# 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.

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.

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.

### Need Graph Paper?

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.

### 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.

### Exercises

**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...

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:

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