Proving/Creating a conjecture on the roots of complex numbers

1. Oct 22, 2012

Daaniyaal

1. The problem statement, all variables and given/known data
Formulate a conjecture for the equation (z^3)-1=0, (z^4)-1=0 (z^5)-1=0
and prove it.

2. Relevant equations
r^n(cosnθ + isinnθ)

3. The attempt at a solution

Well my conjecture is that 2pi/n and 2pi/n + pi are possible values. I'm a bit iffy on how to word it. don't know which way I should go for a proof

2. Oct 22, 2012

tiny-tim

Hi Daaniyaal!

(try using the X2 button just above the Reply box )
Start "if rn(cosnθ + isinnθ) = 1, then …"

3. Oct 23, 2012

Daaniyaal

if rn(cosnθ + isinnθ)=1 then possible values for n are 2pi/n and 2pi/n+pi.

4. Oct 23, 2012

tiny-tim

i was thinking, more like …

if rn(cosnθ + isinnθ) = 1,

then r = 1, and cosnθ = 1,

so nθ = … ?

5. Oct 23, 2012

Daaniyaal

0 most definitely

6. Oct 23, 2012

Staff: Mentor

Aren't there other values for which cos(nθ) = 1?

7. Oct 23, 2012

Daaniyaal

2/3pi

8. Oct 23, 2012

Daaniyaal

By the way how would I put capital pi notation into this?

My conjecture is sqrt(2-2coskpi/n)) in pi notation: n-1, k=1

9. Oct 23, 2012

Staff: Mentor

???
How did you get that?

10. Oct 23, 2012

Staff: Mentor

What do you mean by "capital pi notation?"

11. Oct 24, 2012

tiny-tim

(just got up :zzz:)
Come on, Daaniyaal …

what are all the solutions to cosψ = 1 ?​

(draw a graph of cos)

12. Oct 24, 2012

Daaniyaal

13. Oct 24, 2012

Daaniyaal

I think an alternate form of writing this would be e^(i x) = 1

14. Oct 24, 2012

Daaniyaal

and I can't get the superscript button to work :(

15. Oct 24, 2012

tiny-tim

correct … ψ = 2π times n , for any value of n

but since we're already using n in the question, let's write that as …

ψ = 2π times k , for any value of k

sooo … what are all the solutions (for θ) of cos = 1 ?
yes
do you have javascript turned off?

[noparse]alternatively, you can just type before and after[/noparse]

16. Oct 24, 2012

Daaniyaal

2,4,6,8,10,12 and so on?

17. Oct 24, 2012

Daaniyaal

All even numbers basically

18. Oct 24, 2012

tiny-tim

θ = 2,4,6,8,10,12 … are the solutions to cosnθ = 1 ?

19. Oct 24, 2012

Staff: Mentor

This notation - $\Pi_{i =1}^n a_n$ - has nothing to do with what you're doing. It represents the product a1*a2* ... * an.

20. Oct 24, 2012

Staff: Mentor

Daaniyaal, it would be helpful if you replied with complete statements. tiny-tim is trying to get you to do that with what he says below.