%5E6%2B(1-i)%5E6.jpeg "(1+i)^6+(1-i)^6")
Exploring the Complex Expression: (1+i)^6 + (1-i)^6
This article delves into the intriguing mathematical expression (1+i)^6 + (1-i)^6, where 'i' represents the imaginary unit (√-1). We will explore methods for simplifying this expression and uncover its surprising result.
Understanding Complex Numbers
Before diving into the calculation, let's refresh our understanding of complex numbers. A complex number consists of a real part and an imaginary part, represented in the form a + bi, where 'a' and 'b' are real numbers.
The key to working with complex numbers is understanding the imaginary unit 'i'. Here are some key properties:
- i² = -1
- i³ = i² * i = -i
- i⁴ = (i²)² = 1
These properties are crucial when expanding and simplifying complex expressions.
Simplifying the Expression
We can tackle (1+i)^6 + (1-i)^6 using the binomial theorem or De Moivre's theorem. Let's explore both approaches:
1. Binomial Theorem
The binomial theorem provides a way to expand expressions of the form (a + b)^n:
(a + b)^n = ∑(n choose k) * a^(n-k) * b^k
Where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
Applying this to our expression:
(1+i)^6 = ∑(6 choose k) * 1^(6-k) * i^k
(1-i)^6 = ∑(6 choose k) * 1^(6-k) * (-i)^k
Expanding and simplifying the terms, we notice that many terms cancel out due to the alternating signs of the imaginary unit. The remaining terms sum up to a real number.
2. De Moivre's Theorem
De Moivre's theorem provides a shortcut for calculating powers of complex numbers in polar form:
(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
First, we need to convert (1 + i) and (1 - i) into polar form:
- (1 + i): Magnitude = √2, Angle = 45°
- (1 - i): Magnitude = √2, Angle = -45°
Applying De Moivre's theorem:
(1 + i)^6 = (√2)^6 * (cos 270° + i sin 270°) = 8i
(1 - i)^6 = (√2)^6 * (cos -270° + i sin -270°) = -8i
Adding the results, we get (1+i)^6 + (1-i)^6 = 8i - 8i = 0.
Conclusion
Despite its complex appearance, the expression (1+i)^6 + (1-i)^6 simplifies to zero. This surprising result highlights the elegant interplay between real and imaginary numbers and the power of mathematical tools like the binomial theorem and De Moivre's theorem in simplifying complex expressions.