# Fifth roots of unity help

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Let p = w + w^4 and q = w^2 + w^3, where w = e^2*pi*i/5. Show that p + q = -1 and and pq = -1.

Okay, so my workings out are as follows:

p + q = (w + w^4) + w^2 + w^3

= w + w^4 + w(w + w^2)

= w + w^4 - w

= w^4

w^4 = e^8*pi*i/5

plug x = 8pi/5 into (cos x + i sin x), expecting to get an answer of -1

nope, 0.30902 - 0.95106i.

Where have I gone wrong?

Okay, so my workings out are as follows:

p + q = (w + w^4) + w^2 + w^3

= w + w^4 + w(w + w^2)

= w + w^4 - w

= w^4

w^4 = e^8*pi*i/5

plug x = 8pi/5 into (cos x + i sin x), expecting to get an answer of -1

nope, 0.30902 - 0.95106i.

Where have I gone wrong?

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#2

(Original post by

Let p = w + w^4 and q = w^2 + w^3, where w = e^2*pi*i/5. Show that p + q = -1 and and pq = -1.

Okay, so my workings out are as follows:

p + q = (w + w^4) + w^2 + w^3

= w + w^4 + w(w + w^2)

= w + w^4 - w

= w^4

w^4 = e^8*pi*i/5

plug x = 8pi/5 into (cos x + i sin x), expecting to get an answer of -1

nope, 0.30902 - 0.95106i.

Where have I gone wrong?

**psycholeo**)Let p = w + w^4 and q = w^2 + w^3, where w = e^2*pi*i/5. Show that p + q = -1 and and pq = -1.

Okay, so my workings out are as follows:

p + q = (w + w^4) + w^2 + w^3

= w + w^4 + w(w + w^2)

= w + w^4 - w

= w^4

w^4 = e^8*pi*i/5

plug x = 8pi/5 into (cos x + i sin x), expecting to get an answer of -1

nope, 0.30902 - 0.95106i.

Where have I gone wrong?

This is unjustified.

Instead, use the fact that

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(Original post by

How do you know that ?? (Going from second line of working to the third)

This is unjustified.

Instead, use the fact that

**RDKGames**)How do you know that ?? (Going from second line of working to the third)

This is unjustified.

Instead, use the fact that

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#4

(Original post by

I thought that what true? Or is that just for cube roots of unity?

**psycholeo**)I thought that what true? Or is that just for cube roots of unity?

If then hence hence hence .

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Sorry, how do you know that

And where does the 1 come from?

__[img=273x45]https://www.thestudentroom.co.uk/latexrender/pictures/6a/6aa76ae7212129196979be5d4bcc8bdd.png[/img]__And where does the 1 come from?

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#6

Anyhow, if you're not comfortable with this result, I suggest you learn it and its proof.

https://math.stackexchange.com/quest...x-1xn-1xn-2-x1

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(Original post by

Well to me at A-Level it was given and proven that so I would expect you to have covered it??

Anyhow, if you're not comfortable with this result, I suggest you learn it and its proof.

https://math.stackexchange.com/quest...x-1xn-1xn-2-x1

**RDKGames**)Well to me at A-Level it was given and proven that so I would expect you to have covered it??

Anyhow, if you're not comfortable with this result, I suggest you learn it and its proof.

https://math.stackexchange.com/quest...x-1xn-1xn-2-x1

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#8

(Original post by

Are there other ways to show it without using that? That's not taught to us and where did you get the +1 from when p + q = w + w^2 + w^3 + w^4

**psycholeo**)Are there other ways to show it without using that? That's not taught to us and where did you get the +1 from when p + q = w + w^2 + w^3 + w^4

I'm fairly sure this is the way they'd want you to do it. What is your exam board? I can't see a different nice way about it.

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(Original post by

Just rewrite it as then.

I'm fairly sure this is the way they'd want you to do it. What is your exam board? I can't see a different nice way about it.

**RDKGames**)Just rewrite it as then.

I'm fairly sure this is the way they'd want you to do it. What is your exam board? I can't see a different nice way about it.

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#10

(Original post by

But surely it's show so you can't use p + q = -1 to show p + q = -1. And it's from NEAB June 1998

**psycholeo**)But surely it's show so you can't use p + q = -1 to show p + q = -1. And it's from NEAB June 1998

We begin with in terms of and proeed with logical steps to get down to -1.

If you do not want to use my approach, then you can write out:

and simplify from there.

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Are you sure there's no other way without using the formula you have given?

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#12

(Original post by

Are you sure there's no other way without using the formula you have given?

**psycholeo**)Are you sure there's no other way without using the formula you have given?

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(Original post by

Look at my edit.

**RDKGames**)Look at my edit.

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#14

(Original post by

Thank you for the help (even though you help with all of my student room threads haha). Out of interest once you've put the ws into cos and i sin form, as shown in the first line of your workings. Couldn't you just put it in a calculator and show that equals -1

**psycholeo**)Thank you for the help (even though you help with all of my student room threads haha). Out of interest once you've put the ws into cos and i sin form, as shown in the first line of your workings. Couldn't you just put it in a calculator and show that equals -1

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(Original post by

Dunno, check the mark scheme. If they're posing a question like this when I'd imagine they want you to do a bit more than to just put something into a calculator.

**RDKGames**)Dunno, check the mark scheme. If they're posing a question like this when I'd imagine they want you to do a bit more than to just put something into a calculator.

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#16

Knowledge of geometric series is unnecessary.

Note that .

So

You can use this to deduce something about .

For the second bit about pq, the expansion is quick and easy and you'll get four terms. But the line of thinking that might be helpful from there is something like:

.

Note that .

So

You can use this to deduce something about .

For the second bit about pq, the expansion is quick and easy and you'll get four terms. But the line of thinking that might be helpful from there is something like:

.

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