# Real 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 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 $w, z$.

## Practice Problem 1

Find the conditions on $w$ and $z$ so that $\mathrm{Re}(w\cdot z) = \mathrm{Re}(w) \cdot \mathrm{Re}(z)$.