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

4 min read Jun 16, 2024

Exploring the Relationship: (1-i)^n(1-1/i)^n = 2^n

This article delves into the intriguing relationship between complex numbers and the intriguing identity: (1-i)^n(1-1/i)^n = 2^n. We will explore how this equation unfolds and its implications.

Understanding the Fundamentals

Before we dive into the proof, let's refresh our understanding of complex numbers and some key properties:

Complex Numbers: Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (i² = -1).
Polar Form: A complex number can also be represented in polar form as r(cosθ + isinθ), where 'r' is the magnitude and 'θ' is the angle.
De Moivre's Theorem: This theorem states that (cosθ + isinθ)^n = cos(nθ) + isin(nθ).

Simplifying the Expression

Let's begin by simplifying the left-hand side of the equation:

Simplifying 1 - 1/i:
- Multiplying the numerator and denominator of 1/i by 'i', we get:
- 1/i = (1 * i) / (i * i) = i / -1 = -i
- Therefore, 1 - 1/i = 1 + i.
Substituting:
- Our original equation becomes: (1 - i)^n (1 + i)^n = 2^n
Recognizing a Difference of Squares:
- The left side resembles the difference of squares pattern: (a - b)(a + b) = a² - b²
- In this case, a = 1 - i and b = 1 + i.
Applying the Difference of Squares:
- (1 - i)^n (1 + i)^n = [(1 - i)(1 + i)]^n
Expanding the Square:
- (1 - i)(1 + i) = 1² - i² = 1 + 1 = 2
Final Result:
- (1 - i)^n (1 + i)^n = 2^n, which proves the initial identity.

The Essence of the Proof

The proof hinges on recognizing the difference of squares pattern in the expression. This simplification allows us to easily manipulate the equation, leading us to the straightforward result of 2^n.

Further Exploration

This identity can be further explored by examining its implications in various mathematical fields. For example, it can be used in complex number theory to analyze patterns and relationships within complex spaces.

Conclusion

This article demonstrated the derivation of the identity (1-i)^n(1-1/i)^n = 2^n. By applying fundamental complex number properties and simplifying the expression, we arrived at a clear and concise proof. This identity serves as a fascinating example of how complex numbers can be manipulated to produce insightful results.