Equality of complex numbers

Statement

Let $z$ and $w$ be complex numbers. Then $z = w$ if and only if $\operatorname{Re} z = \operatorname{Re} w$ and $\operatorname{Im} z = \operatorname{Im} w$.

Remarks

Here $\operatorname{Re}$ and $\operatorname{Im}$ are the real and imaginary parts. Thus one equation between complex numbers gives two equations between real numbers. For example, $x + y i = 3 - 2 i$ with $x, y$ real means $x = 3$ and $y = -2$.

Explanations

L1 human

Two complex numbers are equal when their real parts agree and their imaginary parts agree. In the plane, this says that two points are equal when both of their coordinates agree.