Modulus is multiplicative

Statement

For all complex numbers $z$ and $w$, $$|z w| = |z| \, |w|,$$ where $|\cdot|$ is the modulus.

Remarks

Proof: $|z w|^2 = z w \overline{z w} = (z \bar{z})(w \bar{w}) = |z|^2 |w|^2$, and both sides are nonnegative. For $z = a + b i$ and $w = c + d i$, this is the identity $(a^2 + b^2)(c^2 + d^2) = (a c - b d)^2 + (a d + b c)^2$. On the real numbers, it gives that the absolute value is multiplicative.

Explanations

L1 human

The length of a product is the product of the lengths. For example, $|(1 + i)(1 - i)| = |2| = 2$, and $|1 + i| \cdot |1 - i| = \sqrt{2} \cdot \sqrt{2} = 2$.