Problem 58 - Precalculus

Question

When you graph the four fourth roots of \(16 i\) and connect the points, you form a square. Find the roots and draw the square.

Step-by-Step Solution

Answer

The roots are \(2\text{cis}(\frac{\pi}{8})\), \(2\text{cis}(\frac{5\pi}{8})\), \(2\text{cis}(\frac{9\pi}{8})\), \(2\text{cis}(\frac{13\pi}{8})\). Connect them to form a square on the complex plane.

Step 1: Convert to Polar Form

First, convert the number \(16i\) to polar form. Since \(16i = 0 + 16i\), the magnitude is \(r = \sqrt{0^2 + 16^2} = 16\). The angle \(\theta\) in radians is \(\frac{\pi}{2}\) because \(16i\) is on the positive imaginary axis. Thus, the polar form is \(16\text{cis}\left(\frac{\pi}{2}\right)\).

Step 2: Apply De Moivre's Theorem for Roots

Using De Moivre's Theorem, to find the fourth roots, take the fourth root of the magnitude and divide the angle by 4. The magnitude becomes \(16^{1/4} = 2\). The angle for the k-th root is \(\frac{\pi}{2}+ \frac{2k\pi}{4}\) where \(k=0, 1, 2, 3\).

Step 3: Calculate the Roots

The four roots are found by different values of \(k\): - For \(k=0\), the angle is \(\frac{\pi}{8}\), so the root is \(2\text{cis}\left(\frac{\pi}{8}\right)\).- For \(k=1\), the angle is \(\frac{\pi}{8} + \frac{\pi}{2} = \frac{5\pi}{8}\), so the root is \(2\text{cis}\left(\frac{5\pi}{8}\right)\).- For \(k=2\), the angle is \(\frac{\pi}{8} + \pi = \frac{9\pi}{8}\), so the root is \(2\text{cis}\left(\frac{9\pi}{8}\right)\).- For \(k=3\), the angle is \(\frac{\pi}{8} + \frac{3\pi}{2} = \frac{13\pi}{8}\), so the root is \(2\text{cis}\left(\frac{13\pi}{8}\right)\).

Step 4: Calculate Real and Imaginary Parts

Convert each root from polar form to rectangular coordinates:- \(2\text{cis}\left(\frac{\pi}{8}\right)\) has real part \(2\cos\left(\frac{\pi}{8}\right)\) and imaginary part \(2\sin\left(\frac{\pi}{8}\right)\).- \(2\text{cis}\left(\frac{5\pi}{8}\right)\) has real part \(2\cos\left(\frac{5\pi}{8}\right)\) and imaginary part \(2\sin\left(\frac{5\pi}{8}\right)\).- \(2\text{cis}\left(\frac{9\pi}{8}\right)\) has real part \(2\cos\left(\frac{9\pi}{8}\right)\) and imaginary part \(2\sin\left(\frac{9\pi}{8}\right)\).- \(2\text{cis}\left(\frac{13\pi}{8}\right)\) has real part \(2\cos\left(\frac{13\pi}{8}\right)\) and imaginary part \(2\sin\left(\frac{13\pi}{8}\right)\).

Step 5: Sketch the Square

Plot the points \((2\cos(\frac{\pi}{8}), 2\sin(\frac{\pi}{8}))\), \((2\cos(\frac{5\pi}{8}), 2\sin(\frac{5\pi}{8}))\), \((2\cos(\frac{9\pi}{8}), 2\sin(\frac{9\pi}{8}))\), and \((2\cos(\frac{13\pi}{8}), 2\sin(\frac{13\pi}{8}))\) on the complex plane. Connecting these points forms the vertices of a square.

Key Concepts

Polar Form

When dealing with complex numbers, one of the ways to represent them is using the polar form. Unlike the standard form, which is written as a combination of real and imaginary parts (e.g., a + bi), the polar form represents a complex number as a product of its magnitude and a direction angle. Here, the number is expressed as \(r\text{cis}(\theta)\), where \(r\) is the magnitude and \(\theta\) is the angle.

• Magnitude (r): This is the distance from the origin to the point representing the complex number on the complex plane. It can be calculated using the formula \(r = \sqrt{a^2 + b^2}\) for a complex number \(a + bi\).
• Angle (\(\theta\)): This angle is measured from the positive x-axis towards the line connecting the point to the origin and is often given in radians.
• cis: This stands for cosine and sine, as in \( \cos(\theta) + i \sin(\theta) \).

The polar form is especially useful for performing operations like multiplication and division on complex numbers, as it simplifies the calculation process.

Complex Numbers

Complex numbers are an extension of the real numbers and include a component called the imaginary unit, denoted as \(i\), where \(i^2 = -1\). They are often written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here are some important things to remember:

• Every real number is also a complex number with the imaginary part equal to zero (e.g., \(3 = 3 + 0i\)).
• Complex numbers can be plotted on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.
• They are essential in various fields like engineering, physics, and applied mathematics because they facilitate the solution of quadratic equations and occur naturally in the study of oscillations and waves.

Understanding complex numbers provides the foundation for exploring more advanced mathematical concepts, including De Moivre's Theorem for finding roots.

Roots of Complex Numbers

Finding the roots of a complex number can sometimes be challenging, but De Moivre's Theorem provides a systematic approach. This theorem states that if you have a complex number in polar form \(r\text{cis}(\theta)\), you can find the n-th roots by:

• Taking the n-th root of the magnitude \(r^{1/n}\).
• Adjusting the angle \(\theta\) by dividing it by \(n\) and accounting for each possible root by adding \(\frac{2\pi k}{n}\), where \(k = 0, 1, \ldots, n-1\).

For example, to find the fourth roots of a complex number, you would:

• Compute the fourth root of the magnitude.
• Determine four different angles by dividing the original angle by four and adding multiples of \(\frac{\pi}{2}\) for each root.

Each resulting pair of magnitude and angle gives a unique root in polar form. Converting these roots back into standard form provides insight into their position on the complex plane, often exhibiting symmetric properties like forming polygons, as in the given problem where the roots form a square.