Understanding Cube Roots of Unity

The cube roots of unity are special complex numbers that are the solutions to the equation $z^3 = 1$. Finding these roots involves understanding complex numbers in polar form.

Finding the Cube Roots of Unity

We solve the equation $z^3 = 1$. In the complex plane, the number 1 can be written in polar form as $1 = 1 \cdot (\cos(2\pi k) + i \sin(2\pi k))$ for any integer $k$. Using Euler's formula, this is $1 = 1 \cdot e^{i(2\pi k)}$.

To find the cube roots, we take the cube root of both sides:

$z = (1 \cdot e^{i(2\pi k)})^{1/3} = 1^{1/3} \cdot e^{i(2\pi k/3)}$

Since $1^{1/3}$ in real numbers is just 1, we get:

$z_k = e^{i(2\pi k/3)}$, for $k=0, 1, 2$. We use $k=0, 1, 2$ to find the distinct roots.

• For $k=0$: $z_0 = e^{i(0)} = \cos(0) + i\sin(0) = 1 + i(0) = 1$.
• For $k=1$: $z_1 = e^{i(2\pi/3)} = \cos(2\pi/3) + i\sin(2\pi/3) = -1/2 + i\sqrt{3}/2$. This root is often denoted by $\omega$.
• For $k=2$: $z_2 = e^{i(4\pi/3)} = \cos(4\pi/3) + i\sin(4\pi/3) = -1/2 - i\sqrt{3}/2$. This root is often denoted by $\omega^2$.

The three cube roots of unity are $1$, $-1/2 + i\sqrt{3}/2$, and $-1/2 - i\sqrt{3}/2$.

Geometric Representation in the Argand Plane

We can visualize these complex numbers as points in the complex plane (also called the Argand plane). The point corresponding to $x+iy$ is $(x, y)$.

• The root $1$ corresponds to the point $(1, 0)$.
• The root $-1/2 + i\sqrt{3}/2$ corresponds to the point $(-1/2, \sqrt{3}/2)$.
• The root $-1/2 - i\sqrt{3}/2$ corresponds to the point $(-1/2, -\sqrt{3}/2)$.

Let's examine the geometric properties based on these points.

Evaluating the Geometric Properties of Cube Roots of Unity

Option 1: They are collinear

Collinear points lie on a single straight line. The points are $A=(1, 0)$, $B=(-1/2, \sqrt{3}/2)$, and $C=(-1/2, -\sqrt{3}/2)$. Points B and C have the same x-coordinate, $-1/2$. The line passing through B and C is the vertical line $x = -1/2$. Point A has an x-coordinate of 1. Since $1 \neq -1/2$, point A does not lie on the line $x = -1/2$. Therefore, the three points are not collinear.

Option 2: They lie on a circle of radius $\sqrt{3}$

The magnitude of a complex number $z = x+iy$ is $|z| = \sqrt{x^2 + y^2}$, which represents its distance from the origin $(0,0)$ in the complex plane. Let's calculate the magnitudes of the cube roots:

• Magnitude of $1$: $|1| = \sqrt{1^2 + 0^2} = \sqrt{1} = 1$.
• Magnitude of $-1/2 + i\sqrt{3}/2$: $|-1/2 + i\sqrt{3}/2| = \sqrt{(-1/2)^2 + (\sqrt{3}/2)^2} = \sqrt{1/4 + 3/4} = \sqrt{1} = 1$.
• Magnitude of $-1/2 - i\sqrt{3}/2$: $|-1/2 - i\sqrt{3}/2| = \sqrt{(-1/2)^2 + (-\sqrt{3}/2)^2} = \sqrt{1/4 + 3/4} = \sqrt{1} = 1$.

All three cube roots of unity have a magnitude of 1. This means they all lie on a circle centered at the origin with a radius of 1. They do not lie on a circle of radius $\sqrt{3}$.

Option 3: They form an equilateral triangle

To determine if the points form an equilateral triangle, we calculate the distances between each pair of points. Let the points be $P_0(1, 0)$, $P_1(-1/2, \sqrt{3}/2)$, and $P_2(-1/2, -\sqrt{3}/2)$. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

• Distance between $P_0$ and $P_1$: $\sqrt{(-1/2 - 1)^2 + (\sqrt{3}/2 - 0)^2} = \sqrt{(-3/2)^2 + (\sqrt{3}/2)^2} = \sqrt{9/4 + 3/4} = \sqrt{12/4} = \sqrt{3}$.
• Distance between $P_1$ and $P_2$: $\sqrt{(-1/2 - (-1/2))^2 + (-\sqrt{3}/2 - \sqrt{3}/2)^2} = \sqrt{(0)^2 + (-\sqrt{3})^2} = \sqrt{0 + 3} = \sqrt{3}$.
• Distance between $P_2$ and $P_0$: $\sqrt{(1 - (-1/2))^2 + (0 - (-\sqrt{3}/2))^2} = \sqrt{(3/2)^2 + (\sqrt{3}/2)^2} = \sqrt{9/4 + 3/4} = \sqrt{12/4} = \sqrt{3}$.

Since all three distances between the pairs of points are equal (each is $\sqrt{3}$), the three points form an equilateral triangle.

Option 4: None of the above

As we have determined that the cube roots of unity form an equilateral triangle, Option 3 is correct, making this option incorrect.

Conclusion on Geometric Shape of Cube Roots of Unity

The geometric representation of the cube roots of unity in the complex plane shows that they are located at three points on the unit circle, equally spaced at angles of $120^\circ$ from each other. These three points form the vertices of an equilateral triangle inscribed within the unit circle centered at the origin.

Revision Table: Key Properties of Cube Roots of Unity

Additional Information: General nth Roots of Unity

The concept explored with cube roots of unity extends to the $n$th roots of unity. These are the solutions to the equation $z^n = 1$ for any positive integer $n$.

The $n$th roots of unity are given by the formula $z_k = e^{i(2\pi k/n)}$ for $k = 0, 1, 2, \dots, n-1$. There are exactly $n$ distinct $n$th roots of unity.

Geometrically, these $n$ roots are represented by $n$ points in the complex plane that are equally spaced around the unit circle $|z|=1$, centered at the origin. These points form the vertices of a regular $n$-sided polygon (a regular n-gon) inscribed within the unit circle.

• For $n=2$, the square roots of unity are $1$ and $-1$, forming a line segment (a degenerate 2-gon).
• For $n=3$, the cube roots of unity form a regular 3-gon, which is an equilateral triangle.
• For $n=4$, the fourth roots of unity are $1, i, -1, -i$, forming a regular 4-gon, which is a square.
• For $n=5$, the fifth roots of unity form a regular pentagon.

The general principle is that the $n$th roots of unity always form the vertices of a regular $n$-gon inscribed in the unit circle.