Real part

From AoPSWiki

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 real part of $z$ , usually denoted $\Re (z)$ or $\mathrm{Re} (z)$ , is just the value $a$ .

Geometrically, if a complex number is plotted in the complex plane, its real part is its $x$ -coordinate (abscissa).

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

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

Examples

$\mathrm{Re}(3 + 4i) = 3$

$\mathrm{Re}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 4 \cos \frac \pi 6 = 2\sqrt 3$

$\mathrm{Re}(4e^{\frac {\pi i}6}) = \mathrm{Re}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 2\sqrt 3$

$\mathrm{Re}((1 + i)\cdot(2 + i)) = \mathrm{Re}(1 + 3i) = 1$ . Note in particular that $\mathrm {Re}$ is not in general a multiplicative function, $\mathrm{Re}(w\cdot z) \neq \mathrm{Re}(w) \cdot \mathrm{Re}(z)$ for arbitrary complex numbers $w, z$ .