Problem 40 - Precalculus

Question

Find all \(n\) th roots of \(z\). Write the answers in polar form, and plot the roots in the complex plane. $$-\sqrt{128}+\sqrt{128} i, n=4$$

Step-by-Step Solution

Answer

The 4th roots in polar form are: \(2e^{i\frac{3\pi}{16}}, 2e^{i\frac{11\pi}{16}}, 2e^{i\frac{19\pi}{16}}, 2e^{i\frac{27\pi}{16}}\).

Step 1: Convert Complex Number to Polar Form

The given complex number is \(-\sqrt{128} + \sqrt{128}i\). Rewrite this as \(z = re^{i\theta}\) by calculating \(r\) and \(\theta\).Calculate \(r = \sqrt{x^2 + y^2}\), where \(x = -\sqrt{128}\) and \(y = \sqrt{128}\).Thus, \(r = \sqrt{((-\sqrt{128})^2 + (\sqrt{128})^2)} = \sqrt{128 + 128} = \sqrt{256} = 16\).To find \(\theta\), use \(\tan^{-1}\left(\frac{y}{x}\right)\), so \(\theta = \tan^{-1}\left(\frac{\sqrt{128}}{-\sqrt{128}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}\). Since \(\theta\) is in the second quadrant, adjust \(\theta\) to \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).Thus in polar form, \(z = 16e^{i\frac{3\pi}{4}}\).

Step 2: Find the Principal Root

The principal \(n\)-th root of \(z\) is given by \(z_k = r^{1/n} e^{i(\theta/n + 2k\pi/n)}\), where \(k = 0, 1, 2, \ldots, n-1\).With \(n = 4\), calculate \(r^{1/n} = 16^{1/4} = 2\), \(\theta/n = \frac{3\pi}{4}/4 = \frac{3\pi}{16}\).Thus, the principal root is\(z_0 = 2e^{i\frac{3\pi}{16}}\).

Step 3: Calculate All Roots

Calculate the remaining roots using \(z_k = 2e^{i(\frac{3\pi}{16} + \frac{2k\pi}{4})}\).For \(k = 1, 2, 3\), respectively, we have: \[z_1 = 2e^{i(\frac{3\pi}{16} + \frac{\pi}{2})} = 2e^{i\frac{11\pi}{16}},\]\[z_2 = 2e^{i(\frac{3\pi}{16} + \pi)} = 2e^{i\frac{19\pi}{16}},\]\[z_3 = 2e^{i(\frac{3\pi}{16} + \frac{3\pi}{2})} = 2e^{i\frac{27\pi}{16}}.\]

Step 4: Plot the Roots on the Complex Plane

To visualize the roots, plot them as vectors originating from the origin in the complex plane.Each root is located \(2\) units from the origin, at angles \(\frac{3\pi}{16}, \frac{11\pi}{16}, \frac{19\pi}{16}, \frac{27\pi}{16}\) respectively.

Key Concepts

Polar Form

In mathematics, the polar form of a complex number is a way of representing complex numbers in the form of an angle and magnitude. This form makes it easier to understand rotations and scaling in the complex plane.
The polar form is written as \( z = re^{i\theta} \), where

• \( r \) is the magnitude (or modulus) of the complex number.
• \( \theta \) is the angle (or argument) with the positive x-axis.

To convert a complex number from its standard form \( a + bi \) to polar form:

• First calculate the magnitude using the formula \( r = \sqrt{a^2 + b^2} \).
• Next, find the angle \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

If the complex number is not in the first quadrant, adjust \( \theta \) accordingly to ensure it is correctly placed as an angle in standard position.
For example, the given complex number \(-\sqrt{128} + \sqrt{128}i\) was converted to polar form as \( z = 16e^{i\frac{3\pi}{4}} \), after calculating the magnitude \( r = 16 \) and angle \( \theta = \frac{3\pi}{4} \).

Complex Plane

The complex plane is a two-dimensional plane where the horizontal axis represents the real part of complex numbers, while the vertical axis represents the imaginary part. It's similar to the Cartesian plane but tailored for complex numbers.
This visualization helps in understanding various operations on complex numbers, such as addition, scaling, and rotation.

• The x-coordinate represents the real part of the complex number.
• The y-coordinate represents the imaginary part.

The position of a point in the complex plane gives a visual understanding of its magnitude and direction.

For the exercise, plotting the nth roots involves placing points at specific angles and distances from the origin. Each point on this plot represents a root. The roots found for the complex number \(-\sqrt{128} + \sqrt{128}i\) can be visualized as vectors extending from the origin to angles \( \frac{3\pi}{16} \), \( \frac{11\pi}{16} \), \( \frac{19\pi}{16} \), and \( \frac{27\pi}{16} \) at a radius or distance of 2 units from the origin.

Principal Root

The principal root is an important concept when finding the nth roots of complex numbers. It refers to the first or primary root solution in a sequence. When dealing with complex nth roots, we find that they form a regular polygon in the complex plane.

To find the principal root of a complex number

• First, convert the number to its polar form.
• Then apply the formula \( z_0 = r^{1/n}e^{i(\theta/n)} \), where \( n \) is the order of the root.

In our example, with \( n=4 \), the principal root was calculated as \( z_0 = 2e^{i\frac{3\pi}{16}} \).
This is accomplished by determining the nth power of the magnitude \( r = 16^{1/4} = 2 \) and dividing the angle by \( n \). This root is one of the vertices that form a square (since \( n = 4 \)) around the origin in the complex plane.

Complex Number Conversion

Converting a complex number between its rectangular form (like \( a + bi \)) and polar form \( re^{i\theta} \) is a fundamental process in complex number operations. This conversion is essential for understanding complex multiplication and finding nth roots.

• To convert from rectangular to polar, find the magnitude \( r = \sqrt{a^2 + b^2} \).
• Determine the angle \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \) and adjust it based on the quadrant.

Conversely, converting from polar back to rectangular involves calculating:

• The real part as \( r \cos \theta \)
• The imaginary part as \( r \sin \theta \)

This accuracy of conversion ensures precision in operations, such as in our example, where it allowed finding all 4th roots accurately and plotting them on the complex plane. Having a strong grasp of these conversions provides clarity and insight when performing mathematical explorations involving complex numbers.