Cube roots of unity
Cube Roots of Unity
The cube roots of unity are the complex numbers that, when raised to the power of three, equal one. In other words, they are the solutions to the equation $x^3 = 1$. There are exactly three cube roots of unity, one of which is the real number 1, and the other two are complex numbers.
Representation of Cube Roots of Unity
The cube roots of unity can be represented in the complex plane. The complex plane is a two-dimensional plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
The general form of a complex number is $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part. For cube roots of unity, we are looking for complex numbers $z$ such that $z^3 = 1$.
Calculation of Cube Roots of Unity
To find the cube roots of unity, we solve the equation $z^3 = 1$. This can be rewritten as $z^3 - 1 = 0$. Factoring this equation gives us:
$$ z^3 - 1 = (z - 1)(z^2 + z + 1) = 0 $$
This equation has three roots:
- $z_1 = 1$
- $z_2 = \frac{-1 + \sqrt{3}i}{2}$
- $z_3 = \frac{-1 - \sqrt{3}i}{2}$
These roots are equidistant from each other and are at the vertices of an equilateral triangle in the complex plane, centered at the origin.
Table of Cube Roots of Unity
| Root | Expression | Polar Form | Exponential Form |
|---|---|---|---|
| $z_1$ | $1$ | $1(\cos(0) + i\sin(0))$ | $e^{0i}$ |
| $z_2$ | $\frac{-1 + \sqrt{3}i}{2}$ | $1(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))$ | $e^{\frac{2\pi i}{3}}$ |
| $z_3$ | $\frac{-1 - \sqrt{3}i}{2}$ | $1(\cos(\frac{4\pi}{3}) + i\sin(\frac{4\pi}{3}))$ | $e^{\frac{4\pi i}{3}}$ |
Properties of Cube Roots of Unity
- The sum of the cube roots of unity is zero: $1 + \frac{-1 + \sqrt{3}i}{2} + \frac{-1 - \sqrt{3}i}{2} = 0$.
- The product of the cube roots of unity is also one: $(1) \left(\frac{-1 + \sqrt{3}i}{2}\right) \left(\frac{-1 - \sqrt{3}i}{2}\right) = 1$.
- The cube roots of unity form a group under multiplication.
Examples
Example 1: Sum of Cube Roots of Unity
Calculate the sum of all the cube roots of unity.
$$ 1 + \frac{-1 + \sqrt{3}i}{2} + \frac{-1 - \sqrt{3}i}{2} = 1 - \frac{1}{2} - \frac{1}{2} + \frac{\sqrt{3}i}{2} - \frac{\sqrt{3}i}{2} = 0 $$
Example 2: Product of Cube Roots of Unity
Calculate the product of all the cube roots of unity.
$$ (1) \left(\frac{-1 + \sqrt{3}i}{2}\right) \left(\frac{-1 - \sqrt{3}i}{2}\right) = 1 \cdot \frac{1 - \sqrt{3}i + \sqrt{3}i - 3i^2}{4} = \frac{1 + 3}{4} = 1 $$
Example 3: Visual Representation
Plot the cube roots of unity on the complex plane.
The real root $z_1 = 1$ is at the point (1, 0). The other two roots, $z_2$ and $z_3$, are at the points $\left(\frac{-1}{2}, \frac{\sqrt{3}}{2}\right)$ and $\left(\frac{-1}{2}, \frac{-\sqrt{3}}{2}\right)$, respectively. These points form an equilateral triangle with the center at the origin (0, 0).
Understanding the cube roots of unity is essential for solving polynomial equations and for various applications in complex analysis, signal processing, and other fields of mathematics and engineering.