Real and imaginary parts

Remarks

The imaginary part is a real number: $\operatorname{Im}(2 + 3 i) = 3$, not $3 i$. The maps $\operatorname{Re}$ and $\operatorname{Im}$ are additive and $\mathbb{R}$-linear, but they do not respect products. With the conjugate, $\operatorname{Re} z = (z + \bar{z})/2$ and $\operatorname{Im} z = (z - \bar{z})/(2 i)$. A complex number $z$ is real when $\operatorname{Im} z = 0$, and purely imaginary when $\operatorname{Re} z = 0$.