%5E50.jpeg "(1+i)^50")
Exploring the Power of Complex Numbers: (1+i)^50
The expression (1+i)^50 might seem daunting at first glance, but with the right tools and understanding of complex numbers, it becomes a manageable and even fascinating problem.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit 'i', where i² = -1. They are written in the form a + bi, where 'a' and 'b' are real numbers.
Key Properties:
- Magnitude: The magnitude (or modulus) of a complex number a + bi is √(a² + b²).
- Angle: The angle (or argument) of a complex number a + bi is the angle it makes with the positive real axis, measured counterclockwise.
Utilizing De Moivre's Theorem
De Moivre's Theorem is a powerful tool for dealing with powers of complex numbers. It states:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
Where:
- n is an integer
- θ is the angle of the complex number
Solving (1+i)^50
-
Represent (1+i) in polar form:
- Magnitude: |1+i| = √(1² + 1²) = √2
- Angle: θ = arctan(1/1) = π/4
Therefore, 1+i = √2 (cos(π/4) + i sin(π/4))
-
Apply De Moivre's Theorem: (√2 (cos(π/4) + i sin(π/4)))^50 = √2^50 (cos(50π/4) + i sin(50π/4))
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Simplify:
- √2^50 = 2^25
- 50π/4 = 12.5π = 6π + 0.5π = 0.5π
Therefore, (1+i)^50 = 2^25 (cos(0.5π) + i sin(0.5π))
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Calculate:
- cos(0.5π) = 0
- sin(0.5π) = 1
Hence, (1+i)^50 = 2^25 (0 + i) = 2^25i
Conclusion
Through the application of De Moivre's Theorem and the understanding of complex numbers, we successfully determined that (1+i)^50 simplifies to 2^25i. This illustrates the power and elegance of complex number manipulation, revealing a hidden simplicity behind what initially appears to be a complex calculation.