Problem 79 - Algebra and Trigonometry

Question

Find the indicated roots, and graph the roots in the complex plane. The fourth roots of \(-81 i\)

Step-by-Step Solution

Answer

The roots are \(3e^{i(-\pi/8)}, 3e^{i(3\pi/8)}, 3e^{i(7\pi/8)}, 3e^{i(11\pi/8)}\).

Step 1: Convert to Polar Form

First, express \(-81i\) in polar form. Since it's purely imaginary, we have \(z = 0 - 81i\), which means the modulus \(|z| = 81\) and the argument \(\theta = -\frac{\pi}{2}\) (since it's negative and on the imaginary axis). The polar form is \(81e^{-i\frac{\pi}{2}}\).

Step 2: Apply the Formula for Roots

The fourth roots of a complex number in polar form \(re^{i\theta}\) are given by \(\sqrt[4]{r}e^{i(\frac{\theta+2k\pi}{4})}\) for \(k = 0, 1, 2, 3\). We compute \(\sqrt[4]{81} = 3\).

Step 3: Calculate Each Root

Calculate the fourth roots using the formula from Step 2:1. For \(k = 0\): \(3e^{i(-\frac{\pi}{8})} = 3(\cos(-\frac{\pi}{8}) + i\sin(-\frac{\pi}{8}))\)2. For \(k = 1\): \(3e^{i(\frac{3\pi}{8})} = 3(\cos(\frac{3\pi}{8}) + i\sin(\frac{3\pi}{8}))\)3. For \(k = 2\): \(3e^{i(\frac{7\pi}{8})} = 3(\cos(\frac{7\pi}{8}) + i\sin(\frac{7\pi}{8}))\)4. For \(k = 3\): \(3e^{i(\frac{11\pi}{8})} = 3(\cos(\frac{11\pi}{8}) + i\sin(\frac{11\pi}{8}))\).

Step 4: Plot the Roots on the Complex Plane

Plot each root from Step 3 on the complex plane:1. For \(k = 0\), the root lies in the fourth quadrant.2. For \(k = 1\), the root lies in the first quadrant.3. For \(k = 2\), the root lies in the second quadrant.4. For \(k = 3\), the root lies in the third quadrant. All roots should be on a circle with radius 3 centered at the origin.

Key Concepts

Polar Form

When working with complex numbers, the polar form provides a powerful way to express these numbers using a geometric perspective. Instead of writing a complex number as a sum of a real and an imaginary part like in Cartesian form \(a + bi\), the polar form utilizes a radius (known as the modulus) and an angle (known as the argument). This is often represented as \(r e^{i\theta}\), where \(r\) is the modulus and \(\theta\) is the argument.
To convert a complex number into its polar form:

• Calculate the modulus \(|z|\) using \(|z| = \sqrt{a^2 + b^2}\).
• Find the argument \(\theta\) using \(\theta = \text{atan2}(b, a)\), which gives the angle in radians.

Using polar form simplifies operations like multiplication, division, and finding powers and roots. For example, if you need to find the roots of a complex number, the polar form is ideal as it makes use of De Moivre's Theorem.

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. The horizontal axis is known as the real axis, and the vertical axis is referred to as the imaginary axis.
Each complex number \(a + bi\) is plotted as a point where:

• The real part \(a\) determines the horizontal coordinate.
• The imaginary part \(b\) sets the vertical coordinate.

This way of visualizing complex numbers helps in understanding operations such as addition, multiplication, and finding roots. When computing the roots, these roots are not just numbers but positions or points on the complex plane. For example, the fourth roots of a complex number will be symmetrically distributed along a circle centered at the origin.

Roots of Complex Numbers

Finding the roots of complex numbers involves using their polar form, which leverages the modulus and argument to simplify the computation. Using De Moivre's Theorem, which states that the \( ext{n-th}\) root of a complex number in polar form \(r e^{i \theta}\) can be given by \[\sqrt[n]{r} e^{i \frac{(\theta + 2k\pi)}{n}}\] for \(k = 0, 1, ..., n-1\).
Let's break down the steps:

• Find the \( ext{n-th}\) root of the modulus \(|z|\).
• Calculate \((\theta + 2k\pi)/n\) to find each respective root's argument.

Using this formula, the roots will similarly lie on a circle in the complex plane, spaced evenly, providing a neat geometric view of complex roots.

Modulus and Argument

The modulus and argument are essential components of a complex number's polar form. They provide a way to measure and position complex numbers in the complex plane.
The modulus, denoted as \(|z|\), measures the distance of a complex number from the origin of the complex plane, similar to the hypotenuse of a right-angled triangle formed from the real and imaginary parts. It ensures that regardless of the quadrant a complex number is in, its distance from the origin is the same.
The argument, denoted as \(\theta\), is the angle formed with the positive real axis. It is measured in radians and can be determined using \( ext{atan2}(b, a)\), which accurately takes into account the signs of \(a\) and \(b\) to place \(\theta\) correctly in the plane.

• The argument helps in defining the direction of the complex number while the modulus defines its magnitude.

These two components enable the easy manipulation and comprehension of complex numbers, especially when dealing with operations like finding powers and roots.