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Complex Numbers Cheat Sheet

Real and Imaginary parts

Let $z = x + iy$ for $x, y \in \mathbb{R}$. $x$ is the real part of $z$ and denoted by $\Re(z)$. $y$ is the imaginary part of $z$ and denoted by $\Im(z)$.

The modulus is defined as $\abs{z} = \sqrt{x^2 + y^2}$.

Polar notation

$z = r e^{i \theta} = r (\cos \theta + i \sin \theta)$, for $r = \abs{z}$ and $\theta = \tan^{-1}(y/x)$. $\theta$ is called the argument of $z$ and denoted by $\mbox{arg}(z)$.

Conjugate

Let $z = a + ib$ for $a, b \in \mathbb{R}$. The conjugate, denoted by $\overline{z}$ is defined as $z = a - ib$. Use \overline{z} instead of \bar{z} in LaTeX.

Identities

Real (LaTex \Re) and imaginary (LaTeX \Im) parts:

\[\Re(z) = \frac{z + \overline{z}}{2}, \qquad \Im(z) = \frac{z - \overline{z}}{2i}\]

Modulus:

\[\abs{z}^2 = z \overline{z}\]

Properties

Conjugate is invariant with the arithmetic operations of addition, subtraction, multiplication and division:

\[\begin{align} \overline{a + b} &= \overline{a} + \overline{b} \\ \overline{a - b} &= \overline{a} - \overline{b} \\ \overline{a * b} &= \overline{a} * \overline{b} \\ \overline{a / b} &= \overline{a} / \overline{b} \end{align}\]

Modulus

Properties

Triangle inequality:

\[\abs{a + b} \le \abs{a} + \abs{b}\]

Multiplication:

\[\abs{a \cdot b} = \abs{a} \cdot \abs{b}\]