The cube roots of unity

Options

A. are collinear
B. lie on a circle of radius ` sqrt(3)`
C. form an equilateral triangle
D. none of these

Answer

To find the cube roots of unity, we start with the equation \( z^3 = 1 \).

### Step 1: Rewrite the equation
We can express the number 1 in exponential form using Euler's formula:
\[ 1 = e^{i \cdot 0} = e^{i(0 + 2k\pi)} \]
where \( k \) is any integer. This accounts for the periodic nature of the exponential function.

### Step 2: Set up the equation for \( z \)
From the equation \( z^3 = e^{i(0 + 2k\pi)} \), we can take the cube root:
\[ z = e^{i\frac{(0 + 2k\pi)}{3}} = e^{i\frac{2k\pi}{3}} \]

### Step 3: Find the roots for different values of \( k \)
We will find the roots by substituting different integer values for \( k \).

- For \( k = 0 \):
\[ z_0 = e^{i\frac{2 \cdot 0 \cdot \pi}{3}} = e^{i \cdot 0} = 1 \]

- For \( k = 1 \):
\[ z_1 = e^{i\frac{2 \cdot 1 \cdot \pi}{3}} = e^{i\frac{2\pi}{3}} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \]

- For \( k = 2 \):
\[ z_2 = e^{i\frac{2 \cdot 2 \cdot \pi}{3}} = e^{i\frac{4\pi}{3}} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \]

### Step 4: Summarize the cube roots of unity
The three cube roots of unity are:
1. \( z_0 = 1 \)
2. \( z_1 = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \) (often denoted as \( \omega \))
3. \( z_2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \) (often denoted as \( \omega^2 \))

### Step 5: Visualize the roots
These roots can be represented on the complex plane. The points \( 1 \), \( \omega \), and \( \omega^2 \) form an equilateral triangle centered at the origin with a radius of 1.

### Final Result
Thus, the cube roots of unity are:
- \( 1 \)
- \( \omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \)
- \( \omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \)

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