Complex Number

Complex Number

## Definition / Concept

A complex number is a number of the form:

$$z=a+ib$$

where $a,b\in\mathbb R$ and $i=\sqrt{-1}$.

## Terminologies

- Real part → $$a$$ is the real part of $z$.

- Imaginary part → $$b$$ is the imaginary part of $z$.

- Imaginary unit → $i=\sqrt{-1}$.

## Formula

$$\operatorname{Re}(z)=a$$

$$\operatorname{Im}(z)=b$$

$$i^2=-1$$

## Conditions / Special Cases

If:

$$b=0$$

then $z$ is purely real.

If:

$$a=0$$

then $z$ is purely imaginary.

If:

$$a=b=0$$

then:

$$z=0$$

## Usage

- Used to represent numbers involving $\sqrt{-1}$ and solve equations having non-real roots.