Problem 97 - Algebra and Trigonometry

Question

(a) Let \(w=\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\) where \(n\) is a positive integer. Show that \(1, w, w^{2}, w^{3}, \ldots, w^{n-1}\) are the \(n\) distinct nth roots of 1 . (b) If \(z \neq 0\) is any complex number and \(s^{n}=z,\) show that the \(n\) distinct \(n\) th roots of \(z\) are $$ s, s w, s w^{2}, s w^{3}, \ldots, s w^{n-1} $$

Step-by-Step Solution

Answer

(a) 1, w, w^2, ..., w^{n-1} are distinct nth roots of 1; (b) s, sw, sw^2, ..., sw^{n-1} are distinct nth roots of z.

Step 1: Understand the Complex Number

We start with the given complex number \( w = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n} \). This is a standard polar form of a complex number on the unit circle, also known as the complex nth root of unity \( w = e^{i \frac{2\pi}{n}} \).

Step 2: Confirm \( w^n = 1 \)

Using Euler's formula, we know that \( \left ( e^{i\theta} \right )^n = e^{i n \theta} \). Thus, \( w^n = \left(e^{i \frac{2\pi}{n}}\right)^n = e^{i 2\pi} = 1 \). This confirms that \( w \) raised to the power of \( n \) equals 1.

Step 3: Identify All nth Roots of Unity

The nth roots of unity are given by \( w^k = \left(\cos \frac{2\pi k}{n} + i \sin \frac{2\pi k}{n} \right) \) for \( k = 0, 1, 2, \ldots, n-1 \). These correspond to \( 1, w, w^2, \ldots, w^{n-1} \), all distinct due to the periodic nature of the sine and cosine functions.

Step 4: Verify Distinctness

To show that these are distinct, note that if \( w^a = w^b \), for some integers \( 0 \leq a < b < n \), then \( w^{b-a} = 1 \), implying \( \frac{2\pi (b-a)}{n} = 2\pi \times m \) for integer \( m \). Simplifying gives \( (b-a) = mn \). Since \( 0 < b-a < n \), the only possible value for \( m \) is 0, so \( a = b \), proving distinctness.

Step 5: Determine nth Roots of \( z \)

Given that \( s^n = z \), the nth roots of \( z \) can be expressed as \( s w^k \) for \( k = 0, 1, 2, \ldots, n-1 \). Each root \( s w^k \) raised to the nth power gives \( (s w^k)^n = s^n (w^k)^n = z \cdot 1 = z \).

Step 6: Verify Distinctness for \( z \)

Similar to the roots of unity, any two roots \( s w^a \) and \( s w^b \) are distinct if \( w^a eq w^b \), which we have already shown in Step 4. Hence, \( s w^k \) are the distinct nth roots of \( z \).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be beautifully represented using their polar form. This representation is particularly handy when dealing with complex numbers on a unit circle. In polar form, any complex number is written as \( z = r (\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude of the complex number.
• \( \theta \) is the argument or angle, measured from the positive x-axis in the complex plane.
• \( i \) is the imaginary unit, which satisfies \( i^2 = -1 \).

The polar form makes multiplication and division of complex numbers straightforward. When the magnitude \( r \) is 1, we call this the unit circle, where the complex number can be specifically noted as \( \cos \theta + i \sin \theta \). This representation is useful in identifying points on the unit circle in the complex plane. For instance, the number \( w = \cos \frac{2 \pi}{n} + i \sin \frac{2 \pi}{n} \) lies on the unit circle and is an example of a complex number in polar form. This is especially useful when finding complex roots.

Euler's Formula

Euler's formula, named after the Swiss mathematician Leonhard Euler, is a fundamental bridge between trigonometry and complex exponentials. It states that for any real number \( \theta \):\[e^{i\theta} = \cos \theta + i \sin \theta\]
This formula allows us to express complex numbers in exponential form. It is particularly powerful when working with powers and roots of complex numbers. For example, the complex number \( w = \cos \frac{2 \pi}{n} + i \sin \frac{2 \pi}{n} \) can be written using Euler's formula as \( w = e^{i\frac{2\pi}{n}} \).
This form is powerful for calculations involving powers. For instance, raising \( w \) to the \( n \)th power using this exponential form simplifies to:

• \( w^n = \left(e^{i\frac{2\pi}{n}}\right)^n = e^{i 2\pi} = 1 \)
• This confirms that the \( n \)-th power of \( w \) equals 1, making it an \( n \)-th root of unity.

Understanding Euler's formula allows you to manipulate and comprehend complex numbers more efficiently.

Nth Roots of a Complex Number

Exploring the concept of nth roots of complex numbers brings us to the fascinating ideas of symmetry and harmony in the complex plane. The nth root of a number \( z \), denoted by \( s \), satisfies:\[s^n = z\]
To find these roots, start with one known root \( s \), and utilize the nth roots of unity, specifically \( w = e^{i\frac{2\pi}{n}} \), as multiplicative factors. The distinct nth roots of \( z \) are then given by the expressions:

• \( s, sw, sw^2, \ldots, sw^{n-1} \)

Each of these roots, when raised to the nth power, will satisfy the equation \( s^n = z \). Therefore, the values \( sw^k \), for \( k = 0, 1, 2, \ldots, n-1 \), represent distinct roots.
The distinctness arises because the powers of \( w \) are separated evenly around the unit circle, and thanks to the properties of sine and cosine functions, ensure there are no repetitions among \( s, sw, sw^2, \ldots \). This concept is not only mathematically profound but also visually and geometrically satisfying.