If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Precalculus

### Course: Precalculus>Unit 3

Lesson 6: Polar form of complex numbers

# Complex number forms review

Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.

## What are the different complex number forms?

Rectangulara, plus, b, i
Polarr, left parenthesis, cosine, left parenthesis, theta, right parenthesis, plus, i, sine, left parenthesis, theta, right parenthesis, right parenthesis
Exponentialr, dot, e, start superscript, i, theta, end superscript

## Rectangular form

start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, i
The rectangular form of a complex number is a sum of two terms: the number's start color #11accd, start text, r, e, a, l, end text, end color #11accd part and the number's start color #1fab54, start text, i, m, a, g, i, n, a, r, y, end text, end color #1fab54 part multiplied by i.
As such, it is really useful for adding and subtracting complex numbers.
We can also plot a complex number given in rectangular form in the complex plane. The real and imaginary parts determine the real and imaginary coordinates of the number.

## Polar form

start color #e07d10, r, end color #e07d10, left parenthesis, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, plus, i, dot, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, right parenthesis
Polar form emphasizes the graphical attributes of complex numbers: start color #e07d10, start text, a, b, s, o, l, u, t, e, space, v, a, l, u, e, end text, end color #e07d10 (the distance of the number from the origin in the complex plane) and start color #aa87ff, start text, a, n, g, l, e, end text, end color #aa87ff (the angle that the number forms with the positive Real axis). These are also called start color #e07d10, start text, m, o, d, u, l, u, s, end text, end color #e07d10 and start color #aa87ff, start text, a, r, g, u, m, e, n, t, end text, end color #aa87ff.
Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:
start color #e07d10, r, end color #e07d10, left parenthesis, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, plus, i, dot, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, right parenthesis, equals, start overbrace, start color #e07d10, r, end color #e07d10, cosine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #11accd, a, end color #11accd, end superscript, plus, start overbrace, start color #e07d10, r, end color #e07d10, sine, left parenthesis, start color #aa87ff, theta, end color #aa87ff, right parenthesis, end overbrace, start superscript, start color #1fab54, b, end color #1fab54, end superscript, dot, i
This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values start color #e07d10, r, start subscript, 1, end subscript, end color #e07d10 and start color #e07d10, r, start subscript, 2, end subscript, end color #e07d10 and angles start color #aa87ff, theta, start subscript, 1, end subscript, end color #aa87ff and start color #aa87ff, theta, start subscript, 2, end subscript, end color #aa87ff will have an absolute value start color #e07d10, r, start subscript, 1, end subscript, r, start subscript, 2, end subscript, end color #e07d10 and angle start color #aa87ff, theta, start subscript, 1, end subscript, plus, theta, start subscript, 2, end subscript, end color #aa87ff.