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Exploring the Expression (1 + i)^n + (1 - i)^n
This article delves into the fascinating properties of the expression (1 + i)^n + (1 - i)^n, where 'n' is any integer. We'll explore how this expression can be simplified and reveal interesting patterns.
Understanding the Basics
- Complex Numbers: The expression involves complex numbers, where 'i' represents the imaginary unit, defined as √-1.
- De Moivre's Theorem: This theorem is key to simplifying powers of complex numbers in polar form. It states: (cos θ + i sin θ)^n = cos (nθ) + i sin (nθ).
Simplifying the Expression
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Polar Form: Convert (1 + i) and (1 - i) into polar form:
- (1 + i): Magnitude = √(1² + 1²) = √2, Angle = tan⁻¹(1/1) = 45°
- (1 - i): Magnitude = √(1² + (-1)²) = √2, Angle = tan⁻¹(-1/1) = -45°
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Applying De Moivre's Theorem:
- (1 + i)^n = (√2 * (cos 45° + i sin 45°))^n = 2^(n/2) * (cos 45n° + i sin 45n°)
- (1 - i)^n = (√2 * (cos -45° + i sin -45°))^n = 2^(n/2) * (cos -45n° + i sin -45n°)
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Combining the Terms:
- (1 + i)^n + (1 - i)^n = 2^(n/2) * (cos 45n° + i sin 45n° + cos -45n° + i sin -45n°)
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Using Trigonometric Identities: Since cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), the expression simplifies to:
- (1 + i)^n + (1 - i)^n = 2^(n/2) * (2 * cos 45n°)
Analyzing the Result
The final simplified expression 2^(n/2) * 2 * cos 45n° highlights the following:
- Real Value: The expression always evaluates to a real number, as the imaginary component cancels out.
- Periodic Behavior: Due to the cosine function, the value of the expression repeats for different values of 'n'. The period is determined by the angle 45n°.
- Dependence on 'n': The magnitude of the expression is influenced by the power 'n', as indicated by the factor 2^(n/2).
Examples
- n = 1: (1 + i) + (1 - i) = 2
- n = 2: (1 + i)² + (1 - i)² = 0
- n = 3: (1 + i)³ + (1 - i)³ = -4√2
Conclusion
The expression (1 + i)^n + (1 - i)^n offers a captivating exploration of complex numbers and their properties. By leveraging De Moivre's Theorem and trigonometric identities, we can simplify the expression and reveal its real-valued, periodic behavior. Understanding the relationship between the power 'n' and the resulting value allows for further analysis and exploration of this fascinating mathematical concept.