Math Problem Statement

Is it picture 1(on the left) 2(in the middle) or 3 ( on the right)

Is it picture 1 (on the left), 2 (in the middle), or 3 (on the right)?

Solution

The function f(n)=(22+22i)nf(n) = \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \right)^n represents a sequence of complex numbers that lies on the unit circle in the complex plane. Let me break this down step by step:

  1. The term 22+22i\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i is equivalent to eiπ/4e^{i\pi/4} (a point on the unit circle with an angle of π/4\pi/4 radians or 45°).
  2. Raising eiπ/4e^{i\pi/4} to the power nn gives einπ/4e^{in\pi/4}, which represents a point on the unit circle rotated by nπ/4n \cdot \pi/4 radians.
  3. As nn increases, the angle of rotation increases by π/4\pi/4 for each step, creating a sequence of evenly spaced points around the unit circle.

Analysis of the graphs:

  • Graph 1 (Left): It shows a unit circle with 4 points, which corresponds to n=4n = 4, creating quarter-circle rotations.
  • Graph 2 (Middle): It shows a unit circle with 8 points, consistent with n=8n = 8, reflecting eighth-circle rotations.
  • Graph 3 (Right): It shows a unit circle with 16 points, matching n=16n = 16, showing sixteenth-circle rotations.

Since the problem starts from n=1n = 1 and increases nn, the graph on the right (Graph 3) represents the correct pattern as nn grows larger. This is because the number of points doubles with each increase in nn, leading to finer division of the unit circle.

Would you like further clarification? Here are some follow-up questions:

  1. How does the exponential form eiθe^{i\theta} relate to trigonometric functions?
  2. What happens to the graph if 22+22i\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i is replaced with another complex number?
  3. Why does the pattern form a symmetric distribution on the unit circle?
  4. What is the relationship between the power nn and the number of points on the graph?
  5. How could this concept be extended to visualize other complex transformations?

Tip: When visualizing unit circle behavior, remember that the argument (angle) of a complex number controls the rotation, while the magnitude controls the scaling!

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Math Problem Analysis

Mathematical Concepts

Complex numbers
Polar coordinates
Unit circle
Exponential form of complex numbers

Formulas

f(n) = (√2/2 + √2/2 i)^n
Euler's formula: e^(iθ) = cos(θ) + i sin(θ)

Theorems

Unit circle representation of complex numbers
Periodic nature of complex exponentials

Suitable Grade Level

Undergraduate (College Level)