Problem 82 - College Trigonometry

Question

Given any natural number \(n \geq 2\), the \(n\) complex \(n^{\text {th }}\) roots of the number \(z=1\) are called the \(\boldsymbol{n}^{\text {th }}\) Roots of Unity. In the following exercises, assume that \(n\) is a fixed, but arbitrary, natural number such that \(n \geq 2\). (a) Show that \(w=1\) is an \(n^{\text {th }}\) root of unity. (b) Show that if both \(w_{j}\) and \(w_{k}\) are \(n^{\text {th }}\) roots of unity then so is their product \(w_{j} w_{k}\). (c) Show that if \(w_{j}\) is an \(n^{\text {th }}\) root of unity then there exists another \(n^{\text {th }}\) root of unity \(w_{j^{\prime}}\) such that \(w_{j} w_{j^{\prime}}=1 .\) Hint: If \(w_{j}=\operatorname{cis}(\theta)\) let \(w_{j^{\prime}}=\operatorname{cis}(2 \pi-\theta)\). You'll need to verify that \(w_{j^{\prime}}=\operatorname{cis}(2 \pi-\theta)\) is indeed an \(n^{\text {th }}\) root of unity.

Step-by-Step Solution

Answer

(a) Yes, w=1 is an nth root. (b) Product of two roots is also a root. (c) Every root has a reciprocal root such that their product is 1.

Step 1: Define the nth root of unity

The nth roots of unity are the solutions to the equation \(z^n = 1\). These roots can be expressed as: \[ w_k = e^{2k\pi i / n} \quad \text{for} \quad k = 0, 1, 2, \ldots, n-1. \] Notice that these are complex numbers representing evenly distributed points on the unit circle.

Step 2: Show w = 1 is an nth root of unity

Let \(w = 1\). We want to verify that \(w^n = 1\). Substituting \(w = 1\) into the expression, we have: \[ w^n = 1^n = 1. \] Thus, \(w = 1\) is indeed an nth root of unity.

Step 3: Show product of roots is a root

Suppose \(w_j\) and \(w_k\) are nth roots of unity so that \(w_j = e^{2\pi i j / n}\) and \(w_k = e^{2\pi i k / n}\). The product is \(w_j w_k = e^{2\pi i j / n} \cdot e^{2\pi i k / n} = e^{2\pi i (j+k) / n}\). Since \((j+k)\) can be taken modulo \(n\), \(e^{2\pi i (j+k) / n}\) is also of the form \(e^{2k\pi i / n}\), a root of unity.

Step 4: Show existence of complex conjugate root

Let \(w_j = e^{2\pi i j / n}\). The complex conjugate is \(w_j' = e^{-2\pi i j / n}\). Verify that \(w_j'\) is a root by showing \((w_j')^n = 1\). Compute: \[ (w_j')^n = \left(e^{-2\pi i j / n}\right)^n = e^{-2\pi i j} = 1^n = 1. \] Thus, \(w_j'\) is also an nth root of unity and the product \(w_j w_j' = e^{2\pi i j / n} \cdot e^{-2\pi i j / n} = e^0 = 1\).

Key Concepts

Complex Numbers

To understand the concept of roots of unity, it is essential first to grasp the idea of complex numbers. Complex numbers extend regular numbers by including the square root of -1, denoted as the imaginary unit **i**. A typical complex number looks like **a + bi**, where **a** and **b** are real numbers. In the context of roots of unity, complex numbers are used to explore how values can represent directions in a plane. These numbers help us express the roots of unity using Euler's formula, often represented as \(e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta)\). This formula is pivotal for deriving the roots as it allows complex numbers to be plotted on the unit circle, revealing patterns such as symmetry and periodicity.

Unit Circle

The unit circle is a central concept in the understanding of roots of unity. It is a circle in the complex plane that has a radius of 1 and is centered at the origin \((0,0)\). Any complex number on this circle takes the form \(w = e^{i \theta}\), where \(\theta\) is the angle from the positive x-axis. This representation is helpful for understanding how complex roots are distributed. Because the nth roots of unity are equally spaced points on this circle, they form a regular polygon (such as an equilateral triangle, square, etc.) based on the value of **n**. By visualizing roots on the unit circle, it becomes clearer how multiplication of these roots corresponds to rotation around the origin, a key property used in mathematical proofs of roots.

Mathematical Proofs

Mathematical proofs are formal arguments that verify the truth of a statement. In the case of showing properties of nth roots of unity, proofs are used to demonstrate:

• that 1 is an nth root of unity, proven by substituting into the equation \(w^n = 1\) yielding \(1^n = 1\),
• that the product of two nth roots is another nth root, verified by combining exponential forms \(e^{2\pi i j / n}\) and \(e^{2\pi i k / n}\) to form \(e^{2\pi i (j+k) / n}\),
• and the existence of an inverse root \(w_j' = e^{-2\pi i j / n}\), shown via exponentiation to verify it achieves \((w_j')^n = 1\).

These proofs rely heavily on properties of exponents and complex numbers, underlining the interconnected structure of these mathematical elements.

n-th Roots

The concept of nth roots extends square roots to any natural number \(n\). In our specific discussion of roots of unity, an nth root of 1 (unity) is a number that, when raised to the power of \(n\), equals 1. These roots are usually expressed in their complex exponential form as \(w_k = e^{2k\pi i / n}\), where \(k\) ranges from 0 to \(n-1\). The full set of these roots forms a geometric sequence on the unit circle, dividing the circle into equal segments. Understanding these discrete points helps cultivate deeper insights into their properties such as rotation and symmetry. The exploration of nth roots not only broadens comprehension of how numbers behave in higher dimensions but also enriches understanding through concepts like periodicity and resonance, relevant in fields ranging from quantum physics to signal processing.