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III Algebra Roots of Unity

 
 
Picture of Tina Jin
III Algebra Roots of Unity
by Tina Jin - Tuesday, May 21, 2024, 10:58 AM
 

Hello,


For roots of unity, the textbook says "1=z^n=r^n(cis(theta))^n=r^ncis(ntheta). It is easy to see that r=1." Why is r=1? Why can't it be 2 or 1/2? It could just be that 1/2=cis((beta)) and you can solve from there right? I don't really get the proof for Roots of Unity.


Thanks!

Tina Jin

 
Picture of John Lensmire
Re: III Algebra Roots of Unity
by John Lensmire - Tuesday, May 21, 2024, 11:51 AM
 

Let's review polar form quick:

Recall $r\mbox{cis}(\theta) = r\cos(\theta) + i\cdot r\sin(\theta)$. What is the modulus of this? It is$$\sqrt{r^2\cos^2(\theta) + r^2\sin^2(\theta)} = \sqrt{r^2} = r.$$Hence if $r\neq 1$, then the modulus of $r^n$ is not $1$ so it can't equal to $1$. This is the main idea why $r=1$ for the roots of unity.

In general this is one of the main advantages of polar form. If we write a complex number as $r\mbox{cis}(\theta)$ then the $r$ (the modulus) and the $\theta$ (the argument) are somewhat independent from each other. The modulus tells us how far away the number is from the origin and the argument gives the angle.

Hence, if we have two complex numbers $r\mbox{cis}(\theta)$ and $s\mbox{cis}(\phi)$, as long as $r$ and $s$ are positive, then the two complex numbers are equal if and only if $r = s$ and $\phi = \theta \pm 2\pi\cdot k$ for some integer $k$. (The angle isn't unique because sine and cosine have period $2\pi$.)

Note: I highly recommend trying to think of the complex numbers visually, drawing out some examples and pictures as needed. Let us know if you have additional questions.