Solving X^6 - 1 = 0 A Comprehensive Guide In Rectangular Form

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Introduction

In the captivating realm of complex numbers, solving polynomial equations unveils a fascinating interplay between algebra and geometry. This exploration delves into the specific equation x^6 - 1 = 0, aiming to find all its solutions and express them in the rectangular form. This journey will navigate through the fundamental concepts of complex numbers, De Moivre's Theorem, and the geometric interpretation of roots of unity. Understanding these concepts is crucial for grasping the elegance and symmetry inherent in the solutions of polynomial equations within the complex plane. The equation x^6 - 1 = 0, a seemingly simple algebraic expression, holds within it a wealth of mathematical beauty. Its solutions, when plotted on the complex plane, reveal a pattern of perfect symmetry, forming the vertices of a regular hexagon inscribed in the unit circle. This geometric representation provides a visual understanding of the roots and their relationships to each other. The process of finding these solutions involves not just algebraic manipulation but also a deep appreciation for the properties of complex numbers and their geometric interpretations. We will explore how De Moivre's Theorem, a cornerstone of complex number theory, allows us to efficiently calculate the roots of complex numbers. This theorem provides a powerful tool for solving equations of the form x^n = z, where z is a complex number. By applying De Moivre's Theorem, we can systematically find all n distinct roots of the equation. This exploration will not only provide the solutions to the given equation but also offer a broader understanding of the methods used to solve polynomial equations in the complex domain. It will highlight the interconnectedness of different mathematical concepts and demonstrate how they come together to provide a comprehensive solution. The journey of solving x^6 - 1 = 0 is more than just finding numbers; it's an exploration of mathematical harmony and elegance.

Preliminaries: Complex Numbers and Rectangular Form

Before diving into the solution, let's establish a firm foundation in complex numbers and their representation in rectangular form. A complex number, often denoted as z, is expressed as z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., i^2 = -1). The real part of z is a, denoted as Re(z), and the imaginary part of z is b, denoted as Im(z). The rectangular form, also known as the Cartesian form, provides a way to visualize complex numbers on a two-dimensional plane, called the complex plane or Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Thus, the complex number z = a + bi can be plotted as a point with coordinates (a, b) on the complex plane. This geometric representation allows us to visualize complex numbers as vectors originating from the origin, with magnitude and direction. The magnitude of a complex number z = a + bi, denoted as |z|, is the distance from the origin to the point (a, b) in the complex plane, and it is calculated using the Pythagorean theorem: |z| = √(a^2 + b^2). The magnitude is also referred to as the modulus or absolute value of the complex number. The direction of a complex number, also known as the argument, is the angle θ that the vector makes with the positive real axis. The argument of z, denoted as arg(z), can be found using trigonometric functions: θ = arctan(b/a). However, care must be taken when determining the correct quadrant for θ, as the arctangent function has a period of π, not 2π. Understanding the rectangular form is essential because it provides a direct way to represent and manipulate complex numbers. Addition and subtraction of complex numbers are straightforward in rectangular form: (a + bi) + (c + di) = (a + c) + (b + d)i and (a + bi) - (c + di) = (a - c) + (b - d)i. However, multiplication and division are often more conveniently performed using the polar form of complex numbers, which will be discussed later. The rectangular form serves as a crucial stepping stone for understanding the polar form and the deeper properties of complex numbers. It allows us to connect algebraic representations with geometric interpretations, providing a comprehensive understanding of these mathematical entities.

Transforming the Equation and Applying De Moivre's Theorem

To solve the equation x^6 - 1 = 0, we first rewrite it as x^6 = 1. This form highlights that we are seeking the sixth roots of unity, which are complex numbers that, when raised to the power of 6, result in 1. To find these roots, we need to express 1 in its polar form. Any complex number z can be represented in polar form as z = r(cos θ + i sin θ), where r is the magnitude of z and θ is its argument. In the case of 1, the magnitude is 1, and one possible argument is 0. However, we can add any multiple of 2π to the argument without changing the complex number, since the cosine and sine functions are periodic with a period of 2π. Therefore, we can express 1 as 1 = 1(cos(2πk) + i sin(2πk)), where k is an integer. This representation allows us to capture all possible arguments for the complex number 1. Now, we apply De Moivre's Theorem, a fundamental result in complex number theory, which states that for any complex number z = r(cos θ + i sin θ) and any integer n, (r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ)). Using De Moivre's Theorem, we can express the equation x^6 = 1 in polar form as [r(cos θ + i sin θ)]^6 = 1(cos(2πk) + i sin(2πk)). Applying De Moivre's Theorem to the left side, we get r^6(cos(6θ) + i sin(6θ)) = 1(cos(2πk) + i sin(2πk)). For this equation to hold, the magnitudes must be equal, and the arguments must be equal modulo 2π. This gives us two equations: r^6 = 1 and 6θ = 2πk, where k is an integer. Since r is a non-negative real number, the equation r^6 = 1 implies that r = 1. The equation 6θ = 2πk implies that θ = (πk)/3. This means that the arguments of the sixth roots of unity are multiples of π/3. By varying the integer k, we can find all distinct solutions for θ. However, since the cosine and sine functions have a period of 2π, we only need to consider values of k from 0 to 5 to obtain all unique solutions. This approach systematically combines the polar form of complex numbers with De Moivre's Theorem to transform the original algebraic equation into a set of trigonometric equations that can be easily solved. The resulting solutions represent the sixth roots of unity, which are the vertices of a regular hexagon inscribed in the unit circle on the complex plane.

Finding the Six Roots

With the groundwork laid, we can now systematically find the six distinct roots of the equation x^6 = 1. Recall that we found the magnitude r = 1 and the argument θ = (πk)/3, where k is an integer. To find the six distinct roots, we consider the values of k from 0 to 5. For k = 0, we have θ = 0, so the first root is x_0 = 1(cos(0) + i sin(0)) = 1 + 0i = 1. This is the real root 1. For k = 1, we have θ = π/3, so the second root is x_1 = 1(cos(π/3) + i sin(π/3)) = 1/2 + (√3/2)i. This root lies in the first quadrant of the complex plane. For k = 2, we have θ = 2π/3, so the third root is x_2 = 1(cos(2π/3) + i sin(2π/3)) = -1/2 + (√3/2)i. This root lies in the second quadrant of the complex plane. For k = 3, we have θ = π, so the fourth root is x_3 = 1(cos(π) + i sin(π)) = -1 + 0i = -1. This is another real root, -1. For k = 4, we have θ = 4π/3, so the fifth root is x_4 = 1(cos(4π/3) + i sin(4π/3)) = -1/2 - (√3/2)i. This root lies in the third quadrant of the complex plane. For k = 5, we have θ = 5π/3, so the sixth root is x_5 = 1(cos(5π/3) + i sin(5π/3)) = 1/2 - (√3/2)i. This root lies in the fourth quadrant of the complex plane. Thus, we have found all six distinct roots of the equation x^6 = 1, expressed in rectangular form. These roots are 1, 1/2 + (√3/2)i, -1/2 + (√3/2)i, -1, -1/2 - (√3/2)i, and 1/2 - (√3/2)i. When plotted on the complex plane, these roots form the vertices of a regular hexagon inscribed in the unit circle, with one vertex at 1. This geometric interpretation highlights the symmetry inherent in the solutions of polynomial equations in the complex domain. The roots are equally spaced around the unit circle, reflecting the fact that they are the sixth roots of unity. This process demonstrates how De Moivre's Theorem and the polar form of complex numbers provide a powerful method for finding the roots of complex numbers. By systematically varying the argument, we can generate all distinct solutions to the equation. The rectangular form of the roots allows for easy visualization and manipulation, providing a complete understanding of the solutions to the equation x^6 - 1 = 0.

Rectangular Form Solutions

In summary, the six solutions to the equation x^6 - 1 = 0, expressed in rectangular form, are:

  • x_0 = 1
  • x_1 = 1/2 + (√3/2)i
  • x_2 = -1/2 + (√3/2)i
  • x_3 = -1
  • x_4 = -1/2 - (√3/2)i
  • x_5 = 1/2 - (√3/2)i

These complex numbers represent the vertices of a regular hexagon inscribed in the unit circle on the complex plane. The solutions exhibit a beautiful symmetry, reflecting the inherent properties of roots of unity. The real roots are 1 and -1, while the other four roots are complex conjugates, occurring in pairs. This symmetry is a direct consequence of the properties of complex numbers and their geometric representation. The solutions are equally spaced around the unit circle, with an angular separation of π/3 radians between consecutive roots. This uniform distribution of roots is a characteristic feature of roots of unity. The rectangular form of the solutions provides a clear representation of the real and imaginary components of each root. This form is particularly useful for performing arithmetic operations on the roots, such as addition, subtraction, multiplication, and division. For example, the sum of the six roots is 0, which is a general property of the roots of a polynomial equation. The rectangular form also allows for easy visualization of the roots on the complex plane. Each root can be plotted as a point with coordinates corresponding to its real and imaginary parts. This geometric representation provides a visual understanding of the roots and their relationships to each other. The solutions to the equation x^6 - 1 = 0 are not just numbers; they are points in the complex plane that possess specific geometric properties. The symmetry and uniform distribution of these roots reflect the underlying mathematical structure of the equation and its solutions. This exploration of the sixth roots of unity provides a glimpse into the rich and fascinating world of complex numbers and their applications in various fields of mathematics and science. The rectangular form of the solutions serves as a valuable tool for understanding and manipulating these fundamental mathematical entities.

Conclusion

This exploration has successfully unveiled all six solutions of the equation x^6 - 1 = 0 and expressed them in the rectangular form. By leveraging the power of De Moivre's Theorem and the geometric interpretation of complex numbers, we have demonstrated a systematic approach to solving polynomial equations in the complex domain. The solutions, when visualized on the complex plane, form a regular hexagon, a testament to the inherent symmetry in the roots of unity. This journey through complex numbers has not only provided the solutions to a specific equation but has also highlighted the broader concepts and techniques used in complex analysis. The application of De Moivre's Theorem, the understanding of polar and rectangular forms, and the geometric interpretation of complex numbers are all essential tools in the mathematician's arsenal. The solutions to x^6 - 1 = 0 serve as a concrete example of the more general theory of roots of unity. The same methods can be applied to find the nth roots of unity for any positive integer n. The roots will always form the vertices of a regular n-sided polygon inscribed in the unit circle on the complex plane. This geometric representation provides a powerful visual aid for understanding the solutions and their properties. The rectangular form of the solutions allows for easy manipulation and calculation. It is often the preferred form for performing arithmetic operations on complex numbers and for connecting the algebraic and geometric representations. The study of complex numbers and their roots has far-reaching applications in various fields, including physics, engineering, and computer science. Complex numbers are used to model alternating current circuits, quantum mechanics, and signal processing, among other things. The solutions to polynomial equations in the complex domain play a crucial role in these applications. This exploration has provided a comprehensive understanding of the solutions to x^6 - 1 = 0, emphasizing both the algebraic and geometric aspects. The journey has demonstrated the elegance and power of complex number theory and its applications in solving mathematical problems and modeling real-world phenomena. The six roots of unity, expressed in rectangular form, stand as a testament to the beauty and symmetry inherent in mathematics.