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Re: Simplying exponents in nth roots of unity.

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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
22 Sep 2013 08:28 PM
Yes, this is related to programming.
Let us define a complex primitive Nth root of unity omega:

ω = cos(theta) + i*sin(theta)
ω = e^(2*pi/N)
By the definition of an nth root of unity, ω is the second solution to the equation x^N = 1.

Can someone explain to me how we can simplify the solutions in exponents via modular arithmetic and whatnot?
I sort of understand the generic congruence modulo nonsense:

ω^M where M > N
M = Q * N + R where Q is some quotient and R is some remainder.
ω^(Q*N + R)
= ω^(Q*N) + ω^R
= ω^R
R is congruent to M(mod N).
I don't exactly understand this. The congruence modulo is not only slightly confusing, the process is rather odd. This is how my quantum mechanics professor explained it.
I need to input 6 values corresponding to ω (those values are irrelevant to the problem), however I can only use ω and ω^2, even though N = 6. How do I get rid of the other exponents if my problem includes ω^3, ω^6, ω^9, ω^12 and ω^15?
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ElectricBlaze is not online. ElectricBlaze
Joined: 18 Jul 2011
Total Posts: 22930
22 Sep 2013 10:11 PM
My response:

2+2=4

yay im smart

http://wiki.roblox.com/index.php/User:ElectricBlaze
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Jetta765214 is not online. Jetta765214
Joined: 22 Oct 2008
Total Posts: 1855
22 Sep 2013 10:16 PM
^What he said
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awsumpwner27 is not online. awsumpwner27
Joined: 03 Sep 2011
Total Posts: 4389
22 Sep 2013 10:24 PM
Sorry, I'm not a arithmaconologist.
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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
22 Sep 2013 10:45 PM
I figured it out.

w^(QN+R)=w^(QN)×w^R=(w^N)Q×w^R=(1)^Q×w^R=w^R

Thank you ElectricBlaze!
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ElectricBlaze is not online. ElectricBlaze
Joined: 18 Jul 2011
Total Posts: 22930
22 Sep 2013 10:46 PM
No problem, always glad to help people with math!

http://wiki.roblox.com/index.php/User:ElectricBlaze
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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
23 Sep 2013 04:38 PM
<8
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Dr01d3k4 is not online. Dr01d3k4
Joined: 11 Oct 2007
Total Posts: 17916
23 Sep 2013 04:39 PM
I did this in my last FP2 lesson, yet I have no idea what you are talking about DX
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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
23 Sep 2013 04:42 PM
I did this in my last quantum mechanics lesson. I barely understand the identities. Complex nth roots of unity are fine self-contained, but manipulating their properties is a vile act of sorcery.
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Notunknown99 is not online. Notunknown99
Joined: 05 Sep 2008
Total Posts: 25360
23 Sep 2013 04:58 PM
"x^N = 1"

N=0?

Yay, I be smrt!
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AbsoluteLOL is not online. AbsoluteLOL
Joined: 01 Dec 2012
Total Posts: 3939
23 Sep 2013 05:08 PM
Screw math.
I hate math.
Math hates me.
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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
23 Sep 2013 05:33 PM
@Notunknown, by the definition of nth roots of unity, there are n complex solutions. If N were 8, then the following would be the solutions:

1, 1/sqrt(2)*i + 1/sqrt(2), (1/sqrt(2)*i + 1/sqrt(2))^2, (1/sqrt(2)*i + 1/sqrt(2))^3, (1/sqrt(2)*i + 1/sqrt(2))^4, (1/sqrt(2)*i + 1/sqrt(2))^5, (1/sqrt(2)*i + 1/sqrt(2))^6, (1/sqrt(2)*i + 1/sqrt(2))^7

I received 1/sqrt(2)*i + 1/sqrt(2) from the form w = e^((theta*pi*i)/N), which expands and exchanges to cos(pi/4) + i*sin(pi/4).
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Absurdism is not online. Absurdism
Joined: 18 Jul 2013
Total Posts: 2568
23 Sep 2013 05:34 PM
Oh, right: 0 is considered an invalid answer.
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