# 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;

is inθradians; and

`j=sqrt(-1).`

### Example 1

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

Answer

We have `r = 5` from the question.

We must express ` θ = 135^@` in radians.

Recall:

`1^text(o)=pi/180`

So

`135^text(o)=(135pi)/180`

`=(3pi)/4`

`~~2.36` radians

So we can write

`5(cos\ 135^text(o)+j\ sin135^text(o))`

`=5e^((3pij)/4)`

` ~~ 5e^(2.36j)`

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### Example 2

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

Answer

We need to find *θ* in radians (see Trigonometric Functions of Any Angle if you need a reminder about reference angles) and *r*.

`alpha=tan^(-1)(y/x)` `=tan^(-1)(5/1)` `~~1.37text( radians)`

[This is `78.7^@` if we were working in degrees.]

Because our angle is in the second quadrant, we need to apply:

`theta = pi - 1.37 ~~1.77`

And

`r=sqrt(x^2+y^2)`

`=sqrt( (-1)^2 + (5)^2 )`

`= sqrt(26)`

` ~~ 5.10`

So `-1 + 5j` in **exponential** form is `5.10e^(1.77j)`

## 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^@)`

Answer

`282.3^@ = 4.93` radians

So

`4.50(cos\ 282.3^@ + j\ sin\ 282.3^@) ` `= 4.50e^(4.93j)`

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

Answer

This is similar to our `-1 + 5j` example above, but this time we are in the 3rd quadrant.

`r=sqrt(1+25)` `=sqrt(26)~~5.10`

`θ = π + 1.37 = 4.51` radians

The graph for this example:

So

`-1-5j=5.10\ e^(4.51j)`

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

Answer

We can immediately write:

`2.50e^(3.84j) = 2.50\ /_ \ 3.84` [polar form,

θin radians]

OR, if you prefer, since `3.84\ "radians" = 220^@`,

`2.50e^(3.84j) ` `= 2.50(cos\ 220^@ + j\ sin\ 220^@)` [polar form,

θin degrees]

And, using this result, we can multiply the right hand side to give:

`2.50(cos\ 220^@ + j\ sin\ 220^@)` ` = -1.92 -1.61j`

**Summary**

Our complex number can be written in the following equivalent forms:

`2.50e^(3.84j)` [exponential form]

` 2.50\ /_ \ 3.84` `=2.50(cos\ 220^@ + j\ sin\ 220^@)` [polar form]

`-1.92 -1.61j` [rectangular form]

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### Euler's Formula and Identity

The next section has an interactive graph where you can explore a special case of Complex Numbers in Exponential Form:

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