Exponential form

Complex numbers written in terms of eiθ are said to be in the exponential form, as opposed to the polar or Cartesian form we have seen earlier. Using Euler's formula, we can express a complex number, z, as:

So

As you can see, the exponential form is very close to polar form, but now you have θ in one place instead of two!

Exercise 4

Express the following complex numbers in exponential form:

Conjugation

As we have seen, the conjugation of a complex number is represented as a reflection around the real axis. For complex numbers in exponential form, this means we just change the sign of the angle to get the complex conjugate:

Multiplication

Multiplication and division are even easier in exponential form and are one of the reasons why it is so preferred to work with. We can take our steps for multiplication from the polar form and easily restate them in exponential form.

Given the two complex numbers...