Norm of a Gaussian integer

Definition

Let $\alpha = a + bi$ be a Gaussian integer, with integers $a$ and $b$. The norm of $\alpha$ is the integer $N(\alpha) = a^2 + b^2 = \alpha \bar{\alpha}$, where $\bar{\alpha} = a - bi$ is the complex conjugate of $\alpha$.

Remarks

The norm is multiplicative: $N(\alpha \beta) = N(\alpha) N(\beta)$ for all Gaussian integers $\alpha$ and $\beta$. This product rule is the Brahmagupta–Fibonacci identity. The norm is a nonnegative integer, and $N(\alpha) = 0$ only for $\alpha = 0$. The units are the elements of norm $1$. For example, $N(2 + i) = 5$ and $N(3 - 2i) = 13$.

Explanations

L1 human

The norm of $a + bi$ is the square of its distance from $0$ in the complex plane. It turns a question about Gaussian integers into a question about ordinary integers.