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 Concept:
Complex numbers of the form \( e^{i\theta} \) represent points on the unit circle.
The set is finite if and only if the values repeat after a certain integer $n$ (periodicity).

Step 2: Detailed Explanation:
$\bullet$ Set A: The angle is $\theta_{n} = \frac{2n\pi}{5}$.
Values are identical if $\theta_{n} - \theta_{m} = 2k\pi$.
$\frac{2\pi}{5}(n - m) = 2k\pi \implies \frac{n - m}{5} = k \implies n - m = 5k$.
This set repeats every 5 integers ($n=0, 1, 2, 3, 4$). Thus, A contains 5 unique points (roots of unity). A is finite.
$\bullet$ Set B: The angle is $\phi_{n} = \frac{2n}{5}$.
Condition for repetition: $\frac{2}{5}(n - m) = 2k\pi \implies n - m = 5k\pi$.
Since $n, m, k$ are integers and $\pi$ is irrational, this equality can only hold if $k=0$ and $n=m$.
No two different integers $n$ give the same point. Thus, B contains infinite unique points.

Step 3: Final Answer:
A is finite; B is infinite.
This matches option (A).