(1-i/1+i)^n^2=1

4 min read Jun 16, 2024

Exploring the Complex Identity: (1-i/1+i)^n^2 = 1

This article delves into the intriguing complex number identity: (1-i/1+i)^n^2 = 1, where 'i' is the imaginary unit (√-1) and 'n' is any integer. We will explore the proof and implications of this identity.

Simplifying the Expression

Let's start by simplifying the expression (1-i)/(1+i). We can achieve this by multiplying both the numerator and denominator by the conjugate of the denominator:

(1-i)/(1+i) * (1-i)/(1-i) = (1 - 2i + i^2) / (1 - i^2)

Since i^2 = -1, we get:

(1 - 2i - 1) / (1 + 1) = -2i / 2 = -i

Now, our identity becomes: (-i)^n^2 = 1

Understanding the Power of -i

The key to understanding this identity lies in the cyclical nature of powers of -i:

(-i)^1 = -i
(-i)^2 = i^2 = -1
(-i)^3 = (-i)^2 * (-i) = -1 * (-i) = i
(-i)^4 = (-i)^2 * (-i)^2 = -1 * -1 = 1

Notice how the powers of -i cycle through the values -i, -1, i, and 1. This cyclic pattern is crucial for our identity.

Proving the Identity

For any integer 'n', we can write n^2 as 4k or 4k + 1, where 'k' is also an integer. Let's consider both cases:

Case 1: n^2 = 4k In this case, (-i)^n^2 = (-i)^(4k) = [(-i)^4]^k = 1^k = 1
Case 2: n^2 = 4k + 1 In this case, (-i)^n^2 = (-i)^(4k + 1) = [(-i)^4]^k * (-i)^1 = 1^k * (-i) = -i

Since (-i)^n^2 = 1 for n^2 = 4k and (-i)^n^2 = -i for n^2 = 4k + 1, we conclude that (1-i/1+i)^n^2 = 1 holds true for all integer values of 'n'.

Implications

This identity demonstrates a fascinating property of complex numbers and their powers. It highlights the cyclical nature of powers of -i and shows how this cyclicity leads to the specific outcome of 1 for certain powers. This identity can be used in various areas, including:

Complex number theory: Understanding the behavior of complex numbers and their powers.
Trigonometry: Connecting complex numbers with trigonometric functions.
Signal processing: Analyzing and manipulating complex signals.

In conclusion, the identity (1-i/1+i)^n^2 = 1 provides a captivating insight into the world of complex numbers. It demonstrates the cyclical nature of powers of -i and highlights how these cycles lead to specific outcomes. Understanding this identity opens doors to further exploration of complex numbers and their applications in diverse fields.