Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Find all indicated roots and express them in rectangular form. Check your results with a calculator. The fourth roots of \(-1\).

Step-by-Step Solution

Answer

The fourth roots of -1 are \(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\), \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\), \(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\), and \(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).

Step 1: Express in Polar Form

To find the fourth roots of -1, you'll first want to express -1 in polar form. The complex number -1 can be expressed as 1 (magnitude) angle\(\pi\). So in polar form, it's 1 angle\(\pi\): \(re^{i\theta} = e^{i\pi} \).

Step 2: Use De Moivre's Theorem

Use De Moivre’s Theorem, which says that the n-th roots of a complex number are given by:\(z^{1/n} = r^{1/n} (\cos(\frac{\theta + 2k\pi}{n}) + i \sin(\frac{\theta + 2k\pi}{n}))\), where k = 0, 1, ..., n-1.

Step 3: Calculate Magnitude

For the fourth root problem, n=4, \(r = 1\), thus the magnitude \(r^{1/4} = 1^{1/4} = 1\), because any non-zero number raised to any power remains 1.

Step 4: Calculate Angles

We need to find all four roots, so compute angles for k = 0, 1, 2, 3 using:\(\theta_k = \frac{\pi + 2k\pi}{4}\).

Step 5: Calculate for k = 0

For k = 0: \(\theta_0 = \frac{\pi}{4}\),so the first root is \(e^{i\frac{\pi}{4}} = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).

Step 6: Calculate for k = 1

For k = 1: \(\theta_1 = \frac{3\pi}{4}\), thus the second root is \(e^{i\frac{3\pi}{4}} = \cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).

Step 7: Calculate for k = 2

For k = 2: \(\theta_2 = \frac{5\pi}{4}\), so the third root is \(e^{i\frac{5\pi}{4}} = \cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).

Step 8: Calculate for k = 3

For k = 3: \(\theta_3 = \frac{7\pi}{4}\), thus the fourth root is \(e^{i\frac{7\pi}{4}} = \cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers using a magnitude and an angle. Instead of using the standard "a + bi" format known as rectangular form, polar form expresses a complex number as a modulus (r) and an angle (θ).

A complex number like \(-1\) can be rewritten in polar form as \(re^{i\theta}\). In polar form, \(-1\) has a magnitude of 1 (distance from origin), and its angle with respect to the positive x-axis is \(\pi\) radians or 180 degrees.

• The magnitude \(r\) shows the distance from the origin to the point in the complex plane.
• The angle \(\theta\) indicates the counterclockwise rotation from the positive real axis.

So, \(-1\) in polar form is expressed as \(e^{i\pi}\). This conversion is crucial for problems like finding roots, as it simplifies expressions and calculations.

De Moivre's Theorem

De Moivre's Theorem provides a straightforward method for finding roots and powers of complex numbers. It states that if a complex number is expressed in polar form as \(re^{i\theta}\), then the nth roots are given by:

\[ z^{1/n} = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \]

• "n" is the number of roots you are seeking, so for fourth roots, \(n=4\).
• "k" is an integer that runs from 0 to \(n-1\), which allows calculation of all n roots.

To find the fourth roots of \(-1\), De Moivre’s theorem dictates computing angles for each \(k = 0, 1, 2, \text{and}\ 3\). Hence, the angles are calculated using: \(\theta_k = \frac{\pi + 2k\pi}{4}\). The method lets us convert angles conveniently back to trigonometric terms, aiding in finding each root.

Rectangular Form

Rectangular form, also known as Cartesian form, is the standard format to express complex numbers. It is written as \(a + bi\), where:

• "a" denotes the real part
• "b" is the imaginary part

In the rectangular form, each complex number corresponds to a point on the complex plane, using the x-y coordinate system where "a" is along the x-axis (real) and "b" is along the y-axis (imaginary).

Once you've used De Moivre's Theorem to find the roots in polar form, it's often necessary to convert them back to rectangular form for a more familiar interpretation. For each root:

• Convert the polar angle to its corresponding trigonometric values, using cosine for the real part and sine for the imaginary part.
• For example, the root \(e^{i\frac{\pi}{4}}\) converts to \(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\).

This form is especially useful when verifying your results with software or calculators, which often use rectangular form as a standard output.