Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Find the cube roots of each complex number. Leave the answers in trigonometric form. Then graph each cube root as a vector in the complex plane. $$-64$$

Step-by-Step Solution

Answer

The cube roots of \\(-64\\) are \\(4\text{cis}(\pi/3)\\), \\(4\text{cis}(\pi)\\), and \\(4\text{cis}(5\pi/3)\\).

Step 1: Express in Complex Form

The given number is \(-64\). First, express it in complex form. Since it's real, it can be written as \(-64 + 0i\).

Step 2: Convert to Polar Form

Convert \(-64 + 0i\) to polar form. The magnitude (r) is 64, and the angle (𝜃) is \pi\ (or 180°), because it is on the negative real axis. Thus, the polar form is \(64 \text{cis} \pi\).

Step 3: Apply De Moivre's Theorem

To find the cube roots, use De Moivre's Theorem: for \(r \text{cis} \theta\), the nth root is \(r^{1/n} \text{cis} ((\theta + 2k\pi)/n)\), for \(k = 0, 1, 2\). Here, n = 3. Thus, calculate \(64^{1/3} \text{cis} ((\pi + 2k\pi)/3)\).

Step 4: Compute Cube Roots

Calculate the three roots: 1. For \(k = 0\): \(4 \text{cis} (\pi/3)\)2. For \(k = 1\): \(4 \text{cis} ((\pi+2\pi)/3) = 4 \text{cis} (\pi)\)3. For \(k = 2\): \(4 \text{cis} ((\pi+4\pi)/3) = 4 \text{cis} (5\pi/3)\)

Step 5: Graph the Cube Roots

Graph each of the roots on the complex plane as vectors:1. \(4 \text{cis} (\pi/3)\) corresponds to a vector at 60°.2. \(4 \text{cis} (\pi)\) corresponds to a vector at 180°.3. \(4 \text{cis} (5\pi/3)\) corresponds to a vector at 300°.

Key Concepts

Cube Roots

Finding the cube roots of a complex number involves determining three numbers that multiply together to recreate the original number. When dealing with complex numbers, cube roots can be fascinating, as each root represents a rotation and scaling in the complex plane. For a given complex number, like \(-64\), the cube roots are the three solutions to the equation \(z^3 = -64+0i\).

• Why three roots? It's because we are looking for values of \(z\) such that when raised to the power of 3, recreate \(-64\). In general, an \(n\)-th degree polynomial has \(n\) roots in the complex plane.
• Properties: Each cube root will lie equally spaced around a circle centered at the origin when graphed on the complex plane.

Calculating these manually using algebraic methods can be cumbersome, but polar form and De Moivre's Theorem simplify the process.

Polar Form

Polar form expresses a complex number in terms of its distance from the origin (magnitude) and the angle it makes with the positive real axis (argument). Instead of representing complex numbers as \(a + bi\), polar form uses \(r \mathrm{cis} \theta\), where \(r\) is the magnitude and \(\theta\) is the angle.

• Magnitude \(r\): For \(-64 + 0i\), the magnitude is simply \(64\), indicating how far the number is from the origin.
• Angle \(\theta\): Since \(-64\) lies on the negative real axis, the angle is \(\pi\) radians (or 180°).

Converting complex numbers to polar form simplifies the calculation of powers and roots, as it leverages the inherent circular symmetry of complex numbers.

De Moivre's Theorem

De Moivre's Theorem provides a powerful technique for raising complex numbers in polar form to a power or finding roots. For a complex number \(r \mathrm{cis} \theta\), raising it to the power of \(n\) or finding its \(n\)-th root is straightforward.

• Theorem: According to De Moivre, \([r \mathrm{cis} \theta]^n = r^n \mathrm{cis}(n\theta)\).
• Cubes Roots: To find the cube roots, we set \(n = 3\), and calculate \(r^{1/3} \mathrm{cis}((\theta + 2k\pi)/3)\) for \(k = 0, 1, 2\).
This helps to easily find each cube root as a rotation around the origin. It is useful for visualizing complex numbers' behavior over practical transformations like scaling and rotating.

Complex Plane

The complex plane is an imaginative and useful way to visualize complex numbers. It extends two-dimensional geometry by including the imaginary unit \(i\). Real numbers lie on the horizontal axis, while imaginary numbers run along the vertical axis.

• Graphing Cube Roots: For our roots from \(-64\), we plot them as vectors radiating from the origin.
• Vectors: Each of the cube roots corresponds to a vector originating from the center and pointing in the direction corresponding to their angle.

This graphical representation makes it easier to understand how multiplying by \(i\) rotates vectors counterclockwise by 90° and how complex multiplication results in both scaling and rotating these vectors.