This probably is a simple question but i can't do it (1 Viewer)

[Cheezy-G]

I need help with this question.

Find the three cube roots of unity in mod-arg form and plot them on an Argand diagram. I've done this part but i can't do i)




n
i) E w^r = 0,1 or -w^2
r =0

Any help would be great
Thanks

 

[polythenepam]

I need help with this question.

Find the three cube roots of unity in mod-arg form and plot them on an Argand diagram. I've done this part but i can't do i)




n
i) E w^r = 0,1 or -w^2
r =0

Any help would be great
Thanks

i)




n
i) E w^r = 0,1 or -w^2
r =0
????
wat do u mean by this??

 

[Cheezy-G]

sorry
E means the sum of (like the sigma sign) from r = 0 to n of w to the power of r which equals 0, 1 or -w squared

I don't know if u'll get it this time but its the best i could come up with.
I tried copying and pasting it but the symbols won't come up

 

[who_loves_maths]

^ this is a good question... but not difficult to do if you expand out the Sigma sign:

E (w^r) from r = 0 to r = n ; also 'n' is positive integral.

--->(let 'S' denote the sum) S = w^0 + w^1 + w^2 + .... w^n

but note that w^3 =1 in this case, since 'w' denotes the complex cube root of unity with the smallest positive argument.

using this fact, you re-write the series into this:

S = 1 + w + w^2 + 1 + w + w^2 + .... w^n

note how the pattern of three repeat IN THAT ORDER up to the term 'w^n', where there are (n +1) terms in the series.

so 1) if '(n+1)' is divisible by 3, then you shorten the series to:

S = ((n+1)/3)(1 + w + w^2) .

but we know that (1 + w + w^2) = 0 in this case ;
hence, when '(n+1)' is divisible by 3, S = 0.


2) if (n+1) mod(3) = 1 {ie. '(n+1)' divided by 3 gives a remainder of 1}

then you have: S = (n/3)(1 + w + w^2) + 1 = 0 + 1 = 1

hence, when (n+1) mod(3) = 1, S =1.


3) if (n+1) mod(3) = 2 {ie. '(n+1)' divided by 3 gives a remainder of 2}

then you have: S = ((n-1)/3)(1 + w + w^2) + (1 + w) = 0 + (1 + w) = 1 + w

but since (1 + w + w^2) = 0 , then (1 + w) = -w^2 ;
hence, when (n+1) mod(3) = 2, S = -w^2.


so in conclusion: E (w^r) from r = 0 to r = n is equal to {0, 1, -w^2} depending on the value of the integer 'n'.


hope that helps


Edit: replaced 'n' by '(n+1)' in this post thanks to the vigilance of polythenepam (in the post below).

 

[polythenepam]

actually isnt it if n+1 is divisible by 3? cos it starts with n=0? n then if n is divided by 3 then u hav a remainder of 1 and so on??

 

[who_loves_maths]

Originally Posted by polythenepam
actually isnt it if n+1 is divisible by 3? cos it starts with n=0? n then if n is divided by 3 then u hav a remainder of 1 and so on??

haha... yes, nicely picked up polythenepam. my bad, i rushed my answer, so didn't get time to properly think it over.
but thankyou for pointing that out

 

[Cheezy-G]

Thankyou

Thanks. That helped so much!