# Complex Numbers in Polar Form: Computation and Examples

## The Polar Form of Complex Numbers

A **complex number** is a number with an imaginary component. The imaginary component is represented by i, which is the square root of -1. A typical complex number in **standard form** looks like a + bi, where a and b are real numbers. The real part is a, and the imaginary part is b.

The real plane has two dimensions. A point on the real plane is written in **Cartesian form** as an ordered pair (x,y), where x and y are real numbers. The real plane can also be represented through **polar form**. In polar form, a point is represented as (r,x) where r is the radius, which is the distance from the point to the origin, and x is the angle formed between the point, the origin, and the x-axis. For this lesson, all angles will be given in radians. For example, consider the point (1,1) in Cartesian coordinates. This can be converted into polar form by taking the tangent inverse of the y-coordinate over the x-coordinate. Thus, {eq}\arctan\frac{1}{1} = \arctan 1 = \frac{\pi}{4} {/eq}. Then, the radius is the distance from the origin to (1,1). This is {eq}\sqrt{1^2 +1^2} = \sqrt{1+1} = \sqrt{2} {/eq}. Hence, the point in Cartesian form (1,1) is equivalent to the point ({eq}\sqrt{2} {/eq}, {eq}\frac{\pi}{4} {/eq}) in polar form.

Complex numbers both have a Cartesian-like form and a polar form as well. A complex number in standard form has a real part and an imaginary part, which serve as its coordinates on the complex plane. Similar to the real plane, the standard form of a complex number may be turned into polar form.

### How To Write Complex Numbers in Polar Form

To write a complex number in polar form given its standard form, there are a few easy steps. Let z = a + bi be a complex number.

- Take the modulus of z, which is its magnitude. This is calculated via {eq}|z| = |a + bi| = \sqrt{a^2+b^2} {/eq}.

- Find the angle formed between z and the real line. This is {eq}\theta = \arctan\frac{b}{a} {/eq}. Remember to use the appropriate {eq}\theta {/eq} given by its position on the complex plane. If z is in the third or fourth quadrants, then make sure that {eq}\theta {/eq} corresponds to that quadrant.

- Write the complex number in polar form. This is {eq}z = |z|(\cos \theta + i\sin\theta) = |z|e^{i\theta} {/eq}.

It is important to note here that {eq}e^{i\theta} = \cos\theta + i\sin\theta {/eq}. This is identity will not be proven in this lesson, but it is important to know.

To go from the polar form to standard form, do {eq}|z|e^{i\theta} = |z|(\cos\theta +i\sin\theta) = |z|\cos\theta + |z|i\sin\theta {/eq}.

### Example 1: Converting Complex Numbers to Polar Form

Consider the complex number {eq}z = 1 + i\sqrt{3} {/eq}. To turn it into a polar form, first, take its modulus. This is {eq}|z| = \sqrt{1^2 +(\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 {/eq}. Then, find the angle between z and the real line: {eq}\arctan\frac{\sqrt{3}}{1} = \arctan \sqrt{3} = \frac{\pi}{3} {/eq}. Last, put the two pieces together to find that {eq}z = |z|e^{i\theta} = 2e^{\pi i/3} {/eq}.

### Example 2: How To Convert Complex Numbers to Polar Form

Consider the complex number given in standard from z = -1 + i. To write this in polar form, first take the modulus of the complex number: {eq}|z| = \sqrt{(-1)^2 + 1} = \sqrt{1+1} = \sqrt{2} {/eq}. Then, find the angle between z and the real line: {eq}\arctan\frac{1}{-1} = \arctan (-1) = \frac{3\pi}{4} {/eq}. Last, combine the two pieces to find that {eq}z = -1 + i = \sqrt{2}e^{3\pi i/4} {/eq}.

## De Moivre's Formula For Complex Polar Form

Suppose that given a complex number z, it is desired to find {eq}z^n {/eq}. If the standard form is given, z = a + bi, then it is possible to calculate{eq}z^n = (a+bi)^n = (a+bi)(a+bi)\cdots(a+bi) {/eq} by hand. However, this is very complicated and messy. Instead, use **De Moivre's formula**. This states that {eq}e^{(i\theta)n} = e^{i(n\theta)} {/eq}.

So, to calculate {eq}z^n {/eq}, use the polar form: {eq}z^n = (|z|e^{i\theta})^n = |z|^n e^{in\theta} {/eq}. This is much easier to calculate than calculating {eq}z^n = (a+bi)^n {/eq}.

### Example 3: Write the Complex Number in Rectangular Form Using De Moivre's Formula

Consider the complex number {eq}z = \sqrt{2-\sqrt{3}} + \sqrt{\sqrt{3}+2}i {/eq}. It would be too unwieldy to calculate {eq}z^4 {/eq} by hand. So, instead, turn z into its polar form. First, take the modulus of z to find that {eq}|z| = \sqrt{(\sqrt{2-\sqrt{3}})^2 + (\sqrt{\sqrt{3}+2})^2} = \sqrt{2-\sqrt{3} + \sqrt{3} + 2} = \sqrt{4} = 2 {/eq}. Then, find the angle formed between z and the real line. This is {eq}\arctan \frac{\sqrt{\sqrt{3}+2}}{\sqrt{\sqrt{3}-2}} = \frac{5\pi}{12} {/eq}. Now, it is easy to calculate: {eq}z^4 = (|z|e^{i\theta})^4 = |z|^4e^{i4\theta} = 2^4e^{4*5\pi/12} = 16e^{5\pi/3} {/eq}. Then, it is possible to convert this into the standard form: {eq}z = 16e^{5\pi i /3} = 16(\cos \frac{5\pi}{3} + i \sin\frac{5\pi}{3}) = 16(\frac{1}{2} - i\frac{\sqrt{3}}{2}) = 8 - 8i\sqrt{3} {/eq}.

## Lesson Summary

A **complex number** is a number with a real and imaginary part. The imaginary part is represented by i which is the square root of -1. A complex number in **standard form** looks like a + bi where a and b are real numbers. On the real plane, numbers can be represented in **Cartesian** and **polar** forms. Similarly, complex numbers can be written in a standard form or polar form.

To find the polar form of a complex number z = a + bi, first take its modulus, which is {eq}|z| = \sqrt{a^2+b^2} {/eq}. Then, find the angle between z and the real line: {eq}\theta = \arctan\frac{b}{a} {/eq}. It is important to remember which quadrant the complex number lies in because this will be needed to find the proper value of {eq}\theta {/eq} if a calculator is used. Last, combine the two values that have been found in polar form: {eq}z = |z|(\cos\theta + i\sin\theta) = |z|e^{i\theta} {/eq}. **De Moivre's formula** is used for finding the power of a complex number. Instead of finding {eq}z^n = (a+bi)^n {/eq}, do {eq}z^n = (|z|e^{i\theta})^n = |z|^ne^{in\theta} {/eq}.

To unlock this lesson you must be a Study.com Member.

Create your account

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

The polar form of a complex number replaces the standard form by turning it into a product of its radius of the number with e to the power of i theta, where theta is the angle formed between the complex number and the real line.

#### How do you write a complex number in polar form?

Suppose a complex number is a + bi. First, find its modulus which is the square root of the sum of a squared plus b squared. Then, find the angle by doing the arctan of b over a. Then, write the two together as the modulus of z times e to the power of i theta, where theta is the angle found in step 2.

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

BackAlready registered? Log in here for access