# 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)`

(

is therabsolute valueof the complex number, the same as we had before in the Polar Form; and

θis inradians.)

### Example 1

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

### Example 2

Express `-1 + 5j` in exponential form.

Continues below ⇩

## SUMMARY: Forms of a complex number

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

### a. Rectangular form

*x* + *yj *

### b. Polar form

*r*(cos θ +* j* sin θ) = *r* cis θ = *r*∠θ

θ can be in degrees OR radians for Polar form.

### c. Exponential form

re^{j}^{θ}

θ MUST be in radians for Exponential form.

## 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)`

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