%5E20.jpeg "(1-i/1+i)^20")
Exploring the Power of Complex Numbers: (1-i/1+i)^20
This article delves into the fascinating world of complex numbers and explores the intriguing expression (1-i/1+i)^20. We'll unravel the process of simplifying this expression and gain valuable insights into the properties of complex numbers.
Understanding the Basics
Before we embark on this journey, let's revisit some fundamental concepts about complex numbers:
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Complex Numbers: Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
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Polar Form: Complex numbers can be represented in polar form, using their magnitude (distance from origin) and angle (direction). This representation offers a convenient way to understand their multiplication and division.
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De Moivre's Theorem: This crucial theorem states that for any complex number z = r(cos θ + i sin θ) and any integer n, we have z^n = r^n(cos nθ + i sin nθ). This theorem simplifies the calculation of powers of complex numbers.
Simplifying (1-i/1+i)^20
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Rationalizing the Denominator: Begin by simplifying the base of the expression. We rationalize the denominator by multiplying both numerator and denominator by the conjugate of the denominator: (1-i) / (1+i) * (1-i) / (1-i) = (1 - 2i + i^2) / (1 - i^2) Since i^2 = -1, we obtain: (1 - 2i - 1) / (1 + 1) = -2i / 2 = -i
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Expressing in Polar Form: We convert -i into polar form: Magnitude: |-i| = 1 Angle: arg(-i) = -π/2 Therefore, -i = 1(cos(-π/2) + i sin(-π/2))
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Applying De Moivre's Theorem: Now, we can apply De Moivre's theorem to find the 20th power: (-i)^20 = [1(cos(-π/2) + i sin(-π/2))]^20 = 1^20(cos(-20π/2) + i sin(-20π/2))
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Simplifying:
cos(-20π/2) = cos(-10π) = 1 sin(-20π/2) = sin(-10π) = 0 Therefore, (-i)^20 = 1(1 + i * 0) = 1
Conclusion
We have successfully simplified the expression (1-i/1+i)^20 to 1. This journey highlights the power of complex number manipulation, showcasing how understanding their properties, particularly De Moivre's theorem, allows us to efficiently compute powers of complex numbers. This result demonstrates the intriguing nature of complex numbers, where even seemingly complex expressions can be reduced to simple results through the application of fundamental concepts.