Imaginary part

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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 $w, z$ .

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Imaginary part

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