Problem 40 - Precalculus : Building Concepts and Connections

Question

Find the fifth roots of \(i\)

Step-by-Step Solution

Answer

The fifth roots of \(i\) are \(e^{i\frac{\pi}{10}}, e^{i\frac{9\pi}{10}}, e^{i\frac{17\pi}{10}}, e^{i\frac{25\pi}{10}}\), and \(e^{i\frac{33\pi}{10}}\).

Step 1: Convert to Polar Form

Firstly, it's important to express the complex number \(i\) in polar form, or \(re^{i\theta}\). In this case, \(r=1\) (the modulus of \(i\)) and \(\theta=\frac{\pi}{2}\) (the argument of \(i\)). Therefore, \(i\) can be written as \(e^{i\frac{\pi}{2}}\).

Step 2: Use De Moivre's Theorem

De Moivre's theorem asserts if \(z=re^{i\theta}\), then the \(n\) complex roots of \(z\) are given by \(z_k=r^{1/n}e^{i(\theta+2k\pi)/n}\) for \(k=0, 1, 2, …, n-1\). Now, apply De Moivre's theorem to compute the fifth roots of \(i\) (or \(e^{i\frac{\pi}{2}}\)).

Step 3: Calculate the Roots

Applying the formula from De Moivre's theorem, the fifth roots of \(i\) are calculated as \(e^{i\frac{\pi}{2+8k\pi}/5}\) for \(k=0, 1, 2, 3, 4\). After computation, we get five roots: \(e^{i\frac{\pi}{10}}, e^{i\frac{9\pi}{10}}, e^{i\frac{17\pi}{10}}, e^{i\frac{25\pi}{10}}\), and \(e^{i\frac{33\pi}{10}}\).

Key Concepts

De Moivre's Theorem

Understanding De Moivre's Theorem is essential when working with complex numbers, specifically when dealing with problems involving roots of a complex number. Put simply, this theorem provides a formula to raise any complex number written in polar or exponential form to any power.

For a complex number \( z = re^{i\theta} \), where \( r \) is the modulus (the distance from the origin in the complex plane) and \( \theta \) is the argument (the angle made with the positive real axis), De Moivre's Theorem states:

\( z^n = r^n(cos(n\theta) + i\cdot sin(n\theta)) \)
or equivalently,
\( z^n = r^n e^{in\theta} \)

This formula is crucial when finding the \( n \)th roots, as it allows one to find all of the \( n \) evenly spaced roots on the complex plane by just altering the angle \( \theta \). This constructs a regular \( n \)-sided polygon in the complex plane, with the roots at its vertices.

Complex Numbers

At the heart of our problem lies the concept of complex numbers. A complex number can be understood as an extension of the real numbers, and it includes a real part and an imaginary part. It’s denoted as \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit, satisfying the equation \( i^2 = -1 \).

The number \( i \) itself is a complex number, specifically \( 0 + 1i \), emphasizing that complex numbers include pure imaginary numbers. Understanding complex numbers is crucial for fields like electrical engineering, quantum physics, and applied mathematics, as they describe phenomena like oscillations and waves effectively.

Polar Form

The polar form of a complex number is a different way to represent it, focusing on its magnitude and direction rather than its position in terms of the standard x and y axes. It's given by \( z = r(cos\theta + i\cdot sin\theta) \) or, using Euler's formula, as \( z = re^{i\theta} \), where \( r \) is the modulus of \( z \), and \( \theta \) is the argument (or angle).

The polar form is particularly useful when multiplying or dividing complex numbers, as you can simply multiply or divide the moduli and add or subtract the arguments, respectively. This makes operations that can be cumbersome in standard form much more manageable.

Exponential Form of Complex Numbers

Lastly, the exponential form of complex numbers is a streamlined version of the polar form, gained through Euler’s identity, which connects trigonometry and complex exponentials: \( e^{i\theta} = cos(\theta) + i\cdot sin(\theta) \). It represents complex numbers as \( re^{i\theta} \), simplifying the expression of complex numbers and operations on them significantly.

It’s particularly handy with powers and roots of complex numbers, as seen in De Moivre's Theorem. It makes the multiplication of angles evident and the extraction of roots straightforward, as demonstrated by finding the fifth roots of \( i \). Each root represents a point on the complex plane that is equidistant from the origin, forming a shape akin to the slices of a pie or the spokes of a wheel.