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Complex Numbers

Exponential Form

To represent complex numbers in exponential form, we will need to use Euler's formula:

\(e^{i\theta}=cos\theta+isin\theta\)

Recall that the polar form of a complex number \(z=a+bi\) is:

\(z=r(cos\theta+isin\theta)\)

So, by substituting Euler's formula to the polar form formula, we get the exponential form of a complex number:

\(z=re^{i\theta}\)where \(\theta\) is in radians.

Polar Form to Exponential Form

Given the polar form of a complex number, converting it into its exponential form is very straightforward. See the steps required in the below example:

Example: Convert \(z=4\sqrt{2}(cos(\frac{3\pi}{2})+isin(\frac{3\pi}{2}))\) into exponential form.

1. Identify the modulus and argument of the complex number. In this case, we have:

\(r=4\sqrt{2}, \ \theta=\frac{3\pi}{2}\)

*Note: Since the argument is already in radians, we can just substitute it into the exponential form formula (shown in next step). However, if it was in degrees, we would have to first convert it into radians (recall how to do that here) before substituting!

2. Then, substitute these values into the exponential form formula for our answer:

\(z=4\sqrt{2}e^{\frac{3\pi}{2}i}\)

 

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