Problem 88 - Precalculus Mathematics for Calculus

Question

Find the indicated roots, and graph the roots in the complex plane. The fifth roots of \(i\)

Step-by-Step Solution

Answer

The fifth roots of \(i\) produce five points in a regular pentagon on the unit circle.

Step 1: Express in Polar Form

The complex number \(i\) can be expressed in polar form. Since \(i\) is on the imaginary axis with a magnitude of 1, it can be written as \( e^{i\frac{\pi}{2}} \).

Step 2: Use De Moivre’s Theorem

To find the fifth roots of \( i \), use De Moivre's Theorem which states \( z^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right) \) for \( k = 0, 1, 2, 3, 4 \).

Step 3: Calculate the Magnitude

The magnitude of \( i \) is 1, so the magnitude of the fifth root is also \( 1^{1/5} = 1 \).

Step 4: Determine the Angles

Calculate the angles for each of the fifth roots using \( \frac{\pi/2 + 2k\pi}{5} \).- For \( k = 0 \), the angle is \( \frac{\pi}{10} \).- For \( k = 1 \), the angle is \( \frac{\pi}{10} + \frac{2\pi}{5} = \frac{5\pi}{10} = \frac{\pi}{2} \).- For \( k = 2 \), the angle is \( \frac{\pi}{10} + \frac{4\pi}{5} = \frac{9\pi}{10} \).- For \( k = 3 \), the angle is \( \frac{\pi}{10} + \frac{6\pi}{5} = \frac{13\pi}{10} \).- For \( k = 4 \), the angle is \( \frac{\pi}{10} + \frac{8\pi}{5} = \frac{17\pi}{10} \).

Step 5: Convert Angular Form to Rectangular Form

Convert each root from polar to rectangular form.- For \( k = 0 \), the root is \( \cos \frac{\pi}{10} + i \sin \frac{\pi}{10} \).- For \( k = 1 \), the root is \( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = i \).- For \( k = 2 \), the root is \( \cos \frac{9\pi}{10} + i \sin \frac{9\pi}{10} \).- For \( k = 3 \), the root is \( \cos \frac{13\pi}{10} + i \sin \frac{13\pi}{10} \).- For \( k = 4 \), the root is \( \cos \frac{17\pi}{10} + i \sin \frac{17\pi}{10} \).

Step 6: Plot the Roots

Plot these points on the complex plane. They should form a regular pentagon centered at the origin with vertices on the unit circle.

Key Concepts

Fifth Roots

Calculating fifth roots of a complex number like \( i \) involves finding five unique numbers that, when multiplied by themselves five times, yield the original number \( i \). Each root corresponds to a specific angle and is equally spaced on the complex plane. This symmetry is due to the fundamental property of roots in the complex number system: they form regular polygons.

• First, it's crucial to note that the fifth roots have the same magnitude. If the complex number is on the unit circle, like our number \( i \), which has a magnitude of 1, all roots also maintain this magnitude.
• These roots are essentially rotations of one another, spaced evenly around the origin on the complex plane.

Calculating these fifth roots gives us a fascinating insight into how complex numbers behave, showcasing the interplay of angles and magnitudes.

De Moivre's Theorem

De Moivre's Theorem provides a powerful tool for finding roots of complex numbers. By using this theorem, we can transform complex arithmetic into simpler trigonometric work. The theorem is expressed as: \[ z^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right) \]where \( n \) is the degree of the root we are solving for, and \( k \) varies over the integers from 0 to \( n-1 \).

• De Moivre’s Theorem helps in breaking down complex powers and roots into real parts (cosine) and imaginary parts (sine).
• It uses polar coordinates to express complex numbers, allowing for easier multiplication and root extraction.

This formula simplifies the often complex task of extracting roots from numbers expressed in their polar forms.

Polar Form

Understanding the polar form is essential when dealing with complex numbers and calculating their roots or powers. In this coordinate system, a complex number is expressed in terms of a magnitude and an angle.
A number \( z \), which can be normally expressed as \( a + bi \), is often written as \( r e^{i\theta} \) in polar form, where:

• \( r \) is the magnitude or absolute value of the complex number, calculated as \( \sqrt{a^2 + b^2} \).
• \( \theta \) is the argument, or angle, of the complex number with the positive real axis.

For the complex number \( i \), the magnitude is 1 and the angle is \( \frac{\pi}{2} \), leading to the polar form \( e^{i\frac{\pi}{2}} \). Polar form simplifies complex calculations by enabling straightforward computation of roots and powers. This approach streamlines the calculation of fifth roots, which become even increments of angles.

Complex Plane Graphing

Graphing complex numbers and their roots on the complex plane offers a visual representation that aids in understanding their geometric properties. The complex plane consists of a horizontal real axis and a vertical imaginary axis, helping visualize numbers like \( i \) or its roots.
To graph the fifth roots of a complex number \( i \):

• Each root has a magnitude of 1, corresponding to a point on the unit circle.
• The angles for these roots are spaced at equal intervals, forming a regular pentagon centered at the origin.

By plotting these roots, you can observe the symmetry and structure of complex roots, providing insights into their behavior and characteristics, such as regular spacing and identical magnitudes. This graphical approach highlights how mathematical concepts manifest visually, reinforcing understanding through spatial reasoning.