Problem 8 - Advanced Engineering Mathematics

Question

Let \(\omega_{n}=e^{i \pi / n}=\cos (2 \pi / n)+i \sin (2 \pi / n)\). Since \(e^{2 \pi k}=1\), the numbers \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), all have the property that \(\left(\omega_{n}^{k}\right)^{n}=1 .\) Because of this, \(\omega_{n}^{k}, k=0,1,2, \ldots, n-1\), are called the \(n\) th roots of unity and are solutions of the equation \(z^{n}-1=0\). Find the eighth roots of unity and plot them in the \(x y\) -plane where a complex number is written \(z=x+i y\). What do you notice?

Step-by-Step Solution

Answer

Eighth roots of unity are evenly spaced around the unit circle, forming an octagon.

Step 1: Understanding the Problem

We need to find the eighth roots of unity, which are solutions to the equation \( z^8 - 1 = 0 \). These solutions are of the form \( \omega_8^k \) where \( k = 0, 1, 2, \ldots, 7 \) and \( \omega_8 = e^{i \pi / 4} = \cos(\pi / 4) + i \sin(\pi / 4) \).

Step 2: Calculate \( \omega_8 \)

Calculate the base root \( \omega_8 \), which is \( \omega_8 = e^{i\frac{2\pi}{8}} = e^{i\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \).

Step 3: List the Eighth Roots

Compute each power of \( \omega_8 \) up to \( k = 7 \). Each root is \( \omega_8^k \): 1. \( \omega_8^0 = 1 \)2. \( \omega_8^1 = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)3. \( \omega_8^2 = i \)4. \( \omega_8^3 = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)5. \( \omega_8^4 = -1 \)6. \( \omega_8^5 = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)7. \( \omega_8^6 = -i \)8. \( \omega_8^7 = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \).

Step 4: Plotting the Roots on the Complex Plane

Plot each root as a point on the xy-plane where the real part corresponds to x and the imaginary part corresponds to y. The roots distribute evenly around the unit circle centered at the origin, forming an octagon.

Step 5: Observing the Symmetry

Notice that the eighth roots of unity are symmetrical around the center of the unit circle. They form an evenly spaced set of points, indicating rotational symmetry. Each point is \( \frac{\pi}{4} \) radians apart.

Key Concepts

Roots of Unity

The concept of **Roots of Unity** is essential in understanding certain polynomial equations and their solutions. When we refer to the roots of unity, we're talking about the solutions to the equation \( z^n - 1 = 0 \).
These roots are complex numbers located on the unit circle in the complex plane. The equation essentially asks, "Which numbers, when raised to the power of \( n \), yield 1?"
Each root is expressed in the form \( \omega_n^k = e^{2\pi i k / n} \), where \( n \) is the degree of the root, and \( k \) ranges from 0 to \( n-1 \).
For the eighth roots of unity, we solve \( z^8 - 1 = 0 \), obtaining the number \( \omega_8 = e^{i\frac{\pi}{4}} \).

• This represents one of the angles around the circle, and raising it to powers \( k = 0, 1, 2, \, ..., 7 \) uncovers all roots.
• Each of these roots can be visualized as points on the unit circle, showing how they divide the circle into eight equal parts.

Understanding these roots not only helps with polynomial solutions but also with understanding cyclic behaviors in various fields such as signal processing and number theory.

Complex Plane

The **Complex Plane** is a two-dimensional plane used in mathematics to represent complex numbers graphically.
A complex number \( z \) is expressed as \( x + iy \) where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit \( \sqrt{-1} \).
On the complex plane, the horizontal axis (the "real" axis) represents real components, and the vertical axis (the "imaginary" axis) represents imaginary components.

• When plotting the roots of unity, these numbers all lie on the unit circle, which is centered at the origin (0,0) and has a radius of 1.
• For the eighth roots of unity, each point divides the circle into eight equal parts, all equidistant from the center.

Visualizing complex numbers on this plane is crucial for understanding their behavior and properties, like magnitude and argument (angle), both vital for performing operations with complex numbers.

Symmetry in Mathematics

**Symmetry in Mathematics** often refers to an object being invariant under certain transformations, such as rotations and reflections.
In the context of the roots of unity, symmetry plays a significant role. When plotted on the complex plane, the roots exhibit perfect rotational symmetry.
For instance, the eighth roots of unity demonstrate symmetry as their positions on the unit circle are evenly spaced.

• Each root is separated by an angle of \( \frac{\pi}{4} \) radians, illustrating rotational symmetry.
• This means you can rotate the entire circle by \( \frac{\pi}{4} \), and the roots will reposition themselves onto each other.

This symmetry helps simplify many mathematical processes, allowing more straightforward calculations within equations and transformations.
Recognizing symmetry can also reveal underlying patterns in more complex mathematical problems, making it a fundamental tool in both pure and applied mathematics.