Problem 8 - Advanced Engineering Mathematics

Question

Let \(\omega_{n}=e^{i 2 \pi / n}=\cos (2 \pi / n)+i \sin (2 \pi / n)\). Since \(e^{i 2 \pi k}=1\), 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 \(\boldsymbol{n}\) 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

The eighth roots of unity form a regular octagon on the unit circle in the complex plane.

Step 1: Understanding Eighth Roots of Unity

The eighth roots of unity are the solutions to the equation \( z^8 = 1 \). This can be written as \( z^8 - 1 = 0 \). These solutions can be expressed as \( \omega_8^k \) where \( k = 0, 1, 2, \ldots, 7 \) and \( \omega_8 = e^{i 2\pi / 8} \).

Step 2: Calculate \( \omega_8 \)

Calculate \( \omega_8 \) as the primary eighth root of unity: \[ \omega_8 = e^{i 2\pi / 8} = \cos(\pi/4) + i \sin(\pi/4) = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \].

Step 3: Determine Eighth Roots of Unity

The eighth roots of unity can be found by calculating \( \omega_8^k \) for each \( k = 0, 1, 2, \ldots, 7 \). These are: 1. \( \omega_8^0 = 1 \)2. \( \omega_8^1 = \omega_8 = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)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: Plot Complex Numbers

Plot each \( \omega_8^k \) on the complex plane by identifying their coordinates \((x, y)\): 1. \((1, 0)\)2. \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\)3. \((0, 1)\)4. \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\)5. \((-1, 0)\)6. \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\)7. \((0, -1)\)8. \((\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\).

Step 5: Observe the Plot

Notice that the points lie evenly distributed on the unit circle on the complex plane. They form the vertices of a regular octagon centered at the origin.

Key Concepts

Complex Plane

The complex plane is similar to the Cartesian coordinate plane you might already be familiar with, but it is used for plotting complex numbers. A complex number, usually denoted as \(z\), has a real part and an imaginary part and can be written as \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. In the complex plane:

• The real part \(x\) is plotted along the horizontal axis, also called the real axis.
• The imaginary part \(y\) is plotted along the vertical axis, known as the imaginary axis.

Each point \((x, y)\) on this plane represents a complex number \(z = x + iy\).

When plotting the eighth roots of unity, we use the complex plane to visually represent the solutions. These solutions lie on the circumference of the unit circle, evenly spaced because they are all roots of unity. For instance, \( \omega_8^0 = 1\) is represented as the point \((1,0)\) on the complex plane.

Unit Circle

The unit circle plays a crucial role in understanding roots of unity. It is a circle in the complex plane centered at the origin with a radius of one. All points on the unit circle are at a distance of one unit from the origin, which means any complex number lying on this circle has a magnitude (or absolute value) of one.

• The angle each point makes with the positive real axis is known as the argument of the complex number.
• Eighth roots of unity, such as \( \omega_8^k \), are distributed around the unit circle, forming a regular polygon.
• For eighth roots, the polygon has eight vertices, creating an octagon.

The importance of this distribution on the unit circle is that it provides a geometric perspective of multiplication and division in complex numbers, helping in visualizing power equations like \(z^n = 1\).

Each root's angle in degrees and radians on the circle corresponds to the regular division of a full circle (360° or \(2\pi\) radians). For instance, \( \omega_8 = e^{i\pi/4}\) corresponds to a 45° angle from the positive x-axis.

Euler's Formula

Euler's formula is a powerful equation in complex analysis given by \(e^{i\theta} = \cos \theta + i\sin \theta\). This equation bridges the gap between exponential functions and trigonometric functions, simplifying computations involving complex numbers.

• In the formula, \(\theta\) represents the angle in radians, and \(e^{i\theta}\) denotes a complex number on the unit circle with this angle as its argument.
• Euler’s formula is particularly useful when calculating powers and roots of complex numbers.
• For example, finding the eighth roots of unity involves using Euler's formula to express each root as \(\omega_8^k = e^{i2k\pi/8}\).

By interpreting roots of unity using Euler's formula, we can simplify their representation and visualization on the complex plane.

This leads to a deeper understanding of how complex numbers rotate around the origin when multiplied, hence facilitating their interpretation in terms of angles and distances on the unit circle.