# Imaginary part

Any complex number $z$ can be written in the form $z = a + bi$ where $i = \sqrt{-1}$ is the imaginary unit and $a$ and $b$ are real numbers. Then the imaginary part of $z$, usually denoted $\Im (z)$ or $\mathrm{Im} (z)$, is just the value $b$. Note in particular that the imaginary part of every complex number is real.

Geometrically, if a complex number is plotted in the complex plane, its imaginary part is its $y$-coordinate (ordinate).

A complex number $z$ is real exactly when $\mathrm{Im}(z) = 0$.

The function $\mathrm{Im}$ can also be defined in terms of the complex conjugate $\overline z$ of $z$: $\mathrm{Im}(z) = \frac{z - \overline z}{2i}$. (Recall that if $z = a + bi$, $\overline z = a - bi$).

## Examples

• $\mathrm{Im}(3 + 4i) = 4$
• $\mathrm{Im}\left(4\left(\cos \frac \pi6 + i \sin \frac\pi 6\right)\right) = 4 \sin \frac \pi 6 = 2$
• $\mathrm{Im}\left(4e^{\frac {\pi i}6}\right) = \mathrm{Im}\left(4\left(\cos \frac \pi6 + i \sin \frac\pi 6\right)\right) = 2$
• $\mathrm{Im}((1 + i)\cdot(2 + i)) = \mathrm{Im}(1 + 3i) = 3$. Note in particular that $\mathrm Im$ is not in general a multiplicative function, $\mathrm{Im}(w\cdot z) \neq \mathrm{Im}(w) \cdot \mathrm{Im}(z)$ for arbitrary complex numbers $w, z$.