4. Polar Form of a Complex Number

by M. Bourne

We can think of complex numbers as vectors, as in our earlier example. [See more on Vectors in 2-Dimensions].

We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section.

Why Polar Form?

In the Basic Operations section, we saw how to add, subtract, multiply and divide complex numbers from scratch.

However, it's normally much easier to multiply and divide complex numbers if they are in polar form.

Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis.

Complex number on the complex plane

We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and θ (the angle made with the real axis):

From Pythagoras, we have: `r^2=x^2+y^2` and basic trigonometry gives us:

`tan\ theta=y/x``x=r\ cos\ theta` `y = r\ sin\ theta`

Multiplying the last expression throughout by `j` gives us:

`yj = jr\ sin θ`

So we can write the polar form of a complex number as:

`x + yj = r(cos\ θ + j\ sin\ θ)`

r is the absolute value (or modulus) of the complex number

θ is the argument of the complex number.

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There are two other ways of writing the polar form of a complex number:

`r\ "cis"\ θ` [This is just a shorthand for `r(cos\ θ + j\ sin\ θ)`]

`r\ ∠\ θ` [means once again, `r(cos\ θ + j\ sin\ θ)`]

NOTE: When writing a complex number in polar form, the angle θ can be in DEGREES or RADIANS.

Continues below

Example 1

Find the polar form and represent graphically the complex number `7 - 5j`.

Example 2

Express `3(cos\ 232^@+ j\ sin\ 232^@)` in rectangular form.


1. Represent `1+jsqrt3` graphically and write it in polar form.

2. Represent `sqrt2 - j sqrt2` graphically and write it in polar form.

3. Represent graphically and give the rectangular form of `6(cos\ 180^@+ j\ sin\ 180^@)`.

4. Represent graphically and give the rectangular form of `7.32 ∠ -270°`

And the good news is...

Casio calculator

Now that you know what it all means, you can use your calculator directly to convert from rectangular to polar forms and in the other direction, too.

How to convert polar to rectangular using hand-held calculator.

Online Polar Calculator

Also, don't miss this interactive polar converter graph, which converts from polar to rectangular forms and vice-versa, and helps you to understand this concept:

Online polar to rectangular calculator