%5E500.jpeg "(1-i/1+i)^500")
Exploring the Power of Complex Numbers: (1-i/1+i)^500
This article delves into the fascinating world of complex numbers, specifically focusing on the expression (1-i/1+i)^500. We'll explore how to simplify this expression and arrive at a surprising result.
Simplifying the Expression
First, let's simplify the fraction (1-i)/(1+i) using the concept of complex conjugates:
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Multiplying by the complex conjugate:
(1-i)/(1+i) * (1-i)/(1-i) = (1 - 2i + i^2) / (1 - i^2)
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Simplifying using i^2 = -1:
(1 - 2i -1) / (1 + 1) = -2i / 2 = -i
Now, our expression becomes (-i)^500.
Understanding the Cyclic Nature of Powers of 'i'
The key to solving this lies in the cyclical nature of powers of the imaginary unit 'i':
- i^1 = i
- i^2 = -1
- i^3 = -i
- i^4 = 1
Notice that the cycle repeats every four powers.
Finding the Solution
Since 500 is divisible by 4, we can write:
(-i)^500 = ((-i)^4)^125 = 1^125 = 1
Therefore, (1-i/1+i)^500 simplifies to 1. This demonstrates that seemingly complex expressions can be reduced to surprisingly simple results with the understanding of basic mathematical principles.