%5En%2F(1-i)%5En-2.jpeg "(1+i)^n/(1-i)^n-2")
Exploring the Complex Expression: (1+i)^n / (1-i)^n-2
This article delves into the fascinating world of complex numbers and explores the simplification and manipulation of the expression (1+i)^n / (1-i)^n-2.
Understanding the Fundamentals
Before diving into the expression, let's establish a foundation in complex numbers.
- Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).
- Polar form represents a complex number using its magnitude (r) and angle (θ) as r(cosθ + isinθ).
Simplifying the Expression
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Expressing in Polar Form:
- (1+i): The magnitude is √(1^2 + 1^2) = √2 and the angle is tan⁻¹(1/1) = 45°. Therefore, (1+i) = √2(cos45° + isin45°).
- (1-i): The magnitude is √2 and the angle is -45°. Hence, (1-i) = √2(cos(-45°) + isin(-45°)).
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Applying De Moivre's Theorem:
- De Moivre's Theorem states: (cosθ + isinθ)^n = cos(nθ) + isin(nθ).
- We can apply this to our expression:
- (1+i)^n = (√2)^n(cos(45n°) + isin(45n°))
- (1-i)^n = (√2)^n(cos(-45n°) + isin(-45n°))
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Substituting and Simplifying:
- (1+i)^n / (1-i)^n-2 = [(√2)^n(cos(45n°) + isin(45n°))] / [(√2)^(n-2)(cos(-45(n-2)°) + isin(-45(n-2)°))]
- Simplifying the magnitude: (√2)^n / (√2)^(n-2) = (√2)² = 2
- Simplifying the angle: cos(45n°) + isin(45n°) / cos(-45n + 90°) + isin(-45n + 90°) = cos(45n°) + isin(45n°) / sin(45n°) + icos(45n°)
- This can be further simplified using trigonometric identities.
Key Takeaways
The expression (1+i)^n / (1-i)^n-2 can be simplified by:
- Expressing the complex numbers in polar form.
- Applying De Moivre's Theorem.
- Simplifying the magnitude and angle components.
This simplification process showcases the elegance and power of complex numbers and their manipulation through trigonometric identities.
Further Exploration
- Investigate the relationship between the value of 'n' and the resulting complex number.
- Explore the geometric interpretation of this expression on the complex plane.
- Research the applications of complex numbers in various fields, including electrical engineering, quantum mechanics, and fluid dynamics.
By understanding the fundamentals and applying techniques like De Moivre's Theorem, we can navigate the complex world of complex numbers with confidence and delve deeper into their rich and fascinating properties.