U34 Complex Numbers: Essential Formulas

Standard and Cartesian Form

A complex number $z$ is defined as $z = a + bi$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit satisfying $i^2 = -1$. Here, $a$ is the real part, denoted by $\operatorname{Re}(z)$, and $b$ is the imaginary part, denoted by $\operatorname{Im}(z)$.

Polar (Modulus-Argument) Form

A complex number can be expressed in terms of its modulus $r$ and argument $\theta$. The modulus $r = |z| = \sqrt{a^2 + b^2}$. The principal argument $\theta = \arg(z)$ satisfies $-\pi < \theta \leq \pi$.

Euler's Form

Using Euler's formula, a complex number can be written compactly in exponential form.

Addition, Subtraction, and Multiplication

For $z_1 = a + bi$ and $z_2 = c + di$:

In polar form, multiplication becomes simpler: $r_1 e^{i\theta_1} \times r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$.

Conjugate and Division

The conjugate of $z = a + bi$ is $\overline{z} = a - bi$. Important properties: $z \overline{z} = |z|^2 = a^2 + b^2$. For division:

In polar form: $\dfrac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \dfrac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$.

Modulus and Argument Properties

Let $z, z_1, z_2$ be complex numbers.

De Moivre's Theorem

For any integer $n$ and complex number $z = r(\cos \theta + i \sin \theta)$.

Equivalently, $(e^{i\theta})^n = e^{i n\theta}$.

$n$th Roots of a Complex Number

For $z = r(\cos \theta + i \sin \theta)$ and $z \neq 0$, the $n$ distinct $n$th roots are given by:

These roots are equally spaced on a circle of radius $\sqrt[n]{r}$ in the complex plane.