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Chapter 8: Problem 66

Find and graph the sixth roots of \(1 .\)

Short Answer

Expert verified
The sixth roots of 1 are \[1, \frac{1}{2} + i \frac{\sqrt{3}}{2}, -\frac{1}{2} + i \frac{\sqrt{3}}{2}, -1, -\frac{1}{2} - i \frac{\sqrt{3}}{2}, \text{and} \frac{1}{2} - i \frac{\sqrt{3}}{2}\].

Step by step solution

01

- Understand the problem

To find the sixth roots of 1, we need to find all complex numbers that satisfy the equation \(z^6 = 1\). This involves using Euler's formula and converting between polar and Cartesian coordinates.
02

- Use Euler's formula

According to Euler's formula, any complex number can be expressed in the form \(z = re^{i\theta}\). For our problem, \(r = 1\) since we are dealing with the unit circle, and \(\theta\) is the angle. The general solution for \(z^n = 1\) is given by \[\theta = \frac{2\pi k}{n}\], where \( k = 0, 1, 2, \ldots, n-1\).
03

- Calculate the angles

We need to find \(\theta\) for \(n = 6\), so substituting gives us \[\theta_k = \frac{2\pi k}{6} = \frac{\pi k}{3} \]. We have the angles \(\theta_k\) for \(k = 0, 1, 2, 3, 4, 5\): \[\theta_0 = 0, \theta_1 = \frac{\pi}{3}, \theta_2 = \frac{2\pi}{3}, \theta_3 = \pi, \theta_4 = \frac{4\pi}{3}, \theta_5 = \frac{5\pi}{3}\].
04

- Express the roots using the angles

The sixth roots of 1 are then given by \[ z_k = e^{i\theta_k} \]. This results in the following roots: \[ z_0 = e^{i \cdot 0} = 1 \, \ z_1 = e^{i \cdot \frac{\pi}{3}} = \frac{1}{2} + i\frac{\sqrt{3}}{2} \, \ z_2 = e^{i \cdot \frac{2\pi}{3}} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \, \ z_3 = e^{i \cdot \pi} = -1 \, \ z_4 = e^{i \cdot \frac{4\pi}{3}} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \, \ z_5 = e^{i \cdot \frac{5\pi}{3}} = \frac{1}{2} - i\frac{\sqrt{3}}{2} \].
05

- Graph the roots

To graph these roots, plot them on the complex plane. They should be equally spaced on the unit circle at angles of \[0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \text{and } \frac{5\pi}{3}\] radians. Each root lies on the unit circle at the corresponding angle from the positive x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that have both a real and an imaginary part. They are often written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined by the property that \(i^2 = -1\).
Euler's Formula
Euler's formula is a profound equation in mathematics that relates complex exponentials to trigonometric functions. It is given by \ e^{i\theta} = \cos(\theta) + i\sin(\theta) \ where \(\theta\) is an angle in radians. This representation is particularly useful in solving problems involving roots of unity because it allows us to express complex numbers in polar form.
Unit Circle
The unit circle is a circle in the complex plane with a radius of 1 centered at the origin. Any complex number on the unit circle has a magnitude of 1. The unit circle is crucial for understanding the sixth roots of unity because these roots are evenly spaced around the circle. This provides a clear visual representation of the solutions to \(z^6 = 1\).

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Most popular questions from this chapter

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Solve. $$x^{6}+64=0$$
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