Real and Imaginary Parts of a Complex Number

Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$, and let $z\in\mathbb{C}$. By claim 3 of Canonical Form and Arithmetic of Complex Numbers there is exactly one pair of real numbers $a,b$ with $z=a+bi$.

The real part of $z$ is the real number $\operatorname{Re}z=a$, and the imaginary part of $z$ is the real number $\operatorname{Im}z=b$.