(1+i/1-i)^n=1 Find N

2 min read Jun 16, 2024

Finding the Value of n in the Equation (1+i)/(1-i)^n = 1

This problem involves complex numbers and their powers. Let's break down the steps to solve for 'n':

1. Simplifying the Complex Number

First, we need to simplify the complex number (1+i)/(1-i). We can do this by multiplying both the numerator and denominator by the conjugate of the denominator:

(1+i)/(1-i) * (1+i)/(1+i) = (1 + 2i + i²)/(1 - i²)
Since i² = -1, we can substitute to get:

(1 + 2i - 1)/(1 + 1) = 2i/2 = i
Therefore, our equation now becomes: i^n = 1.

2. Understanding Powers of 'i'

Let's examine the powers of 'i':
- i¹ = i
- i² = -1
- i³ = i² * i = -i
- i⁴ = (i²)² = 1
Notice that the powers of 'i' cycle through the values {i, -1, -i, 1}.

3. Finding the Solutions for 'n'

We want to find the values of 'n' where i^n = 1. From our observation above, we see that i^n = 1 whenever 'n' is a multiple of 4.
Therefore, the solutions for 'n' are:

n = 4k, where k is any integer.

In conclusion, the equation (1+i)/(1-i)^n = 1 is satisfied when 'n' is a multiple of 4.