Finite Set of Points in Complex Plane: $\{e^{n r \pi i}\}$

In summary, the conversation discusses how to show that the set \left\{ e^{n r \pi i}: n \in \textbf{Z} \right\} , r \in \textbf{Q} is finite. One approach is to choose a value for r = p/q and look at values of exp(n*r*pi*i) for different values of n. It is observed that when r = 1/q, the points lie on the unit circle and eventually repeat. This can be used to show that the set is finite for any value of r.
  • #1
Somefantastik
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0

Homework Statement



[tex] \left\{ e^{n r \pi i}: n \in \textbf{Z} \right\} , r \in \textbf{Q} [/tex]

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

Homework Equations





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.
 
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  • #2
Since r is a fraction, you could write it as p/q.

Knowing that [itex]e^{2 \pi i} = 1[/itex], can you show that the set has size [itex]{} \le q[/itex]?
 
  • #3
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 [tex] \leq q[/tex].
 
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  • #4
Somefantastik said:
exp(2*pi*i)? from where did the 2 come?

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

Somefantastik said:
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 [tex] \leq q[/tex].
 
  • #5
Mark44 said:
No, that's not true. exp(n*pi*i) alternates between 1 and -1, depending on whether n is even or odd, respectively.

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
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
Just a hint: start with r = 1/q, and plot [itex]e^{nr\pi i}[/itex] in the unit circle. What happens? How could you reduce the the cases where [itex]p\neq 1[/itex] to this one?
 
  • #8
got it, thanks everybody.
 

Related to Finite Set of Points in Complex Plane: $\{e^{n r \pi i}\}$

1. What is a finite set of points in the complex plane?

A finite set of points in the complex plane is a collection of points that can be represented by complex numbers, where both the real and imaginary parts are finite. These points are typically denoted by the symbol $\{z_1, z_2, ..., z_n\}$, where $z_n$ is a complex number.

2. What is the significance of $e^{n r \pi i}$ in the set of points?

The expression $e^{n r \pi i}$ represents the complex numbers that lie on the unit circle in the complex plane. This is because the unit circle can be parameterized by the angle $\theta = nr\pi$, and the point on the unit circle at this angle can be represented by the complex number $e^{i\theta}$. As $n$ and $r$ vary, the points on the unit circle vary, resulting in a finite set of points in the complex plane.

3. How is the set of points $\{e^{n r \pi i}\}$ related to trigonometric functions?

The set of points $\{e^{n r \pi i}\}$ is closely related to the trigonometric functions sine and cosine. This is because $e^{i\theta}$ can be expressed as $\cos(\theta) + i\sin(\theta)$. As a result, the points in the set can be written as $\{\cos(nr\pi) + i\sin(nr\pi)\}$, which are the coordinates of points on the unit circle.

4. How can the set of points $\{e^{n r \pi i}\}$ be useful in mathematics?

The set of points $\{e^{n r \pi i}\}$ has many applications in mathematics, particularly in the study of complex analysis and trigonometry. It can be used to represent periodic functions, solve differential equations, and analyze the behavior of complex numbers. It also has connections to other areas of mathematics, such as number theory and geometry.

5. Can the set of points $\{e^{n r \pi i}\}$ be extended to an infinite set?

Yes, the set of points $\{e^{n r \pi i}\}$ can be extended to an infinite set by allowing $n$ and $r$ to take on any real values. This results in an infinite number of points on the unit circle, forming a continuous curve instead of a finite set. This extension is known as the unit circle or complex exponential function, and it has many important applications in mathematics and engineering.

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