A level further maths Complex numbers

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shivsaransh1
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When w is the 4th root of unity
Why is 1 + w +w^2 +w^3 = 0?

I read this in my textbook but they didnt expand on it could someone please explain it?
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TSRTD
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when w is the 4th root of unity the 4 roots are 1,-1,i,-i. If you take the roots -1,i,-i as the roots of a cubic equation then the resulting equation is 1 + w +w^2 +w^3 = 0.
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old_engineer
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(Original post by shivsaransh1)
When w is the 4th root of unity
Why is 1 + w +w^2 +w^3 = 0?

I read this in my textbook but they didnt expand on it could someone please explain it?
...and another take on this is that 1 + w + w^2 + w^3 is a geometric series summed over the first four terms. Try applying the standard series sum formula.
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B_9710
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Just to add if you have the nth roots of unity  1, \omega,...,\omega^{n-1} then  \sum_{i=0}^{n-1} \omega ^i =1 .
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shivsaransh1
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(Original post by B_9710)
Just to add if you have the nth roots of unity  1, \omega,...,\omega^{n-1} then  \sum_{i=0}^{n-1} \omega ^i =1 .
When you use the series equation you end up getting W^4 -1/ w - 1 why does W^ 4 evaluate to zero? Its that part that I dont seem to understand
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mqb2766
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(Original post by shivsaransh1)
When you use the series equation you end up getting W^4 -1/ w - 1 why does W^ 4 evaluate to zero? Its that part that I dont seem to understand
w^4 = 1
Or
0 = w^4-1 = (w-1)(w^3+w^2+w+1)

w^4 is not 0.

Using the gp series
1+w+w^2+w^3 = (w^4-1)/(w-1)
As above, which is another way of deriving/noticing the factorization. It does not say what w^4 equals as you don't equate it to zero (numerator on right). It sums an arbitrary gp which could equal any value.
For a root of unity w^4=1, so the right hand side is 0.
Last edited by mqb2766; 6 months ago
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shivsaransh1
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O

(Original post by mqb2766)
w^4 = 1
Or
0 = w^4-1 = (w-1)(w^3+w^2+w+1)

w^4 is not 0.

Using the gp series
1+w+w^2+w^3 = (w^4-1)/(w-1)
As above, which is another way of deriving/noticing the factorization. It does not say what w^4 equals as you don't equate it to zero (numerator on right). It sums an arbitrary gp which could equal any value.
For a root of unity w^4=1, so the right hand side is 0.
Sorry I meant why does w^4 evaluate to one making w^4 -1= 0.
What I am asking for a root of unity why is w^4 =1?
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mqb2766
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(Original post by shivsaransh1)
O


Sorry I meant why does w^4 evaluate to one making w^4 -1= 0.
What I am asking for a root of unity why is w^4 =1?
w^4 = 1
w = root(1)
Where root is the fourth root.

The square root of 1 corresponds to solving
w^2 = 1
Last edited by mqb2766; 6 months ago
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