%5E2022.jpeg "(1+i)^2022")
Exploring the Power of Complex Numbers: (1 + i)^2022
The expression (1 + i)^2022 might seem intimidating at first glance. However, understanding the properties of complex numbers allows us to elegantly simplify this seemingly complex calculation.
Complex Numbers and De Moivre's Theorem
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (i² = -1). They can be visualized as points in a complex plane, with the real part ('a') represented on the horizontal axis and the imaginary part ('b') on the vertical axis.
De Moivre's Theorem provides a powerful tool for calculating powers of complex numbers in polar form. It states that for any complex number in polar form, z = r(cos θ + i sin θ), and any integer n:
(z)^n = r^n (cos nθ + i sin nθ)
Applying De Moivre's Theorem
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Polar Form: We first need to express (1 + i) in polar form. The magnitude of (1 + i) is √(1² + 1²) = √2. The angle θ can be found using arctan(1/1) = 45°. Therefore, (1 + i) in polar form is √2 (cos 45° + i sin 45°).
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Applying De Moivre's Theorem: Using De Moivre's theorem, we can calculate (1 + i)^2022 as follows:
(1 + i)^2022 = (√2 (cos 45° + i sin 45°))^2022 = (√2)^2022 (cos (2022 * 45°) + i sin (2022 * 45°))
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Simplifying:
- (√2)^2022 = 2^1011
- 2022 * 45° = 91000° This is a large angle, but we can reduce it by subtracting multiples of 360° to get an equivalent angle within 0° to 360°. 91000° - 252 * 360° = 180°.
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Final Result:
(1 + i)^2022 = 2^1011 (cos 180° + i sin 180°) = -2^1011
Conclusion
By utilizing De Moivre's Theorem and understanding the properties of complex numbers, we have successfully calculated the value of (1 + i)^2022. This seemingly complex calculation becomes manageable with the right approach, demonstrating the elegance and power of complex number theory.