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Prometheus
Post  Post subject: Complex numbers question  |  Posted: Thu Jan 19, 2012 3:12 pm
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Doing a remedial maths course and i just want to check this question.

I need to state the real part $ \displaystyle x$ and the imaginary part $ \displaystyle y$ of:

$ \displaystyle Z=(-i)^9$

I assume they are expecting me to do it by inspection given it's only worth 1 mark, in which case i make it
$ \displaystyle x=0$ and $ \displaystyle y=-1$ since it's already in Cartesian form.

But i wanted to prove it to myself to make sure i understand the concept not just a mechanical application. So i changed it to polar exponential form in order to expand the brackets and i just want to make sure the workings are correct:

$ \displaystyle r=\sqrt0^2+\sqrt-1^2=1$

$ \displaystyle \theta=tan^{-1} \frac{-1}{0}= -\frac{\pi}{2}$

Is the latter correct, even though it is technically undefined, when thinking of the unit circle it equates to $ \displaystyle \frac{\pi}{2}$?

If so, then $ \displaystyle (e^{-i\frac{\pi}{2}})^9=e^-{i\frac{9\pi}{2}}$

and putting back into Cartesian from:

$ \displaystyle cos(-\frac{9\pi}{2})+isin(-\frac{9\pi}{2})$

Which recovers $ \displaystyle Z=0-i1$

Thanks

Edit: - not +


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DrRocket
Post  Post subject: Re: Complex numbers question  |  Posted: Sun Jan 22, 2012 3:43 am
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Prometheus wrote:
Doing a remedial maths course and i just want to check this question.

I need to state the real part $ \displaystyle x$ and the imaginary part $ \displaystyle y$ of:

$ \displaystyle Z=(-i)^9$

I assume they are expecting me to do it by inspection given it's only worth 1 mark, in which case i make it
$ \displaystyle x=0$ and $ \displaystyle y=-1$ since it's already in Cartesian form.

But i wanted to prove it to myself to make sure i understand the concept not just a mechanical application. So i changed it to polar exponential form in order to expand the brackets and i just want to make sure the workings are correct:

$ \displaystyle r=\sqrt0^2+\sqrt-1^2=1$

$ \displaystyle \theta=tan^{-1} \frac{-1}{0}= -\frac{\pi}{2}$

Is the latter correct, even though it is technically undefined, when thinking of the unit circle it equates to $ \displaystyle \frac{\pi}{2}$?

If so, then $ \displaystyle (e^{-i\frac{\pi}{2}})^9=e^-{i\frac{9\pi}{2}}$

and putting back into Cartesian from:

$ \displaystyle cos(-\frac{9\pi}{2})+isin(-\frac{9\pi}{2})$

Which recovers $ \displaystyle Z=0-i1$

Thanks

Edit: - not +


That is correct, but an awfully hard way to do it.

It is a bit easier to note that $ \displaystyle -i^n = -1^ni^n$ and hence $ \displaystyle -i^9 = -1^9i^9= -1 i^8i = -1(i^2)^4i = -1(-1)^4i = -i$

So the real part is 0 and the imaginary part is -1.

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Prometheus
Post  Post subject: Re: Complex numbers question  |  Posted: Sun Jan 22, 2012 1:42 pm
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That's a lot easier, thanks. Love how there can be many ways to get the same answer, though my tendency is to pick the longest. Hopefully that'll change with understanding.


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DrRocket
Post  Post subject: Re: Complex numbers question  |  Posted: Mon Jan 23, 2012 12:24 am
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Prometheus wrote:
That's a lot easier, thanks. Love how there can be many ways to get the same answer, though my tendency is to pick the longest. Hopefully that'll change with understanding.


There is no single right way to solve a mathematical problem. With experience you wil be able to see the shorter and more elegant approaches to a given problem. Usually (not always, but usually) the shortest and most elegant approach is also the most clear.

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