complex numbers problems

[instigator]

Hello everybody,

I'm having some problems with some problems about complex numbers and I thought that you could help me.

This one is about the nth roots of unity.

1. I know that the four 4th roots of unity are 1, i, -1 and -i. I know need to determine if it is possible to write them as 1, w, w ² and w³ where w is defined as cis pi/2.

And then I need to show that 1+w+w²+w³ = 0.


2. The second problem is also about this topic. I need to find the 5th roots of unity and then I have to display them on an Argand diagram. The second part of this is assuming that w is the root with the smallest positive argument, I need to show that the roots are 1, w, w², w³ and w. This will probably be very similar to the first problem.

3. Third: w=x+yi and P(x,y) move in the complex plane, I need the cartesian equation for |w-i| = |w+1+i| [note: |w-i| means modulus of w-i]

4. Next: z1= cos(pi/6) + isin(pi/6) and z2 = cos(pi/4) + isin(pi/4). What is the expression for (z1/z2) in the form of z=a+bi.

5. And the last: What is the fifth root of i ?


Sorry if it just seems too easy for you, but unfortunately for me it's not. Thanks for all your help.

 

[Deleted member 4993]

Hello everybody,

I'm having some problems with some problems about complex numbers and I thought that you could help me.

This one is about the nth roots of unity.

1. I know that the four 4th roots of unity are 1, i, -1 and -i. I know need to determine if it is possible to write them as 1, w, w ² and w³ where w is defined as cis pi/2.

And then I need to show that 1+w+w²+w³ = 0.


2. The second problem is also about this topic. I need to find the 5th roots of unity and then I have to display them on an Argand diagram. The second part of this is assuming that w is the root with the smallest positive argument, I need to show that the roots are 1, w, w², w³ and w. This will probably be very similar to the first problem.

3. Third: w=x+yi and P(x,y) move in the complex plane, I need the cartesian equation for |w-i| = |w+1+i| [note: |w-i| means modulus of w-i]

4. Next: z1= cos(pi/6) + isin(pi/6) and z2 = cos(pi/4) + isin(pi/4). What is the expression for (z1/z2) in the form of z=a+bi.

5. And the last: What is the fifth root of i ?


Sorry if it just seems too easy for you, but unfortunately for me it's not. Thanks for all your help.

Please show us your work - and indicate exactly where you are stuck - so that we would know where to begin to help you.

Have you reviewed the example problems in your book?

 

[pka]

Here is aome guide on #1.
\(\displaystyle \begin{array}{l} \omega = cis\left( {\frac{\pi }{2}} \right) = i \\ \omega ^2 = cis\left( \pi \right) = - 1 \\ \omega ^3 = cis\left( {\frac{{3\pi }}{2}} \right) = - i \\ \omega ^4 = cis\left( {2\pi } \right) = 1 \\ \omega + \omega ^2 + \omega ^3 + 1 = ? \\ \end{array}\)

 

[Deleted member 4993]

for #1 - use the fact that:

\(\displaystyle 1 =e^{(i \cdot\ 2n\pi\ )}\)

 

[PAULK]

Re: complex numbers, roots of unity

You asked to prove that

1 + w + w^2 + w3 = 0, where w is the fourth root of unity.

Generally, we can show that if w is any n-th root of unity (except 1), then

1 + w + w^2 + ... + w^(n-1) = 0

Factor this polynomial:

w^n - 1 = (w - 1)(w^(n-1) + ... + w^2 + w + 1)

Now if w is an n-th root of unity, except 1, then the LHS is zero. But the first factor on the right, w-1, is not zero, so the second factor must be.