%5E32.jpeg "(1+i)^32")
Exploring the Power of Complex Numbers: (1+i)^32
This article delves into the intriguing world of complex numbers, specifically focusing on calculating the value of (1+i)^32. We'll explore the concept of De Moivre's Theorem and utilize it to simplify this seemingly complex calculation.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit i, where i² = -1. They are typically written in the form a + bi, where a and b are real numbers.
De Moivre's Theorem
De Moivre's Theorem provides a powerful tool for simplifying calculations involving powers of complex numbers. It states that for any complex number in polar form (r(cos θ + i sin θ)) and any integer n, the following holds true:
(r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ)
This theorem effectively allows us to raise a complex number to a power by simply raising its magnitude to that power and multiplying its angle by the power.
Calculating (1+i)^32
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Convert to Polar Form: We need to express (1+i) in polar form. The magnitude of (1+i) is √2, and its angle is π/4 (45 degrees). Therefore: (1+i) = √2(cos π/4 + i sin π/4)
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Apply De Moivre's Theorem: Now, we can use De Moivre's Theorem to calculate (1+i)^32: (1+i)^32 = (√2(cos π/4 + i sin π/4))^32 = (√2)^32(cos (32 * π/4) + i sin (32 * π/4))
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Simplify: Simplifying the expression: (1+i)^32 = 2^16(cos 8π + i sin 8π) = 2^16(1 + 0i) = 2^16
Conclusion
By utilizing De Moivre's Theorem, we successfully simplified the calculation of (1+i)^32, finding that it equals 2^16. This demonstrates the power and elegance of complex numbers and their role in simplifying complex calculations. While the initial expression may seem daunting, the theorem provides a straightforward path to the solution.