# 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.

## 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^{jθ} |

**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)`

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