# Magnitude

A magnitude is a measure of the size of a mathematical entity. For example, the magnitude of a complex number is the distance from the number (graphed on the complex plane) to the origin, a measure of the size of a complex number. The magnitude is generally a nonnegative real number.

Formulaically, the magnitude of a real number $x$ is its absolute value $|x|$, sometimes written $\sqrt{x^2}$. The magnitude $|z|$ of a complex number $z$ equals $\sqrt{\mathrm {Re}(z)^2 + \mathrm{Im}(z)^2}$. Both types of magnitude are bound by a form of the Triangle Inequality which states that $|a| + |b| \geq |a + b|$.

## Homomorphism property

For complex numbers $z$ and $\omega$, we have the identity $|z\omega| = |z||\omega|$. Because the absolute value of a real number equals its magnitude when treated as a complex number, the identity also holds for absolute values of real numbers.

### Proof

Let $z = a + bi$ and $\omega = c + di$ be complex numbers.

We have $$z\omega = (a + bi)(c + di) = (ac - bd) + (ad + bc)i,$$ so \begin{align*} |z\omega| &= \sqrt{(ac - bd)^2 + (ad + bc)^2} \\ &= \sqrt{(ac)^2 - 2abcd + (bd)^2 + (ad)^2 + 2abcd + (bc)^2} \\ &= \sqrt{(ac)^2 + (bd)^2 + (ad)^2 + (bc)^2} \\ &= \sqrt{(a^2 + b^2)(c^2 + d^2)} \\ &= \sqrt{(a^2 + b^2)}\sqrt{(c^2 + d^2)} \\ &= |z||\omega|.\\ \end{align*}