Complex Number

number-theory complex-numbers

Definition

Complex Number

The set of complex numbers $\mathbb{C}$ is the set of all numbers containing a real part and an imaginary part.

Parts

An imaginary number is made up of two parts, the real part and the imaginary part.

Real Part

Real Part

The real part of a complex number $z$, denoted as $\Re (z)$, is the part of $z$ containing the real part.

Imaginary Part

Imaginary Part

The imaginary part of a complex number $z$, denoted as $\Im (z)$, is the part of $z$ containing the imaginary part.

Forms

Component Form

Component Form

The component form of a complex number $z$ is denoted as:

$$z=a+bi$$

where $a$ is the real part and $b$ is the imaginary part.

Polar Form

Polar Form

The polar form of a complex number $z$ is denoted as:

$$z=r\cdot e^{i\theta }=[r,\theta ]$$

where $r$ is the radius and $\theta$ is the angle.

Arithmetic

Addition

Addition of Complex Numbers

Given two complex numbers $z_1$ and $z_2$, addition is done in Component Form and is defined as:

$$z_1+z_2:=(\Re z_1+\Re z_2)+(\Im z_1+\Im z_2)i$$

It is done by adding the real parts and the imaginary parts of the complex numbers.

Subtraction is defined analogously.