Problem 93 - 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 \(n\) th 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

For part (a), \(1, w, w^2, \ldots, w^{n-1}\) are the distinct \(n\)th roots of 1. For part (b), \(s, s w, s w^2, \ldots, s w^{n-1}\) are the distinct \(n\)th roots of \(z\).

Step 1: Define the properties of w

The given complex number is \( w = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n} \). By Euler's formula, this can be rewritten as \( w = e^{i \frac{2\pi}{n}} \). \( w \) represents a primitive \( n \)th root of unity, known as the generator of the \( n \)th roots of unity.

Step 2: Prove that w^n = 1

Compute \( w^{n} \): \[ w^{n} = (e^{i \frac{2\pi}{n}})^{n} = e^{i 2\pi} = 1 \]Thus, \( w^{n} = 1 \), verifying that \( w \) is an \( n \)th root of unity.

Step 3: Show distinctness of roots

The sequence \( 1, w, w^2, \ldots, w^{n-1} \) represents the powers of \( w \) starting from 0 up to \( n-1 \). If any two terms in this sequence were equal \( w^a = w^b \) for \( 0 \leq a < b < n \), it would imply \( w^{b-a} = 1 \). Thus, \( b-a = kn \) for some integer \( k \), but \( 0 < b-a < n \) which contradicts \( b-a \) being a multiple of \( n \). Thus, all powers are distinct.

Step 4: Complete part (a) conclusion

1, \( w, w^2, \ldots, w^{n-1} \) are exactly the \( n \) distinct \( n \)th roots of \( 1 \), satisfying \( x^n = 1 \).

Step 5: Define complex powers for part (b)

Assume \( s^n = z \) where \( z eq 0 \) is a complex number. The solution \( s \) can be any particular \( n \)th root of \( z \). One such root is given by \( s = |z|^{1/n} e^{i \arg(z)/n} \).

Step 6: Generate all n roots of z

The \( n \)th roots of \( z \) are obtained by multiplying the root \( s \) by each of the \( n \)th roots of unity: \( s, s w, s w^2, \ldots, s w^{n-1} \). Each powers \( s w^k \) for \( k=0,1,\ldots,n-1 \) yields a distinct solution.

Step 7: Conclude part (b) proof

Thus, the \( n \) distinct \( n \)th roots of \( z \) are \( s, s w, s w^2, \ldots, s w^{n-1} \), completing the solution for part (b).

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, crucial when dealing with equations that have no real solutions. A complex number is expressed in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
Complex numbers can be visualized in the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This allows complex numbers to be plotted similarly to how coordinates are plotted on a Cartesian plane. They help in solving equations such as \( x^2 + 1 = 0 \), which cannot be solved using real numbers alone.
Moreover, every complex number can also be represented in polar form, which is particularly useful when dealing with multiplication, division, and finding powers and roots of complex numbers. In polar form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the argument of the complex number. This polar form can also be rewritten using Euler's formula as \( re^{i\theta} \).

Euler's Formula

Euler's formula is an important mathematical formula establishing the deep relationship between complex numbers and trigonometry. It states that for any real number \( x \):
\( e^{ix} = \cos x + i \sin x \)
This formula is incredibly useful for converting complex numbers from their trigonometric form to their exponential form. For instance, if a complex number is given by \( \cos \theta + i \sin \theta \), it can be expressed using Euler's formula as \( e^{i\theta} \). This simplification helps in operations involving powers and roots, making calculations more straightforward.
Euler's formula also highlights the beauty and interconnectedness of different mathematical areas. It's used extensively in fields like engineering, physics, and number theory. The formula simplifies understanding concepts like rotations in the complex plane and makes complex algebra easier to manage.

Roots of Complex Numbers

The roots of complex numbers extend the concept of finding roots in the realm of real numbers. If you have a complex number \( z \) and you want to find its \( n \)th roots, you seek complex numbers \( r \) that satisfy the equation \( r^n = z \). These roots can have both real and imaginary components.
To find these roots, the complex number is usually represented in polar form \( z = re^{i\theta} \). Then, the \( n \)th roots are calculated as \( r^{1/n}e^{i(\theta + 2k\pi)/n} \) for \( k = 0, 1, 2, ..., n-1 \). This accounts for all possible roots because \( 2\pi \) represents a full rotation in the complex plane.
Each of these roots corresponds to a point on the complex plane, equidistantly spaced on the circle with radius \( r^{1/n} \). The number and arrangement of these roots obey the symmetry dictating that, for instance, the square roots of \(-1\) are \( i \) and \(-i \).

Primitive Roots of Unity

Primitive roots of unity are special complex numbers that satisfy the equation \( x^n = 1 \) and cannot be simplified to a lower power. These numbers are evenly spaced on the unit circle in the complex plane.
Specifically, a primitive \( n \)th root of unity is a complex number \( w = e^{i2\pi/n} \) such that \( w, w^2, ..., w^{n-1} \) are the \( n \) distinct solutions to \( w^n = 1 \), excluding lower powers like \( w^k = 1 \) for \( k < n \). This ensures that each root is unique for \( n \) and is a power of \( w \).
These primitive roots help us understand the fundamental cyclic nature of complex number multiplication. They're crucial in number theory, particularly in modern cryptography, signal processing, and fields where symmetry and periodicity play significant roles.
In essence, primitive roots of unity encapsulate the idea of dividing a circle into equal parts, making them essential not only in pure mathematics but also in applications that rely on uniform distribution and repetition.