Absolute value (modulus) of a number

Definition and basic properties

For real numbers the absolute value is defined piecewise by a simple rule: x if x ≥ 0, and −x if x < 0. This definition guarantees several elementary properties that make the absolute value useful in algebra and analysis. The function is even, meaning |x| = |−x|, and it is positive-definite: |x| ≥ 0 for all x and |x| = 0 exactly when x = 0.

• Multiplicativity: |xy| = |x| |y| for real or complex numbers.
• Triangle inequality: |x + y| ≤ |x| + |y|, a key inequality in analysis and geometry.
• Subtraction form: |x − y| measures the distance between x and y on the number line.

These properties are used repeatedly when solving inequalities, estimating errors, and establishing convergence in sequences and series.

Complex numbers and extensions

For complex numbers z = a + bi the absolute value (often called the modulus) generalizes the idea of distance: |z| = sqrt(a² + b²), the Euclidean distance from the origin in the complex plane. This modulus shares many algebraic properties with the real absolute value, including multiplicativity and the triangle inequality. The modulus leads naturally to the notion of a norm on vector spaces and to metrics that measure distance between points. Complex modulus and vector norms are foundational in linear algebra and functional analysis.

Uses, examples, and problem solving

Absolute value appears in many practical contexts: expressing distances, writing equations for intervals (for example |x − a| < r describes points within distance r of a), formulating error bounds in numerical computation, and describing piecewise-defined behavior in applied models. In elementary algebra, equations like |x| = c have straightforward solutions (x = c or x = −c when c ≥ 0), while inequalities such as |x| < c or |x| ≥ c produce intervals. In programming languages the function is often named abs or fabs; it is used to normalize values and to implement distance calculations.

Connections, distinctions, and notable facts

The absolute value should not be confused with signed magnitude: it always returns a nonnegative result. In abstract settings the absolute value concept generalizes to a norm on a vector space or an absolute value on a field, a structure that satisfies analogous axioms (nonnegativity, multiplicativity, and a form of the triangle inequality). In number theory and algebraic geometry, non-Archimedean absolute values (such as p-adic valuations) satisfy a stronger ultrametric inequality, which alters geometric intuition. See discussions of norms and valuations for further distinctions. Properties and generalizations are central to modern analysis.

Historically, the idea of magnitude independent of sign appears in early arithmetic and geometry; the current formal framework became standard as algebra and analysis were axiomatized. In classroom practice the absolute value is introduced as a basic function and later reinterpreted through distance and norm concepts. For computational references, implementations, and formal definitions consult resources on functions and norms. History and notation pages provide background, while worked examples and interactive tools illustrate solving absolute-value inequalities and equations.

For further reading about applied uses and rigorous generalizations, explore material on metric spaces, complex analysis, and algebraic valuations; the absolute value is a simple but powerful concept connecting arithmetic, geometry, and analysis. Distance, modulus, norm, and absolute value in programming are natural next topics.