# 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.

Continued below

## 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)

### Online Algebra Solver

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

### Algebra Lessons on DVD

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