# Complex Number to Polar Form Calculator

This complex number to polar form calculator is here to serve whenever you need to quickly convert a complex number to polar form. Let's briefly recall what the polar form is and how the conversion works so that you know how to do it by hand if you ever face such a challenge.

## What is the polar form of a complex number?

There are a few ways to represent a given complex number, and the polar form is one of them. You're most probably already familiar with the form `z = a + bi`

, where `a`

and `b`

are the rectangular coordinates.

The polar form uses the fact that `z`

can be identified by two numbers:

- The distance
`r`

from the origin of the plane (i.e., the point`(0,0)`

) to`z`

; and - The angle
`φ`

between the horizontal axis and the radius connecting the origin and`z`

.

We call `r`

the **modulus** (or the **magnitude**) and `φ`

the **argument** of `z`

.

The trigonometric form of a complex number `z`

reads

`z = r × [cos(φ) + i × sin(φ)]`

,

where:

`r`

is the**modulus**, i.e., the distance from`(0,0)`

to`z`

; and`φ`

is the**argument**, i.e., the angle between the x-axis and the radius between`(0,0)`

and`z`

.

Let us discuss how to convert from a rectangular form complex number to polar form.

## Conversion formula for the polar form

To understand how to convert from complex number `a + bi`

to polar form, we need to recall the **Pythagorean theorem and some basics of trigonometry**.

Look at the figure above and note that

`r² = a² + b²`

as well as

`tan(φ) = b / a`

.

The first formula tells us **how to get r** from

`a`

and `b`

:`r = √(a² + b²)`

.

**Getting φ** is slightly more complicated as it is trapped under the tangent function. We need to compute the inverse of the tangent function, that is, the

*arcus tangent*(

`atan`

) function:`φ = atan2(b, a)`

.

The mysterious `atan2`

function is defined simply as `atan2(b, a) = arctan(b / a)`

if `a > 0`

. But if `a < 0`

, then `atan(b / a)`

does not give us the right answer - we must correct it by `±π`

to end up in the correct quadrant of the plane. The definition of `atan2(b, a)`

provides this correction. Formally, `atan2(b, a)`

is equal to:

`atan(b / a)`

if`a > 0`

;`atan(b / a) + π`

if`a < 0 ≤ b`

;`atan(b / a) - π`

if`a,b < 0`

;`π/2`

if `a = 0 < b``;`-π/2`

if`b < 0 = a`

; and- remains undefined if
`x = y = 0`

.

These are exactly the formulas behind Omni's polar form converter.

🙋 The best way to deal with `atan`

is to use Omni's arcus tangent calculator. Or, in the particular context of converting complex numbers to polar form, the best way is, of course, to use our complex number to polar form calculator :)

## How to use this complex number to polar form calculator

This tool is really straightforward to use: given a complex number in its rectangular form, so the `a + bi`

form, you need to **input a and b into their respective fields**.

Our complex number to polar form calculator will immediately display the magnitude `r`

and phase (argument) `φ`

of your number, so that you can write down the polar form `r × exp(iφ)`

.

## Omni tools related to complex numbers

As complex numbers are an essential tool in math and science, the Omni team built a rich collection of tools related to this topic:

## FAQ

### How do I convert a complex number to polar form?

To convert `a + bi`

to polar form, follow these steps:

- Compute
`r = √(a² + b²)`

. This is the**magnitude**(modulus) of your number. - Compute
`φ = atan(b / a)`

. If needed, correct by`±π`

to get into the correct quadrant. This is the**argument**(phase) of your number. - Write down your number as
`r × exp(iφ)`

. - If you have trouble computing
`tan-1`

, do not hesitate to use an online arcus tangent calculator.

### What is the polar form of i?

`i = exp(i π/2)`

. To see why this is the right answer, observe that the distance between `i`

and the points `(0,0)`

is equal to `1`

. Hence, the modulus of `i`

is `1`

. As for the argument, it equals `90° = π/2`

, because the angle between the x-axis and `i`

is the right angle.

### What is the polar form of 0?

`0`

is the only complex number that does not have a unique polar representation: while its **modulus** (magnitude) is obviously `0`

, the **argument** (phase) can be anything. Often it is, by convention, taken as `0`

as well.