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. So in addition to the side lengths of a triangle needing to be positive (, , ), they must additionally satisfy , , .
A generalization is
(3)
See alsoMetric Space,
Ono Inequality,
p-adic Number,
Strong Triangle Inequality,
Triangle,
Triangle Inequalities,
Triangular Inequalities Explore this topic in the MathWorld classroom Explore with Wolfram|Alpha ReferencesAbramowitz, 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. Referenced on Wolfram|AlphaTriangle Inequality Cite this as:Weisstein, Eric W. "Triangle Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleInequality.html
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