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Triangle Inequality

Let ${\bf x}$ and ${\bf y}$ be vectors

\vert{\bf x}\vert-\vert{\bf y}\vert \leq \vert{\bf x}+{\bf y}\vert \leq \vert{\bf x}\vert+\vert{\bf y}\vert.
\end{displaymath} (1)

Equivalently, for Complex Numbers $z_1$ and $z_2$,
\vert z_1\vert-\vert z_2\vert \leq \vert z_1+z_2\vert \leq \vert z_1\vert+\vert z_2\vert.
\end{displaymath} (2)

A generalization is
\left\vert{\,\sum_{k=1}^n a_k}\right\vert \leq \sum_{k=1}^n \vert a_k\vert.
\end{displaymath} (3)

See also p-adic Number, Strong Triangle Inequality


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

© 1996-9 Eric W. Weisstein