The Imaginary Unit

Definition: The Imaginary Unit

The imaginary unit $\mathrm{i}$ is a number with the property

$${\mathrm{i}}^2=-1$$

Imaginary Numbers

Definition: Imaginary Number

An imaginary number has the form $\mathrm{i}b$, where $\mathrm{i}$ is the imaginary unit and $b\in \mathbb{R}$ is a real number.

Complex Numbers

Definition: Complex Number

A complex number has the form $a+\mathrm{i}b$, where $\mathrm{i}$ is the imaginary unit and $a,b\in \mathbb{R}$ are real numbers.

NOTATION

Complex numbers are usually denoted by $z$.

The set of all complex numbers is denoted by $\mathbb{C}$.

Definition: Real Part

The real part of a complex number $z=a+\mathrm{i}b$ is $a$.

NOTATION

$$\mathrm{Re}(z)\Re (z)$$

Definition: Imaginary Part

The imaginary part of a complex number $z=a+\mathrm{i}b$ is $b$.

NOTATION

$$\mathrm{Im}(z)\Im (z)$$

The Complex Plane

Complex numbers can be plotted on a plane where the horizontal axis contains the real numbers and the vertical axis contains the imaginary numbers.

Modulus

Definition: Modulus (Absolute Value) of a Complex Number

The modulus (or absolute value) of a complex number $z\in \mathbb{C}$ is defined as the square root of the sum of the squares of its real and imaginary parts

$$\sqrt{{\mathrm{Re}}^2(z)+{\mathrm{Im}}^2(z)}$$

NOTATION

$$|z|$$

Theorem: Triangle Inequality for Complex Numbers

For the moduli of all complex numbers $z_1$ and $z_2$

$$|z_1+z_2|\le |z_1|+|z_2|$$

PROOF

TODO

Argument

Definition: Argument of a Complex Number

The argument $\arg (z)$ of a complex number $z=a+\mathrm{i}b$ is the angle between $z$ and the real axis in the the complex plane.

NOTATION

$$\arg (z)$$

NOTE

By convention, the argument lies in the range $(-\pi ;\pi ]$. Positive angles are assigned to numbers of the real axis and negative angles are assigned to numbers below it.

The Forms of a Complex Number

Definition: Cartesian Form of Complex Numbers

The cartesian form of a complex number $z$ is just its usual form $z=a+\mathrm{i}b$.

Definition: Polar Form of a Complex Number

The polar form of a complex number $z$ is

$$z=r(\cos \phi +\mathrm{i}\sin \phi ),$$

where $r$ is the magnitude of $z$ and $\phi$ is its argument.

Definition: Exponential Form of a Complex Number

The exponential form of a complex number $z$ is $r{\mathrm{e}}^{\mathrm{i}\phi }$, where $r$ is the magnitude of $z$ and $\phi$ is its argument.

Form Conversions

Theorem: Cartesian $\to$ Polar

Let $z=a+\mathrm{i}b$ be a complex number in cartesian form.

The polar form of $z$ is $r(\cos \phi +\mathrm{i}\sin \phi )$, where

$$\begin{array}{rl}r& =\sqrt{a^2+b^2}\\ \phi & =\{\begin{array}{l}\arccos \left(\frac{a}{r}\right)b\ge 0\\ -\arccos \left(\frac{a}{r}\right)b<0\end{array}\end{array}$$

PROOF

TODO

Theorem: Polar $\to$ Cartesian

Let $z=r(\cos \phi +\mathrm{i}\sin \phi )$ be a complex number in polar form.

The cartesian form of $z$ is $a+\mathrm{i}b$, where

$$\begin{array}{rl}a& =r\cos (\phi )\\ b& =r\sin (\phi )\end{array}$$

PROOF

TODO