Complex Numbers

COMPLEX NUMBER

The set of complex numbers is

\begin{equation*}\mathbb{C}=\{a+ib:a,b\in\mathbb{R},\ i^2=-1\}.\end{equation*}

Complex numbers extend the real line by adjoining a number

i

whose square is

-1

. Every complex number has the form

\begin{equation*}z=a+ib,\end{equation*}

where

a

and

b

are real numbers. The number

a

is the real part, and

b

is the imaginary part.

Polar form

The same point can be described by Cartesian coordinates

(a,b)

or by polar coordinates

(r,\theta)

For

z\neq 0

, let

\theta

be an angle measured counterclockwise from the positive real axis to the ray through

z

. This angle is called an argument of

z

, and it is determined up to integer multiples of

2\pi

Its distance from the origin is called its modulus:

\begin{equation*}r=|z|=\sqrt{a^2+b^2}.\end{equation*}

Writing

r=|z|

, we have

\begin{equation*}a=r\cos\theta,\qquad b=r\sin\theta,\end{equation*}

\begin{equation*}z=r(\cos\theta+i\sin\theta).\end{equation*}

The polar form of a complex number is based on the Euler formula.

Euler's formula places

e^{i\theta}

on the unit circle and

re^{i\theta}

on the same ray.

EULER FORMULA

For every real number

\theta

\begin{equation*}e^{i\theta}=\cos\theta+i\sin\theta.\end{equation*}

Thus a complex number with modulus

r

and argument

\theta

can be written as

\begin{equation*}z=re^{i\theta}.\end{equation*}

[Needham2023]Needham, Tristan. Visual Complex Analysis. 25th anniversary edition. Oxford University Press, 2023.