Why do we Bother to Find Roots of Unity?

Date: 7/23/96 at 21:25:29
From: Alan Yu
Subject: Why do we Bother to Find Roots of Unity? 

Dr. Math,

As a math teacher, sometimes I think from the students' point of 
view.  Next academic year I am going to touch on complex numbers. One 
of the topics is about roots of unity. The technicality of solving 
those problems may be difficult.  

But the first question in my mind is, why do we bother to find the 
roots of unity?  What is the point in looking at the cube root, the 
fourth root, fifth root, etc. of the unity, just a simple number?  
Any significance there?

I just can't figure out even after searching some libraries. I look 
forward to your suggestions.

Best wishes,
Alan

Date: 7/24/96 at 5:33:54
From: Doctor Pete
Subject: Re: Why do we Bother to Find Roots of Unity? 

I'm not quite sure what you mean by "finding" - do you refer to actual 
computation, such as

     z[3] = -1/2 + i*Sqrt[3]/2
          = Cos[2*Pi/3] + i*Sin[2*Pi/3] ,

or are you more interested in representation and manipulation?  

Either way, there are very good reasons to "find" roots of unity.  For 
example,

     "What regular n-gons are constructible by the classical Greek
      methods; i.e., with straightedge and compass?"
     
This question has an intriguing answer:  When n is precisely the 2^k 
times a product of distinct Fermat primes, for nonnegative integers k 
(recall a Fermat prime is a prime of the form 2^(2^m) + 1). The 
surprising result is that regular 3-, 5-, 17-, 257-, and 65537-gons 
can be constructed using only straightedge and compass. This, in turn, 
is related to the fact that the tools of straightedge and compass 
correspond to the algebraic operations of addition, multiplication, 
the extraction of square roots, and no others.  Therefore, the value 
of cos(2*Pi/17) is expressible as a series of the above operations on 
a finite number of integers. So the computation of the 17th root of 
unity is extremely important in devising an actual method of 
construction, for example. Conversely, if an n(th) root of unity is 
expressible using only these operations, a regular polygon of n sides 
is constructible (so from the value I gave for z[3], the equilateral 
triangle is constructible). The math behind all of this is intimately 
related to Galois theory; see a book on abstract algebra for more.

There are a number of other interesting features of the n(th) roots of 
unity. For example, the function x^n - 1 has as roots the n(th) roots 
of unity. A common method for numerical approximation of these roots 
is Newton's method, generalized to the complex plane. But since there 
are multiple roots for n > 1, the process will converge on different 
roots depending on the initial guess. People familiar with Newton's 
method might be led to believe that the closer the initial guess is to 
a particular root, the more likely its iteration will converge to that 
root. But the story isn't as simple as that - in the complex plane, 
what do the boundaries look like?  That is, if we plot the set of all 
points in the complex plane that, when used as an initial guess, 
converge to one particular root, what will the boundary of that set 
look like?  Indeed, the boundary is not smooth, but fractal.

These are just a few things that are related to n(th) roots of unity; 
perhaps other Math Doctors can add more connections.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/24/96 at 13:50:41
From: Doctor Robert
Subject: Re: Why do we Bother to Find Roots of Unity? 

The reason that we look at the roots of unity is that mathematicians 
desire to know the solution sets to equations.  If we look at the 
equation   x^6 = 1 , mathematicians know that there are 6 roots to 
this equation.  (Only two of them are real roots).  It's nice to know 
the other roots.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/