(1+i)^3 Demoivre's Theorem

2 min read Jun 16, 2024

DeMoivre's Theorem: Exploring (1 + i)³

DeMoivre's Theorem is a powerful tool for simplifying complex number calculations, especially when dealing with exponentiation. It states that for any complex number in polar form, z = r(cos θ + i sin θ), and any integer n, the following holds true:

zⁿ = rⁿ(cos nθ + i sin nθ)

Let's use this theorem to find the value of (1 + i)³.

1. Convert to Polar Form

First, we need to convert the complex number (1 + i) into its polar form.

Magnitude (r):
- |1 + i| = √(1² + 1²) = √2
Angle (θ):
- θ = tan⁻¹(1/1) = 45° (or π/4 radians)

Therefore, (1 + i) in polar form is √2 (cos 45° + i sin 45°).

2. Apply DeMoivre's Theorem

Now, we can apply DeMoivre's Theorem with n = 3:

(1 + i)³ = [√2 (cos 45° + i sin 45°)]³ = (√2)³ (cos (3 * 45°) + i sin (3 * 45°)) = 2√2 (cos 135° + i sin 135°)

3. Convert back to Rectangular Form

Finally, we can convert the result back to rectangular form:

2√2 (cos 135° + i sin 135°) = 2√2 (-√2/2 + i √2/2) = -2 + 2i

Summary

Using DeMoivre's Theorem, we found that (1 + i)³ = -2 + 2i. This demonstrates how the theorem efficiently handles exponentiation of complex numbers, making calculations much simpler and providing a clearer understanding of the results.