(1+i/1-i)^4n+1

3 min read Jun 16, 2024

Exploring the Complex Power: (1 + i / 1 - i)^(4n + 1)

This article delves into the intriguing expression (1 + i / 1 - i)^(4n + 1), where 'i' represents the imaginary unit (√-1), and 'n' is any integer. We'll investigate the simplification and properties of this expression, unraveling its fascinating mathematical nature.

Simplifying the Base

First, let's simplify the base of the expression:

(1 + i) / (1 - i)

To do this, we multiply both the numerator and denominator by the conjugate of the denominator (1 + i):

(1 + i) / (1 - i) * (1 + i) / (1 + i) = (1 + 2i + i²) / (1 - i²)

Since i² = -1, we get:

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

Therefore, the expression becomes: i^(4n + 1)

Utilizing the Cyclicity of i

Now, let's explore the powers of 'i':

i¹ = i
i² = -1
i³ = i² * i = -i
i⁴ = i² * i² = (-1) * (-1) = 1

Notice that the powers of 'i' cycle through these four values. Every four powers, the cycle repeats.

Finding the Pattern

With the cyclic nature of 'i' in mind, let's analyze the exponent (4n + 1):

4n is always divisible by 4, resulting in i⁴, which equals 1.
+ 1 shifts the cycle by one step, leaving us with i¹.

Therefore, regardless of the value of 'n', the expression i^(4n + 1) always simplifies to i.

Conclusion

The seemingly complex expression (1 + i / 1 - i)^(4n + 1) simplifies to a remarkably consistent result: i. This exploration highlights the power of understanding complex number properties, especially the cyclic nature of the imaginary unit 'i'. Such simplification is not only interesting but can be highly valuable in various mathematical fields and applications.