Problem 32 - Precalculus

Question

Find all \(n\) th roots of \(z\). Write the answers in polar form, and plot the roots in the complex plane. $$2+2 \sqrt{3} i, n=2$$

Step-by-Step Solution

Answer

The roots are \(2(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6})\) and \(2(\cos\frac{7\pi}{6} + i\sin\frac{7\pi}{6})\).

Step 1: Convert to Polar Form

First, convert the complex number from its rectangular form to polar form. The given complex number is \(2 + 2 \sqrt{3} i\). To find the magnitude \(r\), calculate:\[ r = \sqrt{2^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \]Next, find the argument \(\theta\) by using the tangent formula:\[ \theta = \tan^{-1}\left(\frac{2\sqrt{3}}{2}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]Thus, the polar form of the complex number is \( z = 4 \left( \cos\frac{\pi}{3} + i \sin\frac{\pi}{3} \right) \).

Step 2: Use the n-th root formula

To find the \(n\)-th roots of the complex number, use the formula for roots of complex numbers:\[ z^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right) \]where \(k = 0, 1, \ldots, n-1\). Here, \(r = 4\), \(\theta = \frac{\pi}{3}\), and \(n = 2\).

Step 3: Compute the magnitude of the roots

Calculate \( r^{1/n} \):\[ r^{1/2} = 4^{1/2} = 2 \]The magnitude of each root is 2.

Step 4: Compute the angles for each root

Compute the angles for each of the \(n\)-roots using \(k = 0\) and \(k = 1\):- For \(k = 0\):\[ \theta_0 = \frac{\frac{\pi}{3} + 2 \times 0 \times \pi}{2} = \frac{\pi}{6} \]- For \(k = 1\):\[ \theta_1 = \frac{\frac{\pi}{3} + 2 \times 1 \times \pi}{2} = \frac{\pi}{6} + \pi = \frac{7\pi}{6} \]

Step 5: Write roots in polar form

The two roots in polar form are:- Root 1: \( 2 \left( \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} \right) \)- Root 2: \( 2 \left( \cos\frac{7\pi}{6} + i \sin\frac{7\pi}{6} \right) \)

Step 6: Plot the roots

Plot the roots on the complex plane:- The first root is positioned at angle \(\frac{\pi}{6}\) with a distance 2 units from the origin.- The second root is positioned at an angle of \(\frac{7\pi}{6}\) with a distance 2 units from the origin. These points are symmetric about the origin.

Key Concepts

Polar Form

Complex numbers can be represented in two forms: rectangular (or Cartesian) and polar. Rectangular form uses a real part and an imaginary part, expressed as \( a + bi \), while polar form is more of an angular representation, leveraging the magnitude of the vector and an angle, or argument, representing its direction.

To convert a complex number from rectangular to polar form, you first calculate the magnitude \( r \). This is the length of the vector from the origin of the complex plane to the point \( (a, b) \), where \( a \) is the real part and \( b \) is the imaginary part. The formula is:

• \( r = \sqrt{a^2 + b^2} \)

Next, find the argument \( \theta \), which shows the angle that the vector makes with the positive x-axis. Use the arctangent function:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

Thus, the polar form of a complex number \( z \) is expressed as:

• \( z = r \left( \cos \theta + i \sin \theta \right) \)

In problems involving roots, polar form simplifies multiplying and dividing complex numbers by transforming multiplication to just adding angles.

Complex Plane

The complex plane is a visual representation of complex numbers, similar to a coordinate plane but designed specifically for complex numbers. Each point in the complex plane is represented by a pair \( (a, b) \), where \( a \) is the real part and \( b \) is the imaginary part.

The horizontal axis of this plane, called the real axis, represents the real component of the numbers, while the vertical axis, called the imaginary axis, represents the imaginary component. Here’s how the complex plane helps visualize complex numbers:

• Every complex number \( z = a + bi \) can be thought of as a vector starting from the origin \( (0,0) \) to the point \( (a, b) \).
• The magnitude of this number is the length of the vector, while the argument is the direction it points, measured as an angle in radians from the positive real axis.

Plotting complex numbers in this plane provides a clear, visual understanding of their properties and relations. When dealing with roots or powers of complex numbers, the complex plane helps by offering a graphical view where these transformations can often be seen as rotations or reflections.

Magnitude and Argument

Magnitude and argument are two critical components when dealing with the polar form of complex numbers. Understanding both is essential for tasks like finding roots or performing multiplications and divisions.

The **magnitude** \( |z| \) of a complex number \( z = a + bi \) tells you how far the number is from the origin in the complex plane—it’s essentially the "length" of the vector. Calculating magnitude is straightforward:

• \( |z| = \sqrt{a^2 + b^2} \)

The **argument** \( \arg(z) \), also known as the "angle" or "phase" of the complex number, indicates the direction of its vector. This is found using the arctangent function:

• \( \theta = \tan^{-1}\left( \frac{b}{a} \right) \)

Together, the magnitude and argument give a full picture of where the complex number "lives" in the plane. This understanding is crucial when finding \( n \)th roots because it helps visualize these roots as evenly spaced points on a circle, all with the same magnitude.