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Triangle Inequality

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

|x|-|y|<=|x+y|<=|x|+|y|.
(1)

Equivalently, for complex numbers z_1 and z_2,

|z_1|-|z_2|<=|z_1+z_2|<=|z_1|+|z_2|.
(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 (a>0, b>0, c>0), they must additionally satisfy a+b>c, b+c>a, a+c>b.

A generalization is

|sum_(k=1)^na_k|<=sum_(k=1)^n|a_k|.
(3)

See also

Metric Space, Ono Inequality, p-adic Number, Strong Triangle Inequality, Triangle, Triangle Inequalities, Triangular Inequalities Explore this topic in the MathWorld classroom

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References

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.

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Triangle Inequality

Cite this as:

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

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