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Magazine Chapter 9: Problem 26
Show that the \(n\)th roots of \(z \in \mathcal{C}\) can be represented geometrically as \(n\) equally spaced points on the circle \(x^{2}+y^{2}=|z|^{2}\).
Short Answer
Expert verified
The nth roots of \( z \) form equally spaced points on the circle \( x^2 + y^2 = |z|^2 \).
Step by step solution
01
Identify the Problem
Our goal is to show that the nth roots of a complex number \( z \) form equally spaced points on a circle with radius \( |z| \).
02
Represent the Complex Number in Polar Form
Express the complex number \( z \) in polar form: \( z = |z|e^{i\theta} \). Here, \( |z| \) is the magnitude and \( \theta \) is the argument of \( z \).
03
Calculate the nth Roots Formulaically
The formula for the nth roots of the complex number \( z \) is given by: \[ w_k = |z|^{1/n} e^{i(\theta + 2k\pi)/n} \] for \( k = 0, 1, 2, \, \ldots, \, n-1 \).
04
Understand the Geometrical Implication
Each \( w_k \) is on a circle centered at the origin with radius \( |z|^{1/n} \), due to the magnitude of each root being \( |z|^{1/n} \).
05
Analyze the Angles Between Roots
Since the roots are derived as: \[ \text{Angle}(w_k) = \frac{\theta + 2k\pi}{n} \] The difference between successive angles \( w_k \) and \( w_{k+1} \) is \( \frac{2\pi}{n} \). Hence, all roots are \( \frac{2\pi}{n} \) radians apart.
06
Conclude with Circle Equation Argument
Thus, all the nth roots fit into the circle equation \( x^2 + y^2 = |z|^2 \) because their radius \( |z|^{1/n} \) satisfies the condition provided. These points create a regular n-sided polygon inscribed in this circle.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Representation
A complex number can be effectively represented using polar coordinates, which provide an insightful way to understand its properties. In polar representation:
- The complex number is expressed as: \( z = |z|e^{i\theta} \)
- \( |z| \) is the magnitude, indicating the distance from the origin in the complex plane.
- \( \theta \) is the argument, representing the angle the vector makes with the positive x-axis.
Geometrical Interpretation
The geometrical interpretation of complex numbers provides a visual understanding of their behavior and interaction. Each root of a complex number can be seen as a point in the plane.
- These points lie on a circle centered at the origin.
- The radius of the circle is the nth root of the magnitude of the complex number, \( |z|^{1/n} \).
Complex Number Magnitude
The magnitude of a complex number, often denoted as \( |z| \), is a crucial element in understanding its position on the complex plane. It is akin to the distance from the origin to the point \( z = a + bi \), calculated as:
- \( |z| = \sqrt{a^2 + b^2} \)
- A non-negative value representing the "size" of the complex number.
Circle Equation
The circle equation plays a pivotal role when visualizing the roots of a complex number. The traditional circle equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius. For complex roots:
- This equation helps illustrate how the roots are distributed on the circle.
- Each root satisfies the condition set by the circle with radius \( |z|^{1/n} \).
Equally Spaced Points
When considering nth roots of a complex number, a fascinating result is how these roots are arranged geometrically. They manifest as equally spaced points on a circle, facilitating their comparison and examination.
- The spacing is defined by the angle between consecutive roots: \( \frac{2\pi}{n} \) radians apart.
- This equal angular distribution means the roots form a regular polygon inscribed within the circle.
