Roots of Unity

• Jun 1st 2011, 02:33 PM
imagemania
Roots of Unity
Hi,
Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
Here's an example:
$\displaystyle z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}$
$\displaystyle z^{6}=e^{\frac{7i\pi}{5}+2\pi in}$
$\displaystyle z=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}$

Where n is an integer, i is the complex number and e an exponential.

Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.

However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?

Hope you can help me :)
• Jun 1st 2011, 03:12 PM
Plato
Quote:

Originally Posted by imagemania
Hi,
Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
Here's an example:
$\displaystyle z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}$
$\displaystyle z^{6}=e^{\frac{7i\pi}{5}+2\pi in}$
$\displaystyle z^{6}=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}$
Where n is an integer, i is the complex number and e an exponential.
Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.
However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?

Can you please explain how this question has anything to do with roots of unity/
It does not make any sense to me.
• Jun 1st 2011, 03:32 PM
imagemania
[Ignore- it double posted for some reason] :\
• Jun 1st 2011, 03:33 PM
imagemania
Well it might not be, but its under the same area in my notes, the section starts of with:
$\displaystyle z=e^{2\pi ik} = 1$ where k is an integer.
$\displaystyle z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}$
Then it extrapolates by bringing in n roots

Oh i also made a mistake there, the last z should not be raised to the 6.
• Jun 1st 2011, 04:39 PM
Plato
Quote:

Originally Posted by imagemania
Well it might not be, but its under the same area in my notes, the section starts of with:
$\displaystyle z=e^{2\pi ik} = 1$ where k is an integer.
$\displaystyle z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}$
Then it extrapolates by bringing in n rootsOh i also made a mistake there, the last z should not be raised to the 6.

That is still nonsense as far as I can read it.
Why don't you simply post a clear question?