² + b²).

Conceptually, absolute value does not sort lists, compute percentiles, or solve equations such as |x − 2| = 5 — those tasks need branch analysis or a separate solver. This calculator answers one question: what is the distance of the entered number from zero. Signum, floor, fractional part, and vector length live on neighboring pages because they answer different questions about the same number or about several coordinates.

In engineering and statistics |x| appears in absolute errors, deviations from a target, and in denominators (for example |old| in percentage change) where direction should stay in the numerator, not in the division base.

Formal properties

Domain: x ∈ ℝ. Range: [0, ∞). |x| = 0 if and only if x = 0. Evenness: |−x| = |x|. Positive homogeneity: |λx| = |λ|·|x| for λ ∈ ℝ. Triangle inequality: |x + y| ≤ |x| + |y|. Reverse triangle inequality: ||x| − |y|| ≤ |x − y|. Multiplicativity: |xy| = |x|·|y|. For x ≠ 0: |x| = x · sgn(x) and sgn(x) = x / |x|.

Links to max and min: max(a, b) = (a + b + |a − b|)/2 and min(a, b) = (a + b − |a − b|)/2. Hence, max − min = |a − b|. On the line |x| is distance from zero; |a − b| is distance between a and b. The function f(x) = |x| is convex, continuous everywhere, and differentiable away from zero (derivative sgn(x) for x ≠ 0).

The identity √(x²) follows because x² ≥ 0 and the principal square root is non-negative: for x ≥ 0 one has √(x²) = x; for x < 0 one has √(x²) = −x.

Full derivation

Step 1. Let x ≥ 0. The piecewise definition gives |x| = x. Also, x² = x·x and √(x²) = x because x is non-negative. Hence, |x| = √(x²).

Step 2. Let x < 0. The definition gives |x| = −x (a positive number). The square x² = (−x)·(−x) = (−x)², so √(x²) = −x. Thus, again |x| = √(x²).

Step 3. When x = 0 both branches give 0. |0| = 0. The distance from zero is zero.

Step 4. The calculator computes |x|. The √(x·x) check in the panel should match |x| in floating-point arithmetic up to square-root representation. For typical school data (for example −7, 12.5) both paths give the same displayed result after formatting.

Step 5 (metric). The function d(x, y) = |x − y| satisfies the metric axioms on ℝ: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. The case y = 0 reduces to |x|.

Special cases

Examples: |−7| = 7; |7| = 7; |12.5| = 12.5; |0| = 0; |−3.2| = 3.2; |100| = 100; |−0.5| = 0.5; |−12| = 12. The sum |−7| + |7| is 14, not 0 — you drop signs first, then add. The raw sum −7 + 7 would be zero, but that is a different operation.

An undefined value (NaN) passes through |x| unchanged; this calculator requires a finite input and shows an error on unparsable text. ±Infinity: |±∞| = +∞ in computer arithmetic, but the form expects finite values.

Negative zero (−0): |−0| = 0 in computer arithmetic (result +0). The calculator does not show a separate “negative zero” label in the result — the distance is still 0.

Numerical notes

The comparison x ≥ 0 is exact for finite numbers other than NaN. The product x·x can overflow to Infinity for huge |x|, while |x| still returns a finite magnitude up to the floating-point arithmetic limit — so the built-in absolute value is safer than √(x²) at extremes; the panel still shows both paths on typical data.

A period and a comma are equivalent in the parser. The axis diagram marks zero, the point x, and the segment of length |x|. The unit setting do not change the mathematical result.

Display rounding (locale en-US / pl-PL) does not change the internal value: |−7| is exactly 7, not “about 7.”

Related notions and sources

Related: sign of a number (signum), square root (because |x| = √(x²)), maximum of two numbers (identity with |a−b|), vector length, percentage change with |old|, fractional part and floor. Complex modulus is a separate geometric model on the plane.

Sources: Wikipedia — Absolute value; MathWorld — Absolute Value; Rudin, W., Principles of Mathematical Analysis; ProofWiki — definitions and the triangle inequality. Educational tool — not a textbook substitute or a teacher’s check.