Problem 33 - Precalculus

Question

Find all \(n\) th roots of \(z\). Write the answers in polar form, and plot the roots in the complex plane. $$\sqrt{18}-\sqrt{18} i, n=2$$

Step-by-Step Solution

Answer

The square roots of \( \sqrt{18} - \sqrt{18}i \) are \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \) and \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \).

Step 1: Convert the Complex Number to Polar Form

The given complex number is \( \sqrt{18} - \sqrt{18}i \). To convert it to polar form, we need the modulus \( r \) and the argument \( \theta \). The modulus is calculated as \( r = \sqrt{x^2 + y^2} \), where \( x = \sqrt{18} \) and \( y = -\sqrt{18} \). Thus, \( r = \sqrt{(\sqrt{18})^2 + (-\sqrt{18})^2} = \sqrt{18 + 18} = 6 \). The argument is given by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-\sqrt{18}}{\sqrt{18}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \). Therefore, the polar form is \( z = 6\text{cis}(-\frac{\pi}{4}) \).

Step 2: Find the Principal Root

To find the square roots of a complex number, we use the formula \( z^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right) \) for each integer \( k \). Here, \( r = 6 \), \( \theta = -\frac{\pi}{4} \), and \( n = 2 \). The principal root corresponds to \( k = 0 \):\[ z_0 = 6^{1/2}\text{cis}\left(\frac{-\pi/4}{2}\right) = \sqrt{6}\text{cis}\left(-\frac{\pi}{8}\right) \].

Step 3: Find the Second Root

To find the second root, use the formula from Step 2 with \( k = 1 \):\[ z_1 = 6^{1/2}\text{cis}\left(\frac{-\pi/4 + 2\pi}{2}\right) = \sqrt{6}\text{cis}\left(\frac{7\pi}{8}\right) \].

Step 4: Plot the Roots in the Complex Plane

In the complex plane, each root has a magnitude of \( \sqrt{6} \). The angles to plot are \( -\frac{\pi}{8} \) for the principal root and \( \frac{7\pi}{8} \) for the second root. In polar coordinates, these respectively correspond to the points:1. \( \sqrt{6}\text{cis}\left(-\frac{\pi}{8}\right) \) - located just below the x-axis in the first quadrant.2. \( \sqrt{6}\text{cis}\left(\frac{7\pi}{8}\right) \) - located just above the x-axis in the second quadrant, leftward.

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers using a combination of modulus and angle. When you have a number like \( z = a + bi \), converting it to polar form involves calculating two things:

• Modulus \( r \): This is the distance from the origin to the number in the complex plane. It's found using the formula \( r = \sqrt{a^2 + b^2} \).
• Argument \( \theta \): This is the angle formed with the positive x-axis. You can find it using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

The polar form is then denoted as \( z = r\text{cis}(\theta) \), where \( \text{cis}(\theta) \) represents \( \cos(\theta) + i\sin(\theta) \).
For the given complex number, \( \sqrt{18} - \sqrt{18}i \), converting to polar form involves calculating \( r = 6 \) and \( \theta = -\frac{\pi}{4} \), resulting in \( z = 6\text{cis}(-\frac{\pi}{4}) \).

n-th Roots

Finding the \( n \)-th roots of a complex number means finding all the numbers that, when raised to the power of \( n \), give the original complex number. The formula used is:

• \( z^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right) \)

where \( k \) is an integer from 0 to \( n-1 \). For \( n = 2 \), you find two roots. In our example, \( r = 6 \) and \( \theta = -\frac{\pi}{4} \).
Here are the steps to follow:

• Calculate \( r^{1/2} \): This gives the magnitude of the roots, in our case, \( \sqrt{6} \).
• Find each root: Start with \( k = 0 \) for the principal root, resulting in \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \). Use \( k = 1 \) for the second root, leading to \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \).

Finding the roots in polar form is often simpler than using the standard form because you only adjust angles by \( 2\pi \) increments.

Complex Plane

The complex plane is a way of graphically representing complex numbers. It involves a two-dimensional grid where:

• The real part of the number is plotted on the x-axis.
• The imaginary part of the number is plotted on the y-axis.

Each complex number corresponds to a unique point on this plane. In this exercise, the roots of the complex number \( \sqrt{18} - \sqrt{18}i \) were plotted:

• First root \( \sqrt{6}\text{cis}(-\frac{\pi}{8}) \): Plotted below the x-axis in the first quadrant.
• Second root \( \sqrt{6}\text{cis}(\frac{7\pi}{8}) \): Plotted above the x-axis in the second quadrant.

The real benefit of using the complex plane is visualizing operations like addition, multiplication, and finding roots as geometric transformations or rotations.

Modulus and Argument

The modulus and argument are essential concepts when dealing with complex numbers, particularly in polar form.

• Modulus \( r \): Represents the magnitude or absolute value of the complex number, calculated with \( r = \sqrt{a^2 + b^2} \). It measures how far from the origin the number is located in the complex plane.
• Argument \( \theta \): This is the angle between the positive x-axis and the line joining the origin with the number in the complex plane. Calculated with \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \), it helps in defining the direction of the number.

For the given problem, the calculated modulus is 6, representing the root's distance from the origin. The argument \( \theta = -\frac{\pi}{4} \) helps determine the direction on the complex plane where this distance is measured. Understanding the modulus and argument not only helps in polar conversion but also gives insight into how complex numbers behave geometrically.