(1+i)^6n+(1-i)^6n

4 min read Jun 16, 2024

Exploring the Complex Arithmetic of (1+i)^6n + (1-i)^6n

This article delves into the intriguing mathematical expression (1+i)^6n + (1-i)^6n, where 'i' represents the imaginary unit (√-1) and 'n' is any integer. We'll explore its simplification and uncover its interesting properties.

DeMoivre's Theorem: The Key to Unlocking the Expression

To simplify the expression, we employ DeMoivre's Theorem. This theorem states that for any complex number in polar form (r(cosθ + i sinθ)) and any integer n, the following holds:

(r(cosθ + i sinθ))^n = r^n(cos(nθ) + i sin(nθ))

First, let's convert (1+i) and (1-i) into polar form:

(1 + i):
- Magnitude (r) = √(1² + 1²) = √2
- Angle (θ) = arctan(1/1) = π/4
- Polar form: √2(cos(π/4) + i sin(π/4))
(1 - i):
- Magnitude (r) = √(1² + (-1)²) = √2
- Angle (θ) = arctan(-1/1) = -π/4
- Polar form: √2(cos(-π/4) + i sin(-π/4))

Now, let's apply DeMoivre's theorem to our expression:

(1 + i)^6n = (√2(cos(π/4) + i sin(π/4)))^6n = 2^(3n) (cos(6nπ/4) + i sin(6nπ/4)) (1 - i)^6n = (√2(cos(-π/4) + i sin(-π/4)))^6n = 2^(3n) (cos(-6nπ/4) + i sin(-6nπ/4))

Simplifying the Expression

Notice that the angles in both terms are multiples of π/4. Also, since cosine is an even function (cos(-x) = cos(x)) and sine is an odd function (sin(-x) = -sin(x)), we can simplify further:

(1 + i)^6n + (1 - i)^6n = 2^(3n) (cos(3nπ/2) + i sin(3nπ/2)) + 2^(3n) (cos(3nπ/2) - i sin(3nπ/2))

This simplifies to:

(1 + i)^6n + (1 - i)^6n = 2^(3n+1) cos(3nπ/2)

Interesting Observation

Examining the simplified expression, we notice that the imaginary component vanishes. The result depends only on the cosine of 3nπ/2, which can take values of 0, 1, or -1 depending on the value of 'n'. This means the expression always evaluates to a real number, even though it involves complex numbers initially.

For example:

When n = 1, the expression becomes 2^4 cos(3π/2) = -16.
When n = 2, the expression becomes 2^7 cos(3π) = -128.

Conclusion

The seemingly complex expression (1+i)^6n + (1-i)^6n, when simplified using DeMoivre's Theorem, reveals a fascinating property: it always evaluates to a real number. This demonstrates the power of complex number manipulation and the elegance of mathematical relationships.