# Thread: Complex numbers and polar form

1. ## Complex numbers and polar form

Hello,

We're doing a bit of an introduction to complex numbers and their polar form in my math unit. I have a question that according to the unit notes, I have incorrect, but I would think that I am actually, correct. Could someone clarify the correct answer here?

$\displaystyle (2e^{i\frac{\Pi}{5}}})^{25}$

$\displaystyle =(2)^{25}(e^{i\frac{\Pi}{5})^{25}$

$\displaystyle =2^{25}e^{5i\Pi}$

Now, according to the unit notes, the answer I SHOULD have gotten is

$\displaystyle 2^{25}e^{i\Pi}$

Can somebody explain which answer is correct?

2. ## Re: Complex numbers and polar form

Remember that:

$\displaystyle \mathrm{Z=e^{i\theta}=\cos(\theta)+i\sin(\theta)=\ cos(2k\pi+\theta)+i\sin(2k\pi+\theta)=e^{i(2k\pi+\ theta)}\qquad k \in \mathds{Z}}$

3. ## Re: Complex numbers and polar form

Your answer is INCORRECT, because we define complex numbers in the region \displaystyle \displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*}. If you end up with an angle outside that region you need to add/subtract multiples of \displaystyle \displaystyle \begin{align*} 2\pi \end{align*} until you ARE in that region.

4. ## Re: Complex numbers and polar form

Originally Posted by Prove It
Your answer is INCORRECT, because we define complex numbers in the region \displaystyle \displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*}. If you end up with an angle outside that region you need to add/subtract multiples of \displaystyle \displaystyle \begin{align*} 2\pi \end{align*} until you ARE in that region.
but the two answers are okay, right?

As far as I've understood it is a convention to not have two identical answers. I'm not sure...

5. ## Re: Complex numbers and polar form

Originally Posted by darthjavier
but the two answers are okay, right?

As far as I've understood it is a convention to not have two identical answers. I'm not sure...
The two answers are identical, but you are EXPECTED to use the convention that \displaystyle \displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*} to make life easier for everyone.