Problem 84 - Precalculus

Question

Refer to the following: According to the \(n\) th root theorem, the first of the \(n\) th roots of the complex number \(z=r(\cos \theta+i \sin \theta)\) is given by \(w_{1}=r^{1 / n}\left[\cos \left(\frac{\theta}{n}+\frac{2 \pi}{n}\right)+i \sin \left(\frac{\theta}{n}+\frac{2 \pi}{n}\right)\right],\) with \(\theta\) in radians or \(w_{1}=r^{1 / n}\left[\cos \left(\frac{\theta}{n}+\frac{360^{\circ}}{n}\right)+i \sin \left(\frac{\theta}{n}+\frac{360^{\circ}}{n}\right)\right]\) with \(\theta\) in degrees. Using the graphing calculator to plot the \(n\) roots of a complex number \(z,\) enter \(r_{1}=r, \theta \min =\frac{\theta}{n}, \theta \max =2 \pi+\frac{\theta}{n}\) or \(360^{\circ}+\frac{\theta}{n}, \theta\) step \(=\frac{2 \pi}{n}\) or \(\frac{360^{\circ}}{n}, \mathrm{xmin}=-r, \mathrm{xmax}=r\) ymin \(=-r,\) ymax \(=r,\) and \([\mathrm{MODE}]\) in radians or degrees. Find the fourth roots of \(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i,\) and plot the roots with a calculator.

Step-by-Step Solution

Answer

The fourth roots of \\(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\\) are \\(w_0, w_1, w_2, \\) and \\(w_3\\). Plot them using a calculator in polar mode.

Step 1: Convert Complex Number to Polar Form

The complex number is \(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\). First, find its modulus using \(r = \sqrt{a^2 + b^2}\), where \(a = -\frac{\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\). Calculate \( r = \sqrt{\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1\). Find the argument \(\theta\) using tangent: \(\theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}(-1)\). Since the number is in the second quadrant, \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\). Hence, the polar form is \(1\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\).

Step 2: Find the 4th Roots

Use the \(n\)th root theorem: \(w_k = r^{1/n} \left[ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right]\) for \(k = 0, 1, 2, 3\). \(r^{1/4} = 1^{1/4} = 1\).1. For \(k=0\), \(w_0 = 1 \left[ \cos \left( \frac{3\pi}{16} \right) + i \sin \left( \frac{3\pi}{16} \right) \right]\).2. For \(k=1\), \(w_1 = 1 \left[ \cos \left( \frac{3\pi}{16} + \frac{\pi}{2} \right) + i \sin \left( \frac{3\pi}{16} + \frac{\pi}{2} \right) \right]\).3. For \(k=2\), \(w_2 = 1 \left[ \cos \left( \frac{3\pi}{16} + \pi \right) + i \sin \left( \frac{3\pi}{16} + \pi \right) \right]\).4. For \(k=3\), \(w_3 = 1 \left[ \cos \left( \frac{3\pi}{16} + \frac{3\pi}{2} \right) + i \sin \left( \frac{3\pi}{16} + \frac{3\pi}{2} \right) \right]\).

Step 3: Plot the Roots on Graphing Calculator

Set the calculator to 'polar' mode and 'radians' for angle measurement. Input the following parameters for plotting:- \(r_1 = 1\)- \(\theta\min = \frac{3\pi}{16}\)- \(\theta\max = 2\pi + \frac{3\pi}{16}\)- \(\theta\) step = \(\frac{\pi}{2}\)- \([xmin, xmax] = [-1, 1]\)- \([ymin, ymax] = [-1, 1]\)These settings should allow you to correctly plot four evenly spaced points on the unit circle, representing the fourth roots.

Key Concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics and engineering. They extend the idea of the number line, used for real numbers, into a plane via a complex plane. A complex number takes the general form of \( z = a + bi \) where \( a \) is the real part and \( b \) is the imaginary part. The imaginary part is expressed using \( i \), which is defined as the square root of negative one. This allows complex numbers to be represented as points or vectors on the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part.

For example, the complex number \( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \) has a real part \( -\frac{\sqrt{2}}{2} \) and an imaginary part \( \frac{\sqrt{2}}{2}i \). Its position on the complex plane is determined by these components. Understanding complex numbers is crucial for exploring operations such as addition, subtraction, multiplication, and particularly root extraction in cases involving imaginary components, as highlighted in the nth root theorem.

Polar Form

Polar form is a powerful way of expressing complex numbers, especially when dealing with multiplication, division, and finding roots. Instead of using the rectangular form \( a + bi \), we can express a complex number in polar form as \( r(\cos \theta + i \sin \theta) \), sometimes abbreviated as \( r \operatorname{cis} \theta \).

Where:

• \( r \) is the modulus or magnitude of the complex number, representing the distance from the origin to the point on the complex plane.
• \( \theta \) is the argument, which is the angle between the positive real axis and the line representing the complex number in the plane.

Using polar form simplifies many operations. For instance, rotating or scaling a vector becomes straightforward when calculated in polar form. In our example, the complex number \(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\) converts to polar form as \(1(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4})\) because it lies on the unit circle (distance equals one), with an angle of \(\frac{3\pi}{4}\) radians from the positive x-axis.

Graphing Calculator

Using a graphing calculator can make visualizing complex numbers and their transformations like nth roots much easier. When dealing with polar coordinates, graphing calculators often contain functions specifically designed for polar plotting in either degrees or radians.

Here's how you can set up your graphing calculator to plot complex numbers like the fourth roots of our given complex number:

• Switch the calculator to 'polar mode' for plotting complex conjugates.
• Ensure it is in 'radians mode' if you're working with radian measures for precision.
• Set \( r = 1 \) owing to the magnitude found earlier.
• Define \( \theta \min \) and \( \theta \max \) to cover the complete unit circle with increments to capture each root evenly.
• Bound your graph within a range, say \([-1, 1]\) for both axes, to capture the unit circle visually.

This configuration allows you to plot the four roots, which should appear evenly spaced around a circle, offering intuitive geometric insight into their distribution.

Modulus and Argument

To fully grasp complex numbers, understanding the modulus and argument is essential. The modulus of a complex number \( z = a + bi \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \), representing the distance from the origin to the point \( (a, b) \) on the complex plane. This is akin to finding the hypotenuse in a right triangle where \( a \) and \( b \) are the legs.

In our specific example, the modulus is \( 1 \), stemming from \( \sqrt{\left(-\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} = 1 \).

The argument \( \theta \) is the angle of rotation from the positive real axis to the line representing the complex number, measured in radians or degrees. It is crucial for expressing the direction of the complex number. For the given complex number, the argument, after adjusting for its position in the second quadrant, is \( \frac{3\pi}{4} \) radians, allowing the further conversion to polar form and exploration of roots.