Question

Consider the following sets of points in the complex plane \[ A = \left\{ \cos \left( \frac{2n\pi}{5} \right) + i \sin \left( \frac{2n\pi}{5} \right) : n \in \mathbb{Z} \right\} \text{ and} \] \[ B = \left\{ \cos \left( \frac{2n}{5} \right) + i \sin \left( \frac{2n}{5} \right) : n \in \mathbb{Z} \right\} . \] Which of the following statements is TRUE?

• A. $A$ is finite but $B$ is infinite.
• B. $A$ is finite and $B$ is also finite.
• C. $A$ is infinite but $B$ is finite.
• D. $A$ is infinite and $B$ is also infinite.

Hint:

An exponential expression of the form $e^{i \alpha n}$ (where $n \in \mathbb{Z}$) is periodic and represents a finite set of points if and only if $\alpha/\pi$ is a rational number.
If $\alpha/\pi$ is irrational, the points are dense on the unit circle and form an infinite set.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two sets of complex numbers, $A$ and $B$, represented in polar form.
The elements of these sets are generated by varying the parameter $n$ over all integers $\mathbb{Z}$.
We need to determine if each set contains a finite or an infinite number of unique elements.

Step 2: Key Formula or Approach:
Euler's formula states that:
\[ e^{i\theta} = \cos \theta + i \sin \theta \]
Using this, we can rewrite the sets as:
\[ A = \left\{ e^{i \theta_n} : \theta_n = \frac{2n\pi}{5}, n \in \mathbb{Z} \right\} \]
\[ B = \left\{ e^{i \phi_n} : \phi_n = \frac{2n}{5}, n \in \mathbb{Z} \right\} \]
Two angles $\theta_1$ and $\theta_2$ yield the same complex number on the unit circle if and only if their difference is an integer multiple of $2\pi$.

Step 3: Detailed Explanation:

• Let us analyze set $A$ first:
The angle is given by $\theta_n = \frac{2n\pi}{5}$.
For two integers $n_1$ and $n_2$ to represent the same complex point:
\[ \theta_{n_1} - \theta_{n_2} = 2k\pi \quad (\text{where } k \in \mathbb{Z}) \]
\[ \frac{2n_1\pi}{5} - \frac{2n_2\pi}{5} = 2k\pi \]
Dividing both sides by $2\pi$:
\[ \frac{n_1 - n_2}{5} = k \implies n_1 - n_2 = 5k \]
This indicates that the values repeat with a period of $5$.
Thus, the set $A$ contains exactly $5$ unique elements corresponding to $n = 0, 1, 2, 3, 4$:
\[ A = \left\{ 1, e^{i2\pi/5}, e^{i4\pi/5}, e^{i6\pi/5}, e^{i8\pi/5} \right\} \] Hence, $A$ is a finite set.

• Let us now analyze set $B$:
The angle is given by $\phi_n = \frac{2n}{5}$.
For two integers $n_1$ and $n_2$ to represent the same complex point:
\[ \phi_{n_1} - \phi_{n_2} = 2k\pi \quad (\text{where } k \in \mathbb{Z}) \]
\[ \frac{2n_1}{5} - \frac{2n_2}{5} = 2k\pi \]
\[ \frac{2(n_1 - n_2)}{5} = 2k\pi \implies n_1 - n_2 = 5k\pi \]
Since $n_1, n_2,$ and $k$ must be integers, and $\pi$ is an irrational number, this equality can hold if and only if $k = 0$.
This implies $n_1 = n_2$.
Thus, no two distinct integers $n_1$ and $n_2$ can produce the same complex number in set $B$.
Hence, all elements in $B$ for different $n \in \mathbb{Z}$ are distinct, making $B$ an infinite set.

Step 4: Final Answer:
Thus, $A$ is finite and $B$ is infinite.