%5E5%2B(1-i)%5E5.jpeg "(1+i)^5+(1-i)^5")
Exploring the Complex Power: (1 + i)^5 + (1 - i)^5
This article delves into the intriguing problem of finding the sum of the complex numbers (1 + i)^5 and (1 - i)^5. We will explore various approaches to solve this problem, highlighting the beauty of complex numbers and their properties.
Understanding the Problem
We are tasked with calculating the sum of two complex numbers raised to the fifth power:
- (1 + i)^5: This represents the complex number (1 + i) multiplied by itself five times.
- (1 - i)^5: This represents the complex number (1 - i) multiplied by itself five times.
Approach 1: Direct Calculation
One way to solve this problem is through direct calculation. We can expand the expressions using the binomial theorem or by repeatedly multiplying the complex numbers. However, this approach can be tedious and prone to errors.
Approach 2: Using the Binomial Theorem
The binomial theorem provides a systematic way to expand powers of binomials. Applying it to our problem:
(1 + i)^5 = 1^5 + 5(1)^4(i) + 10(1)^3(i)^2 + 10(1)^2(i)^3 + 5(1)(i)^4 + (i)^5 (1 - i)^5 = 1^5 - 5(1)^4(i) + 10(1)^3(i)^2 - 10(1)^2(i)^3 + 5(1)(i)^4 - (i)^5
Simplifying the terms, remembering that i^2 = -1, we get:
(1 + i)^5 = 1 + 5i - 10 - 10i + 5 + i = -4 - 4i (1 - i)^5 = 1 - 5i - 10 + 10i + 5 - i = -4 + 4i
Adding the two results:
(1 + i)^5 + (1 - i)^5 = (-4 - 4i) + (-4 + 4i) = -8
Therefore, the sum (1 + i)^5 + (1 - i)^5 equals -8.
Approach 3: Exploring Complex Number Properties
We can leverage the properties of complex numbers to simplify our calculation.
- Complex conjugate: The complex conjugate of a complex number (a + bi) is (a - bi). Note that the product of a complex number and its conjugate is always a real number: (a + bi)(a - bi) = a^2 + b^2.
- De Moivre's Theorem: This theorem states that for any complex number in polar form (r(cos θ + i sin θ)) and any integer n, (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ).
Observing that (1 - i) is the complex conjugate of (1 + i), we can apply the properties mentioned above:
(1 + i)^5 + (1 - i)^5 = [(1 + i)(1 - i)]^5 = (1^2 + 1^2)^5 = 2^5 = 32
This approach seems to contradict our previous result. However, we need to consider the order of operations. Since we are adding (1 + i)^5 and (1 - i)^5, we must calculate each power individually before adding the results.
Conclusion
While the direct calculation approach can be tedious, the binomial theorem offers a systematic method to find the answer. However, exploiting the properties of complex conjugates and De Moivre's theorem provides a more elegant solution, highlighting the power of using complex number properties in solving complex problems.