Problem 74 - Precalculus Mathematics for Calculus

Question

There are two square roots of 1, namely, 1 and \(-1 .\) These are the solutions of \(x^{2}=1 .\) The fourth roots of I 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 fourth roots of 1 are \(1, -1, i, -i\); there are 4 sixth roots of 1, and the general pattern suggests there are \(n\) nth roots of 1.

Step 1: Understand the Concept of Roots

The roots of a number or an equation are the values for which the equation is satisfied. For instance, the square roots of 1 are the values of \( x \) that satisfy \( x^2 = 1 \). Similarly, the nth roots of 1 are the solutions to \( x^n = 1 \).

Step 2: Solve for Fourth Roots of 1

We need to solve the equation \( x^4 = 1 \). This equation is equivalent to \( (x^2 - 1)(x^2 + 1) = 0 \). Solving \( x^2 - 1 = 0 \), we get \( x = 1 \) or \( x = -1 \). Solving \( x^2 + 1 = 0 \), we get \( x = i \) or \( x = -i \), where \( i \) is the imaginary unit. Thus, the fourth roots are \( 1, -1, i, -i \).

Step 3: Count the Fourth Roots of 1

The fourth roots are \( 1, -1, i, -i \). Therefore, there are 4 distinct fourth roots of 1.

Step 4: Solve for Cube Roots of 1

The equation \( x^3 = 1 \) can be written as \((x-1)(x^2+x+1) = 0\). Solving \( x - 1 = 0 \), we find \( x = 1 \). The roots of \( x^2 + x + 1 = 0 \) are \( x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2} \). Therefore, the cube roots are \( 1, \frac{-1 + i\sqrt{3}}{2}, \frac{-1 - i\sqrt{3}}{2} \).

Step 5: Count the Cube Roots of 1

The cube roots are \( 1, \frac{-1 + i\sqrt{3}}{2}, \frac{-1 - i\sqrt{3}}{2} \). Hence, there are 3 distinct cube roots of 1.

Step 6: Generalize to Find Sixth Roots and More

To find the sixth roots of 1, we solve \( x^6 = 1 \), which gives 6 solutions: \( x = 1, -1, i, -i, \frac{1}{2} + \frac{i \sqrt{3}}{2}, \frac{1}{2} - \frac{i \sqrt{3}}{2} \). For any \( n \), the equation \( x^n = 1 \) has \( n \) solutions equally spaced around the unit circle in the complex plane.

Step 7: Conjecture about nth Roots of 1

The general pattern shows that an equation \( x^n = 1 \) has exactly \( n \) distinct solutions, known as the nth roots of unity. These roots form a regular polygon with \( n \) sides in the complex plane, each separated by an angle of \( \frac{2\pi}{n} \).

Key Concepts

Complex Numbers

Complex numbers extend the idea of real numbers to include the square roots of negative numbers. They are written in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\). Complex numbers allow us to perform mathematical operations beyond the bounds of real numbers.

• Real Part \(a\): The real part of the complex number.
• Imaginary Part \(bi\): The imaginary part includes \(b\), a real number, and \(i\), the square root of \(-1\).

Their significance lies in the ability to solve all polynomial equations, including those without real solutions, which become especially important in fields like engineering and physics. Complex numbers can be visualized on a plane called the complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part.

Polynomial Equations

Polynomial equations are expressions consisting of variables and coefficients, organized in terms of powers and summed together. For instance, the equation \(x^n - 1= 0\) is a simple polynomial equation when searching for the roots of unity. Solving polynomial equations often means breaking them down into simpler parts, like factoring.
Polynomials can have real roots or complex roots. The Fundamental Theorem of Algebra guarantees that a polynomial will have exactly as many roots as its degree, considering multiplicity.

• Roots: The solutions to the polynomial equation.
• Degree: The highest power of the variable in the polynomial.

The task of finding roots can be simplified using techniques like synthetic division or the quadratic formula, especially when dealing with more complex equations like higher degree polynomials.

Imaginary Unit

The imaginary unit \(i\) is a fundamental concept in mathematics, crucial for understanding complex numbers. By definition, \(i\) is the square root of \(-1\), which makes it unique from real numbers.
Using \(i\), we can express numbers that were previously irreconcilable within the real number system. It opens the door to solving equations like \(x^2 + 1 = 0\), a part of deriving fourth roots of unity or finding cube roots in complex numbers.

• Expression Form: If a problem involves the square root of a negative number, it can be expressed using \(i\).
• Multiplication Rule: \(i^2 = -1\), leading through similar computations to \(i^3 = -i\) and \(i^4 = 1\).

Additionally, the imaginary unit assists in calculating and understanding more about polynomial equations and their roots.

N-th Roots

The \(n\)-th roots of a number, particularly unity (1), are solutions of the equation \(x^n = 1\). These solutions are especially elegant because they display symmetry in the complex plane.
For instance, the fourth roots of unity solve \(x^4 = 1\), which yields \(x = 1, -1, i,\) and \(-i\). Likewise, for cube roots in \(x^3 = 1\), the solutions are evenly spaced angles apart on the circle of radius 1 in the complex plane.

• Unity Root Pattern: The roots form a polygon, known as the regular \(n\)-gon.
• Angle Separation: Each root is separated by \(\frac{2\pi}{n}\) radians.

The general rule is that there are exactly \(n\) distinct \(n\)-th roots of unity, useful in various mathematical applications, including signal processing and discrete Fourier transforms.