Formula for the modulus

Statement

Let $x$ and $y$ be real numbers. Then the modulus of the complex number $x + y i$ is $$|x + y i| = \sqrt{x^2 + y^2}.$$

Remarks

This follows from the definition of the modulus, because $\operatorname{Re}(x + y i) = x$ and $\operatorname{Im}(x + y i) = y$ for real $x$ and $y$. The hypothesis that $x$ and $y$ are real is necessary. For example, $|i + i \cdot i| = |-1 + i| = \sqrt{2}$, but $\sqrt{i^2 + i^2} = \sqrt{-2}$ is not even real.

Explanations

L1 human

To find the length of a complex number, square its two real coordinates, add them, and take the square root. For $1 - i$, this gives $\sqrt{1 + 1} = \sqrt{2}$.