Problem 65 - Algebra and Trigonometry

Question

Find the four complex fourth roots of unity, \(1,\) and plot them.

Step-by-Step Solution

Answer

The four complex fourth roots of unity are \( 1, i, -1, -i \).

Step 1: Understand the Problem

We need to find all the complex numbers that satisfy the equation \[ z^4 = 1 \]. These are called the fourth roots of unity.

Step 2: Represent 1 in Polar Form

The number 1 can be represented in polar form as \[ 1 = e^{i \times 2\text{π}k} \] where \( k \) is any integer. For our purposes, we can use \( 0, 1, 2, 3 \).

Step 3: Solve for the Roots

To find the fourth roots, solve \[ z^4 = e^{i \times 2\text{π}k} \]. This leads us to \[ z = e^{i \times \frac{2\text{π}k}{4}} \]. For \( k=0, 1, 2, 3 \), we get four roots: \( z_0, z_1, z_2, z_3 \).

Step 4: Compute Each Root

Substitute \( k = 0, 1, 2, 3 \): \[ z_0 = e^{i \times \frac{2\text{π} \times 0}{4}} = e^{i \times 0} = 1 \]\[ z_1 = e^{i \times \frac{2\text{π} \times 1}{4}} = e^{i \times \frac{\text{π}}{2}} = i \]\[ z_2 = e^{i \times \frac{2\text{π} \times 2}{4}} = e^{i \times \text{π}} = -1 \]\[ z_3 = e^{i \times \frac{2\text{π} \times 3}{4}} = e^{i \times \frac{3\text{π}}{2}} = -i \]

Step 5: Plot the Roots

Plot these roots on the complex plane. These points \( 1, i, -1, -i \) will lie on the unit circle at the angles \( 0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2} \).

Key Concepts

roots of unity

Roots of unity are special complex numbers that are solutions to equations of the form \( z^n = 1 \). In our case, we are looking for the fourth roots of unity. These are the complex numbers where \( z^4 = 1 \).
To find these roots, we use the polar form of complex numbers and solve for the angles that make this equation true. The general form for the nth roots of unity is given by:
\[ z = e^{i \times \frac{2\text{π}k}{n}} \] where \( k \) can be any integer from \( 0 \) to \( n-1 \).
In our problem, with \( n=4 \), we have the roots:
\begin{align*} z_0 &= e^{i \times 0} = 1 \ z_1 &= e^{i \times \frac{\text{π}}{2}} = i \ z_2 &= e^{i \times \text{π}} = -1 \ z_3 &= e^{i \times \frac{3\text{π}}{2}} = -i \end{align*}
These four roots are evenly spaced around the unit circle in the complex plane.

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
In the context of roots of unity, we encounter several basic complex numbers:

• 1 (a real number)
• i (the imaginary unit)
• -1 (a real number)
• -i (negative of the imaginary unit)

These complex numbers can be represented graphically on the complex plane, with real parts on the x-axis and imaginary parts on the y-axis.
Understanding these representations helps in visually comprehending how they relate to the unit circle and how they form the vertices of a polygon in the complex plane.

polar coordinates

Polar coordinates provide a different way to describe complex numbers, using their magnitude and angle, rather than their real and imaginary parts directly.
Any complex number can be written as \( z = re^{i\theta} \), where \( r \) (the magnitude) is the distance from the origin to the point, and \( \theta \) (the angle) specifies the direction of the point from the positive x-axis.
For the fourth roots of unity, the magnitude \( r \) is 1, because they lie on the unit circle. The angles \( \theta \) are multiples of \( \frac{\text{π}}{2} \).

• \( z_0 = e^{i \times 0} = 1 \)
• \( z_1 = e^{i \times \frac{\text{π}}{2}} = i \)
• \( z_2 = e^{i \times \text{π}} = -1 \)
• \( z_3 = e^{i \times \frac{3\text{π}}{2}} = -i \)

unit circle

The unit circle is a vital concept in understanding complex roots of unity. It's a circle in the complex plane with a radius of 1 centered at the origin. Any complex number on this circle has a magnitude of 1.
When finding roots of unity, the solutions are points on the unit circle. For our fourth roots of unity, the points are:

• \( 1 \) at an angle of \( 0 \) radians
• \( i \) at an angle of \( \frac{\text{π}}{2} \) radians
• \( -1 \) at an angle of \( \text{π} \) radians
• \( -i \) at an angle of \( \frac{3\text{π}}{2} \) radians

These points divide the unit circle into four equal segments, each corresponding to one of the fourth roots of unity.
Understanding the unit circle and its properties is crucial for grasping how complex numbers behave under multiplication and rotation.