Magnitude (mathematics)

Magnitude is a numerical measure of size, such as a number’s absolute value or a vector’s length. Its precise definition depends on the mathematical object being measured.

Absolute value: A real number’s distance from zero, extended to complex numbers by modulus. It measures a scalar’s size without regard to its sign.
Norm: A function assigning vectors nonnegative lengths while satisfying homogeneity and the triangle inequality. A norm is the principal framework for defining vector magnitudes.
Manhattan distance: A distance measured by summing absolute coordinate differences. Its associated vector length differs from Euclidean magnitude except in special cases.
Triangle inequality: An inequality stating that the length of one side of a triangle cannot exceed the sum of the other two. Norm-based magnitudes obey this bound, which constrains sums and distances.
Euclidean norm: The square root of the sum of squared coordinates, giving ordinary length in Euclidean space. It is the standard rule for measuring vector magnitude in Euclidean geometry.
Complex modulus: The nonnegative square root of the sum of a complex number’s squared real and imaginary parts. It computes complex magnitude as distance from zero.
Chebyshev distance: A distance between coordinate vectors given by their largest coordinate difference. It measures size by the largest component, not by Euclidean length.
Error analysis: The study of how numerical errors arise, propagate, and affect computed results. Absolute magnitudes quantify errors without allowing positive and negative deviations to cancel.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Distance generalizes the size-measuring role of absolute value to pairs of points.
Inner product: A bilinear or sesquilinear operation that encodes angles and lengths in a vector space. An inner product induces a norm through the square root of a vector’s self-product.

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