Triangle inequality

The triangle inequality says the length of one side of a triangle is at most the sum of the other two. For complex numbers $z_1,z_2\in \mathbb{C}$:

$|z_1+z_2|\le |z_1|+|z_2|.$

Geometrically, $z_1+z_2$ is the third side of the triangle with sides $z_1$, $z_2$ in the complex plane — and that side can’t be longer than the sum of the other two.

Equality holds iff $z_1$ and $z_2$ are non-negative real multiples of each other (same direction).

Reverse triangle inequality

A frequently-used corollary:

$||z_1|-|z_2||\le |z_1-z_2|.$

Proof: $|z_1|=|(z_1-z_2)+z_2|\le |z_1-z_2|+|z_2|$, so $|z_1|-|z_2|\le |z_1-z_2|$. By symmetry, $|z_2|-|z_1|\le |z_1-z_2|$ as well, giving the absolute value form.

The reverse inequality is the tool for bounding $|p(z)|$ from below on a contour. Example: on the circle $|z|=R$ with $R>1$,

$|z^2+1|\ge |z{|}^2-1=R^2-1.$

This is used in the ML estimate when bounding contour integrals from above — you need a lower bound on a denominator’s modulus.

Generalization to finite sums

By induction,

$|z_1+z_2+\cdots +z_n|\le |z_1|+|z_2|+\cdots +|z_n|.$

Together with the reverse form, this is the workhorse for all magnitude estimates in complex analysis: bounding contour integrals, proving series converge absolutely, controlling errors in Laurent expansions.

Other settings

The inequality holds in any normed vector space — real Euclidean space ${\mathbb{R}}^n$, function spaces with $L^p$ norms, anywhere with a metric satisfying the standard axioms. In each case, the geometric picture is the same: the “side from $0$ to $z_1+z_2$” is at most the sum of the sides ”$0$ to $z_1$” and ”$z_1$ to $z_1+z_2$.”