patterns from complex numbers

Printable View

April 28th 2011, 02:21 AM
domenfrandolic

patterns from complex numbers

i have been given to obtain solutions to z^n=i for n=3,4,5 by using the Moivre's Theorem.

then the teacher is saying you need to prove and generalize your results for z^n= a+bi, where |a+bi|=1
i dont know what do to...any ideas would be much appreciated...
April 28th 2011, 06:47 AM
TheEmptySet

Quote:

Originally Posted by domenfrandolic

i have been given to obtain solutions to z^n=i for n=3,4,5 by using the Moivre's Theorem.

then the teacher is saying you need to prove and generalize your results for z^n= a+bi, where |a+bi|=1
i dont know what do to...any ideas would be much appreciated...

Note that

$|a+bi|=1 \iff z = e^{i\theta+2\pi k}, \theta \in \mathbb{R},k \in \mathbb{Z}$

So if that is the case then

$z^n= e^{i\theta+2\pi k} \iff ...$

Hint: use your expoential laws and there are n solutions!
April 28th 2011, 08:36 AM
domenfrandolic

hmm..yeah..but then how can i go further...basically..

z^n= a+bi, where |a+bi|=1...this is what i need to prove...by using moivre's theorem...

sorry if i bother you again..but i did not quite get how to solve..

thanks
April 28th 2011, 08:45 AM
Plato

Quote:

Originally Posted by domenfrandolic

hmm..yeah..but then how can i go further...basically..
z^n= a+bi, where |a+bi|=1...this is what i need to prove...by using moivre's theorem...

Frankly, I do not understand what the original question is asking you to show. You see, we do not know what you have shown in previous problems.

What do you think you are to show?
April 28th 2011, 10:52 AM
topsquark

Quote:

Originally Posted by domenfrandolic

i have been given to obtain solutions to z^n=i for n=3,4,5 by using the Moivre's Theorem.

then the teacher is saying you need to prove and generalize your results for z^n= a+bi, where |a+bi|=1
i dont know what do to...any ideas would be much appreciated...

Perhaps we should start at the beginning. You had to obtain solutions for z^n = i for n = 3, 4, and 5 using De Moivre's theorem. What have you been able to do with this part?

-Dan
April 28th 2011, 11:46 AM
domenfrandolic

OK..so basically...at the beginning of the task i had to obtain solutions by moivre's theorem for the equation z^n=1...find a pattern...so i came up with the conjecture that works for it..it was quite easy....now...i am stuck in obtaining solutions to solve Z^n=i for n=3,4,5 by using moivre's theorem...it says...represent the solutions on the argand diagram and the generalize and prove your result for z^n=a+bi, where |a+bi|=1...finally..what happens when |a+bi|≠1..this is the whole task..but is quite ridiculous...the teacher is gonna mark it...and does not care much....and it will count for the final mark...dont really know how to do it...thanks that you are here..
April 28th 2011, 01:18 PM
topsquark

Quote:

Originally Posted by domenfrandolic

OK..so basically...at the beginning of the task i had to obtain solutions by moivre's theorem for the equation z^n=1...find a pattern...so i came up with the conjecture that works for it..it was quite easy....now...i am stuck in obtaining solutions to solve Z^n=i for n=3,4,5 by using moivre's theorem...it says...represent the solutions on the argand diagram and the generalize and prove your result for z^n=a+bi, where |a+bi|=1...finally..what happens when |a+bi|≠1..this is the whole task..but is quite ridiculous...the teacher is gonna mark it...and does not care much....and it will count for the final mark...dont really know how to do it...thanks that you are here..

(sighs) Actually, as of now you probably won't thank me that I'm here. We cannot help you with problems that count toward the final grade.

Thread closed.

-Dan