%5En%3D1.jpeg "(1+i/1-i)^n=1")
Solving the Equation (1 + i / 1 - i)^n = 1
This article explores the equation (1 + i / 1 - i)^n = 1, where 'i' is the imaginary unit (√-1) and 'n' is an integer. We aim to determine the values of 'n' that satisfy this equation.
Simplifying the Expression
First, we need to simplify the complex expression inside the parenthesis:
- (1 + i / 1 - i)
To simplify, we multiply both the numerator and denominator by the complex conjugate of the denominator:
- (1 + i / 1 - i) * (1 + i / 1 + i) = (1 + 2i - 1) / (1 + 1) = 2i / 2 = i
Now our equation becomes:
- (i)^n = 1
Finding Solutions for 'n'
We know that 'i' raised to the power of 4 is 1:
- i^4 = (i^2)^2 = (-1)^2 = 1
Therefore, we can deduce the following:
- i^1 = i
- i^2 = -1
- i^3 = i^2 * i = -i
- i^4 = 1
This pattern repeats for higher powers of 'i'. Therefore, for any integer 'n' that is a multiple of 4, (i)^n = 1.
Conclusion
The equation (1 + i / 1 - i)^n = 1 is satisfied when n is a multiple of 4. In other words, the solutions for 'n' are:
- n = 0, 4, 8, 12, ...
This indicates that the equation has an infinite number of solutions.