%5E20.jpeg "(1+i)^20")
Exploring the Power of Complex Numbers: (1 + i)^20
The expression (1 + i)^20 might seem intimidating at first, but it can be elegantly solved using the principles of complex numbers. Let's delve into the process.
De Moivre's Theorem - Our Key
The core of solving this problem lies in De Moivre's Theorem. This theorem states:
[cos(θ) + i sin(θ)]^n = cos(nθ) + i sin(nθ)
This theorem provides a powerful tool for raising complex numbers in polar form to a power.
Breaking Down (1 + i)
First, we need to express (1 + i) in its polar form.
- Magnitude: |1 + i| = √(1² + 1²) = √2
- Angle: arctan(1/1) = π/4
Therefore, (1 + i) = √2 [cos(π/4) + i sin(π/4)]
Applying De Moivre's Theorem
Now we can apply De Moivre's Theorem to (1 + i)^20:
(1 + i)^20 = [√2 (cos(π/4) + i sin(π/4))]^20
= (√2)^20 [cos(20π/4) + i sin(20π/4)]
= 2^10 [cos(5π) + i sin(5π)]
= 1024 [-1 + 0i]
= -1024
Conclusion
By utilizing De Moivre's Theorem, we effectively simplified (1 + i)^20 to -1024. This illustrates the elegance and power of complex numbers in solving seemingly complicated expressions.