Complex numbers and polar form

**astuart** · November 18th 2012, 05:38 PM

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?

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

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

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

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

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

Can somebody explain which answer is correct?

**darthjavier** · November 18th 2012, 06:01 PM

your answer is correct too...

Remember that:

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

**Prove It** · November 18th 2012, 06:09 PM

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

**darthjavier** · November 18th 2012, 06:18 PM

Originally Posted by Prove It

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

**Prove It** · November 18th 2012, 06:27 PM

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 $\displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*}$ to make life easier for everyone.

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