Problem 74 - Algebra and Trigonometry

Question

There are two square roots of \(1,\) namely, 1 and \(-1 .\) These are the solutions of \(x^{2}=1 .\) The fourth roots of 1 are the solutions of the equation \(x^{4}=1\) or \(x^{4}-1=0 .\) How many fourth roots of 1 are there? Find them. The cube roots of 1 are the solutions of the equation \(x^{3}=1\) or \(x^{3}-1=0\) . How many cube roots of 1 are there? Find them. How would you find the sixth roots of 1\(?\) How many are there? Make a conjecture about the number of \(n\) th roots of \(1 .\)

Step-by-Step Solution

Answer

The number of \(n\)th roots of 1 is \(n\). The roots are given by \(e^{2k\pi i / n}\) where \(k = 0, \, 1, \, ..., \, n-1\).

Step 1: Understanding the Problem

We need to find the number of roots and the specific roots for the equations concerning the roots of 1: fourth roots (for the equation \(x^4 = 1\)), cube roots (for \(x^3 = 1\)), and sixth roots (for \(x^6 = 1\)). Finally, we will find a general pattern or conjecture for the \(n\)th roots of 1.

Step 2: Finding Fourth Roots of 1

To solve \(x^4 = 1\), we set the equation as \((x^2)^2 = 1\). The solutions to this are \(x^2 = 1\) and \(x^2 = -1\). Solving \(x^2 = 1\), the roots are \(x = 1\) or \(x = -1\). Solving \(x^2 = -1\), the roots are \(x = i\) or \(x = -i\), where \(i\) is the imaginary unit. Therefore, the fourth roots of 1 are 1, -1, \(i\), and \(-i\). There are 4 fourth roots.

Step 3: Finding Cube Roots of 1

To solve \(x^3 = 1\), rewrite the equation as \((x - 1)(x^2 + x + 1) = 0\). The solutions are \(x = 1\), and the roots of \(x^2 + x + 1 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = 1\), we find the other roots: \(x = \frac{-1 \pm \sqrt{-3}}{2}\) or \(x = \frac{-1 \pm i\sqrt{3}}{2}\). So, the cube roots of 1 are 1, \(\frac{-1 + i\sqrt{3}}{2}\), and \(\frac{-1 - i\sqrt{3}}{2}\). There are 3 cube roots.

Step 4: Finding Sixth Roots of 1

To solve \(x^6 = 1\), one way is to use the exponential form: write 1 as \(e^{2k\pi i}\) for integer \(k\). The sixth roots are \(x = e^{2k\pi i / 6}\), where \(k = 0, 1, 2, 3, 4, 5\). Therefore, the sixth roots of 1 are \(e^{0}\), \(e^{\pi i / 3}\), \(e^{2\pi i / 3}\), \(e^{\pi i}\), \(e^{4\pi i / 3}\), and \(e^{5\pi i / 3}\). There are 6 sixth roots.

Step 5: Making a Conjecture

For any positive integer \(n\), the number of \(n\)th roots of 1 is \(n\). These roots can be found using the expression \(e^{2k\pi i / n}\) for \(k = 0, 1, 2, \ldots, n-1\). Therefore, there are \(n\) \(n\)th roots of 1.

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers and are written in the form of \(a + bi\). Here, \(a\) is the real part, and \(bi\) is the imaginary part. Both \(a\) and \(b\) are real numbers, making them part of this complex number format.

• The real part of a complex number is the component without the imaginary unit \(i\).
• The imaginary part is the component with \(i\), where \(i\) is the imaginary unit that is defined as the square root of -1.

Complex numbers are incredibly useful in solving polynomial equations, especially when roots are not real. They allow us to find and express roots that do not exist within the realm of real numbers.
They play a vital role in advanced mathematical theories and applications, including electrical engineering and quantum mechanics. Understanding complex numbers sets the ground for deeper exploration into roots of unity and other advanced mathematical concepts.

Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental element in complex numbers. It is defined by the property \(i^2 = -1\). This definition breaks the rules of multiplication in the realm of real numbers since no real number squared can be negative.
Using \(i\), we can express the square root of negative numbers, which is impossible with just real numbers. For example:

• \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2i\)

This concept introduces us to complex numbers and is critical in solving equations like \(x^2 = -1\), which cannot be solved using only real numbers.
The imaginary unit is vital as it allows us to work with and interpret polynomial equations that have solutions outside the real-number spectrum. It effectively doubles the dimensionality of our number system, providing a useful tool in numerous scientific and engineering calculations.

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of powers of variables. In simple terms, they are equations like \(x^2 - 1 = 0\) or \(x^4 - 1 = 0\), where the variable is raised to different powers, and coefficients can be numbers or parameters.

• A polynomial of degree \(n\) has up to \(n\) solutions or roots.
• The Fundamental Theorem of Algebra states that every non-zero polynomial equation has as many complex solutions as its degree when counted with multiplicity.

Every polynomial can be broken down into factors corresponding to its roots. By solving these equations, we find the roots, which can be real or complex, such as the roots of unity.
Understanding polynomial equations and their solutions is crucial as it allows us to predict and calculate important values in algebra, geometry, and various applications across science and engineering.

Euler's Formula

Euler's formula is a cornerstone of complex analysis, establishing a profound link between complex numbers and exponential functions. It states that for any real number \( \theta \), \[ e^{i\theta} = \cos \theta + i\sin \theta \] This formula elegantly combines trigonometric functions with exponential growth.
It’s particularly powerful when dealing with roots of unity. For example, the \(n\)th roots of unity are given by the expression \(e^{2k\pi i/n}\), where \(k = 0, 1, 2, \ldots, n-1\). With Euler’s formula, one can express these roots in terms of sines and cosines, making it much easier to visualize them on the complex plane as points on the unit circle.
Using Euler's Formula not only simplifies the operations involving complex numbers but also provides essential insights into wave behaviors, signal processing, and other areas of applied science.