# 5. Exponential Form of a Complex Number

by M. Bourne

IMPORTANT:

In this section, θ MUST be expressed in radians.

We use the important constant

e = 2.718 281 8...

in this section.

We first met e in the section Natural logarithms (to the base e).

The exponential form of a complex number is:

r e^(\ j\ theta)

(r is the absolute value of the complex number, the same as we had before in the Polar Form; and

### Example 1

Express 5(cos 135^@ +j\ sin\ 135^@) in exponential form.

### Example 2

Express -1 + 5j in exponential form.

## SUMMARY: Forms of a complex number

These expressions have the same value. They are just different ways of expressing the same complex number.

Rectangular form Polar form Exponential form
x + yj r(cos\ θ + j\ sin\ θ) = r\ "cis"\ θ = r∠ θ re

NOTE:

1. For Polar Form, θ can be in degrees OR radians.

2. For Exponential Form, θ MUST be in radians.

## Exercises

1. Express in exponential form:

4.50(cos\ 282.3^@+ j\ sin\ 282.3^@)

2. Express in exponential form: -1 - 5j

3. Express in polar and rectangular forms: 2.50e^(3.84j)

Didn't find what you are looking for on this page? Try search:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Play a math game.

(Well, not really a math game, but each game was made using math...)

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

### Algebra Lessons on DVD

Easy to understand algebra lessons on DVD. See samples before you commit.