%5Em%3D1.jpeg "(1+i/1-i)^m=1")
Solving the Equation (1+i)/(1-i))^m = 1
This article will explore the equation (1+i)/(1-i))^m = 1, where i is the imaginary unit (√-1) and m is an integer. We will analyze the equation, solve for m, and discuss its implications.
Simplifying the Equation
Firstly, we need to simplify the complex fraction:
(1+i)/(1-i) = (1+i)(1+i)/(1-i)(1+i) = (1+2i+i²)/(1-i²) = (1+2i-1)/(1+1) = i
Therefore, the equation simplifies to i^m = 1.
Finding the Solutions
To solve for m, we need to understand the cyclic nature of powers of i:
- i¹ = i
- i² = -1
- i³ = i² * i = -i
- i⁴ = i² * i² = 1
Notice that the powers of i repeat in a cycle of four. This means that i^m = 1 when m is a multiple of 4.
Therefore, the solutions to the equation are m = 4k, where k is any integer.
Implications and Conclusion
The solution m = 4k tells us that the equation (1+i)/(1-i))^m = 1 holds true for every fourth power of i. This highlights the cyclical nature of powers of imaginary units and their importance in complex number theory.
In essence, this equation and its solution demonstrate how manipulating complex numbers can lead to interesting patterns and relationships. It emphasizes the importance of simplifying expressions and recognizing cyclic patterns when dealing with complex numbers.