Simplifying Complex Number Expressions: A Step-by-Step Guide

Understanding and simplifying expressions involving complex numbers is essential in various mathematical and engineering applications. In this article, we will explore the simplification of a specific expression and demonstrate the underlying principles of complex number manipulation. The expression we will focus on is:

Expression Analysis

The given expression is

(frac{2^n}{1i^{2n}} - frac{1i^{2n}}{2^n})

Step 1: Simplify the Complex Number 1i2n

First, let us simplify 1i2n to understand its behavior.

1i represents the imaginary unit i with a modulus magnitude of 1. The argument angle of 1i is (frac{pi}{4}).

Expression in polar form: 1i sqrt{1^2 1^2} left( cos(frac{pi}{4}) i sin(frac{pi}{4}) right) sqrt{2} left( cos(frac{pi}{4}) i sin(frac{pi}{4}) right)) Using De Moivre's Theorem: 1i2n left( sqrt{2} right)^{2n} left( cos{(frac{2npi}{4})} i sin{(frac{2npi}{4})} right) 2^n left( cos{(frac{npi}{2})} i sin{(frac{npi}{2})} right))

Step 2: Substitute and Simplify the Original Expression

Next, we substitute (1i^{2n}) into the original expression:

(frac{2^n}{1i^{2n}} - frac{1i^{2n}}{2^n} frac{2^n}{2^n left( cos{(frac{npi}{2})} i sin{(frac{npi}{2})} right)} - frac{2^n left( cos{(frac{npi}{2})} i sin{(frac{npi}{2})} right)}{2^n})

This simplifies to:

(frac{1}{cos{(frac{npi}{2})} i sin{(frac{npi}{2})}} left( cos{(frac{npi}{2})} i sin{(frac{npi}{2})} right))

Let (z cos{(frac{npi}{2})} i sin{(frac{npi}{2})}). Then the expression becomes:

(frac{1}{z} z)

Step 3: Further Simplify Using the Complex Number Property

We can use the property of complex numbers, (frac{1}{z} frac{overline{z}}{z^2}), where (overline{z}) is the conjugate of (z).

Since (z^2 cos^2{(frac{npi}{2})} - sin^2{(frac{npi}{2})} 1), we have:

(frac{1}{z} frac{overline{z}}{1} overline{z} cos{(frac{npi}{2})} - i sin{(frac{npi}{2})})

Thus, the expression simplifies to:

((cos{(frac{npi}{2})} - i sin{(frac{npi}{2})}) (cos{(frac{npi}{2})} i sin{(frac{npi}{2})}) 2 cos{(frac{npi}{2})})

Conclusion

The original expression simplifies to:

(frac{2^n}{1i^{2n}} - frac{1i^{2n}}{2^n} 2 cos{(frac{npi}{2})})

Alternate Approach

An alternative method to simplify the same expression involves recognizing 1i2n as:

(1i^{2n} (1i^2)^n 1i^2 cdot 2^n 2^n cdot i^n)

Therefore, the expression becomes:

(frac{2^n}{1i^{2n}} - frac{1i^{2n}}{2^n} frac{2^n}{2^n cdot i^n} - frac{2^n cdot i^n}{2^n} frac{1}{i^n} i^n - i^n 1 - i^n)

When n is an odd integer, the expression simplifies to zero:

(1 - i^n 0)

When n is an even integer, particularly a multiple of 4, the expression becomes:

(frac{2}{-1^m} frac{2}{-1^{n/2}} -2^{1-n/2})

This conclusion is summarized through the use of De Moivre's theorem and the properties of complex numbers.