# Finite set

1. Apr 30, 2010

### Somefantastik

1. The problem statement, all variables and given/known data

$$\left\{ e^{n r \pi i}: n \in \textbf{Z} \right\} , r \in \textbf{Q}$$

I'm trying to show that this set is finite.

2. Relevant equations

3. The attempt at a solution

Other than the fact that these points lie on the unit circle in the complex plane, I'm not sure where to start. Any direction would be helpful. clearly there's a way to choose r or n and use periodicity to show a finite set of points for this sets. But I'm not sure how r and n could be chosen.

2. Apr 30, 2010

### CompuChip

Since r is a fraction, you could write it as p/q.

Knowing that $e^{2 \pi i} = 1$, can you show that the set has size ${} \le q$?

3. Apr 30, 2010

### Somefantastik

exp(2*pi*i)? from where did the 2 come?

I know that exp(n*pi*i) = 1 for n integer. if r= p/q, then exp(n*r*pi*i) = exp(2*p*pi*i) if n = 2*q... but no, I don't know why the set has size $$\leq q$$.

Last edited: Apr 30, 2010
4. Apr 30, 2010

### Staff: Mentor

No, that's not true. exp(n*pi*i) alternates between 1 and -1, depending on whether n is even or odd, respectively.

5. Apr 30, 2010

### Somefantastik

well in this case n = 2q which is even right? And exp(2*p*pi*i) = 1 since 2*p is even as well since p is an integer. Later I can set n = 2q+1 to handle the odd cases, but I'm still trying to figure out why the set is finite. so I still need some help please.

6. Apr 30, 2010

### Staff: Mentor

Pick a value for r = p/q, then look at values of exp(n*r*pi*i) for n = 1, 2, 3, and so on. What is it that eventually happens at some value of n and thereafter?

7. Apr 30, 2010

### JSuarez

Just a hint: start with r = 1/q, and plot $e^{nr\pi i}$ in the unit circle. What happens? How could you reduce the the cases where $p\neq 1$ to this one?

8. May 4, 2010

### Somefantastik

got it, thanks everybody.