Question

Find all the values of the indicated roots and plot them. $$ \sqrt[3]{1} $$

Short answer

The roots are \(1\), \(-\frac{1}{2} + i\sqrt{3}/2\), and \(-\frac{1}{2} - i\sqrt{3}/2\). Plot them on the complex plane.

Step-by-step solution

Understand the Problem

Given is \(\root[3]{1}\), which requires finding all complex numbers \(z\) such that \(z^3 = 1\).

Express 1 in Polar Form

The number 1 can be expressed in polar form as \(1 = e^{i \theta}\). Since \(e^{i \theta} = \cos \theta + i \sin \theta \), we can choose \(\theta = 0 + 2k\frac{\pi}{3}\) for \(\root[n]{1}\).

Identify the Principal Root

For \(\theta = 0\), the principal root of the equation is \(z_0 = e^{i \cdot 0} = 1\).

Calculate the Remaining Roots

The remaining roots are obtained by adding \(\frac{2\pi}{3}\) and \(2 \cdot \frac{2\pi}{3}\) to the principal argument and evaluating: \(z_1 = e^{i \frac{2\pi}{3}} = \cos(\frac{2\pi}{3}) + i \sin(\frac{2\pi}{3}) = -\frac{1}{2} + i \cdot \frac{\root{3}{4}}\) and \(z_2 = e^{i \frac{4\pi}{3}} = \cos(\frac{4\pi}{3}) + i \sin(\frac{4\pi}{3}) = -\frac{1}{2} - i \cdot \frac{\root{3}{4}}\).

Plot the Roots on the Complex Plane

The roots \(z_0 = 1, z_1 = -\frac{1}{2} + i \sqrt{3}/2,\) and \(z_2 = -\frac{1}{2} - i \sqrt{3}/2\ \) should be plotted on the complex plane, forming the vertices of an equilateral triangle centered at the origin.

Key concepts

Complex Numbers

Complex numbers are numbers formed by combining real and imaginary parts. A complex number is typically written as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
These numbers allow for calculations that extend beyond the real number system and can solve equations that have no real solutions. For example, the equation \( x^2 + 1 = 0 \) has no real solution but has complex solutions \( x = i \) and \( x = -i \), where \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).

• Real Part: The real part \( a \) of a complex number \( z = a + bi \) is the component along the real axis.
• Imaginary Part: The imaginary part \( b \) is the coefficient of \( i \) and lies along the imaginary axis.
• Complex Conjugate: The complex conjugate of \( z = a + bi \) is \(\bar{z} = a - bi\). It reflects the complex number across the real axis.

Understanding complex numbers is crucial for diving into advanced topics like the polar form and roots of unity.

Polar Form

Polar form is another way to represent complex numbers. Instead of using rectangular coordinates \( (a, b) \), it uses a magnitude (or modulus) \( r \) and an angle (or argument) \( \theta \).
In polar form, a complex number \( z = a + bi \) can be written as \( z = r e^{i\theta} \), where:

• \( r = \sqrt{a^2 + b^2} \) is the magnitude, or distance from the origin to the point \( (a, b) \).
• \( \theta = \text{atan2}(b, a) \) is the angle formed with the positive real axis.

Using Euler's formula, the polar form is also given by \( z = r ( \cos \theta + i \sin \theta ) \). This representation is particularly useful for multiplying and dividing complex numbers, as well as finding roots.
For example, to find the roots of \( 1 \), we express \( 1 \) in its polar form as \( e^{i \cdot 0} \), since its magnitude is 1 and its angle is 0 radians.

Roots of Unity

Roots of unity are the solutions to the equation \( z^n = 1 \). They are complex numbers that, when raised to a certain power (\( n \)), yield 1.
To find these roots, we express 1 in its polar form as \( e^{i \cdot 2k\pi} \) where \( k \) is an integer. The \( n \)-th roots of unity are then given by:
\[ z_k = e^{i \cdot \frac{2k\pi}{n}} \text{ for } k = 0, 1, ..., n-1 \]For \( n = 3 \), the cube roots of unity are:
\[ z_0 = e^{i \cdot 0} = 1 \]
\[ z_1 = e^{i \cdot \frac{2\pi}{3}} \approx -\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2} \]
\[ z_2 = e^{i \cdot \frac{4\pi}{3}} \approx -\frac{1}{2} - i \cdot \frac{\sqrt{3}}{2} \]
These roots are equidistant and symmetrically distributed on the unit circle in the complex plane, forming vertices of a regular polygon.

Complex Plane

The complex plane is a two-dimensional plane used to visualize complex numbers. It consists of a horizontal axis (real axis) and a vertical axis (imaginary axis).
Each complex number \( z = a + bi \) is represented as a point \( (a, b) \) on this plane:

• The horizontal axis represents the real part \( a \).
• The vertical axis represents the imaginary part \( b \).

For example, to plot the roots of \( 1 \) on the complex plane:

• \( z_0 = 1 \) lies on the positive real axis.
• \( z_1 = -\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2} \) is plotted above the real axis, forming a 120° angle with it.
• \( z_2 = -\frac{1}{2} - i \cdot \frac{\sqrt{3}}{2} \) is plotted below the real axis, forming a 240° angle with it.

Together, these points form an equilateral triangle centered at the origin. This visualization helps to understand the symmetrical properties and geometric interpretation of complex numbers.