%5E10000.jpeg "(1+i)^10000")
Exploring the Complex Power: (1 + i)^10000
The expression (1 + i)^10000 presents a fascinating challenge in complex number arithmetic. While directly calculating this power might seem daunting, we can employ clever techniques to arrive at a simplified and understandable solution.
De Moivre's Theorem: The Key to Unlocking the Power
De Moivre's Theorem provides a powerful tool for simplifying expressions involving complex numbers raised to a power. 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:
z^n = r^n(cos nθ + i sin nθ)
This theorem allows us to efficiently calculate powers of complex numbers by manipulating their magnitude and angle.
Applying De Moivre's Theorem to (1 + i)^10000
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Converting to Polar Form: We begin by expressing (1 + i) in polar form. Its magnitude is |1 + i| = √(1² + 1²) = √2. The angle, θ, is found using arctangent: θ = arctan(1/1) = π/4. Therefore, (1 + i) in polar form is √2(cos π/4 + i sin π/4).
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Applying the Theorem: Applying De Moivre's Theorem, we get:
(1 + i)^10000 = (√2)^10000 (cos(10000 * π/4) + i sin(10000 * π/4))
- Simplifying: Since (√2)^10000 = 2^5000, we have:
(1 + i)^10000 = 2^5000 (cos(2500π) + i sin(2500π))
- Final Result: Recognizing that 2500π is a multiple of 2π, the cosine and sine terms simplify to:
(1 + i)^10000 = 2^5000 (1 + 0i) = 2^5000
Conclusion
We have successfully determined that (1 + i)^10000 equals 2^5000, a surprisingly simple result considering the initial complexity of the expression. This demonstrates the power of De Moivre's Theorem in simplifying calculations involving complex numbers. It highlights the elegance and efficiency of working with these numbers in polar form, revealing hidden patterns and providing insightful solutions.