Simplifying Complex Expressions with Loci and De Moivre’s Theorem

When dealing with complex numbers, simplifying expressions can often seem daunting. However, by utilizing a few key concepts such as De Moivre’s Theorem and understanding polar coordinates, we can simplify these expressions more efficiently. This article will delve into these strategies and provide a detailed example.

Simplifying Complex Expressions

Simplification of complex expressions typically involves converting the given expressions into a form that is easier to manipulate and understand. In this article, we will look at the transformation of expressions involving complex numbers, particularly using De Moivre’s Theorem and trigonometric identities.

The Problem at Hand

Consider the expression:
1 / i^31 - i^5

At first glance, this expression might appear complex, but by breaking it down and applying the appropriate transformations, we can simplify it significantly.

Using De Moivre’s Theorem

Let's start by applying De Moivre’s Theorem to each term in the expression. Recall that a complex number (z r(cos{theta} isin{theta})) raised to the power (n) can be simplified using De Moivre’s Theorem as (z^n r^n(cos{ntheta} isin{ntheta})).

Step 1: Simplifying (i^3)

The complex number (i) can be expressed in polar form as (i sqrt{1}(cos{frac{pi}{2}} isin{frac{pi}{2}})). Thus, applying De Moivre’s Theorem:

[i^3 left(sqrt{1}right)^3 left(cos{3cdotfrac{pi}{2}} isin{3cdotfrac{pi}{2}}right) 1 left(cos{frac{3pi}{2}} isin{frac{3pi}{2}}right) -i]

Step 2: Simplifying (i^5)

Similarly, for (i^5), we can write:

[i^5 left(sqrt{1}right)^5 left(cos{5cdotfrac{pi}{2}} isin{5cdotfrac{pi}{2}}right) 1 left(cos{2pi frac{pi}{2}} isin{2pi frac{pi}{2}}right) i]

Step 3: Combining the Results

We now have the simplified terms:

[frac{1}{i^3} frac{1}{-i} quad text{and} quad 1 - i^5 1 - i]

Combining these, we get:

[frac{1}{i^3}(1 - i^5) frac{1}{-i}(1 - i)]

Step 4: Simplifying the Expression

Now, let's simplify the expression further:

[frac{1}{-i}(1 - i) frac{1 - i}{-i} frac{1}{-i} - frac{i}{-i} -i cdot frac{1}{i} 1 -i cdot -1 1 i 1]

Alternative Method Using Binomial Multiplication

Another method involves breaking down the expression step-by-step and combining the results:

[1 / i^31 - i^5 1 / i1i1i(1 - i)1 - i1 - i1 - i1 - i1 - i]

Step 1: Grouping the terms as pairs:

[ [1 / i1 - i][1 / i1 - i][1 / i1 - i][1 - i1 - i]]

Step 2: Recognizing the pattern (frac{a}{b}(a - b) a^2 - b^2), with (a 1) and (b i):

[ 1^2 - i^2 cdot 1^2 - i^2 cdot 1^2 - i^2 [1 - i] / (1 - i) 1 - (-1)1 - (-1)1 - (-1) [1 - i] / (1 - i)]

Step 3: Simplifying the expression:

[ 1 11 11 [1 - i / (1 - i)] 222 [1 - i / (1 - i)] 8 [1 - i / (1 - i)] 4(1 - i) -16i]

De Moivre’s Theorem for Generalization

Given the expression (1 / i^31 - i^5), we can use De Moivre’s Theorem to express the general form of complex numbers in polar coordinates and exponential form. For a complex number (z a bi) in rectangular coordinates, its polar form is (z r(cos{theta} isin{theta})) and its exponential form is (z re^{itheta}). For our specific example:

The complex number (i) can be written as (1(cos{frac{pi}{2}} isin{frac{pi}{2}})) or (e^{ifrac{pi}{2}}), and (i^3 e^{ifrac{3pi}{2}}), and (i^5 e^{ifrac{5pi}{2}} e^{ifrac{pi}{2}}). Simplifying, we get:

[1 / i^31 - i^5 1 / e^{ifrac{3pi}{2}}1 - e^{ifrac{pi}{2}} e^{-ifrac{3pi}{2}}1 - e^{ifrac{pi}{2}} -16i]