Let and be vectors. Then the triangle inequality is given by

(1)

Equivalently, for complex numbers and ,

(2)

Geometrically, the right-hand part of the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.

A generalization is

(3)

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, p. 42, 1967.

Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 12, 1999.

CITE THIS AS:

Weisstein, Eric W. "Triangle Inequality." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TriangleInequality.html