5. Exponential Form of a Complex Number

by M. Bourne


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
θ is in radians.)

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


θ MUST be in radians for Exponential form.


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)`