Problem 38 - Precalculus

Question

Find all \(n\) th roots of \(z\). Write the answers in polar form, and plot the roots in the complex plane. $$4 \sqrt{2}+4 \sqrt{2} i, n=3$$

Step-by-Step Solution

Answer

The 3rd roots are \(2 \text{cis} \frac{\pi}{12}\), \(2 \text{cis} \frac{3\pi}{4}\), and \(2 \text{cis} \frac{17\pi}{12}\).

Step 1: Convert to Polar Form

The complex number given is \(4\sqrt{2} + 4\sqrt{2} i\). First, find the modulus \(r\) of the complex number:\[ r = \sqrt{(4\sqrt{2})^2 + (4\sqrt{2})^2} = \sqrt{32 + 32} = \sqrt{64} = 8\]Next, determine the argument \(\theta\) using \( \tan \theta = \frac{\text{Imaginary part}}{\text{Real part}} = 1 \):\[\theta = \frac{\pi}{4} + 2k\pi \quad \text{for some integer } k\]Thus, the polar form is \(8 \text{cis} \frac{\pi}{4}\).

Step 2: Apply De Moivre's Theorem

Use De Moivre's Theorem to find the 3rd roots of the complex number in polar form \( r \text{cis} \theta \). For this, the 3 roots are given by:\[z_k = \sqrt[3]{r} \text{cis} \left(\frac{\theta + 2k\pi}{3}\right)\]where \( k = 0, 1, 2 \).

Step 3: Calculate the Modulus of the Roots

Calculate \(\sqrt[3]{r}\):\[\sqrt[3]{8} = 2\]So each root will have the modulus 2.

Step 4: Find the Angles for Each Root

Substitute \(\theta = \frac{\pi}{4}\) and \( r = 8 \):- For \(k=0\):\[\theta_0 = \frac{\pi}{12}\]- For \(k=1\):\[\theta_1 = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{9\pi}{12} = \frac{3\pi}{4}\] - For \(k=2\):\[\theta_2 = \frac{\pi}{12} + \frac{4\pi}{3} = \frac{17\pi}{12}\]

Step 5: Write the Roots in Polar Form

Using the moduli and angles calculated, write the roots as:\[z_0 = 2 \text{cis} \frac{\pi}{12}\]\[z_1 = 2 \text{cis} \frac{3\pi}{4}\]\[z_2 = 2 \text{cis} \frac{17\pi}{12}\]

Step 6: Plot the Roots on the Complex Plane

To plot these on the complex plane, each root can be represented as a point at - a distance of 2 units from the origin- forming angles \(\frac{\pi}{12}\), \(\frac{3\pi}{4}\), and \(\frac{17\pi}{12}\) with the positive real axis.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in the field of complex numbers. It helps us find the powers and roots of complex numbers when they're expressed in polar form. The theorem is given by:

• 
  

Consider the complex number's polar form as \( z = r(\cos \theta + i \sin \theta) \), which can be simplified to \( z = r \text{cis} \theta \). According to De Moivre's Theorem, the \( n \)th power of \( z \) is:\[z^n = r^n \text{cis} (n\theta)\]When finding the \( n \)th roots, the process is slightly reversed. The \( n \)th roots of \( z \) are found using the formula:\[z_k = \sqrt[n]{r} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right)\]where \( k = 0, 1, 2, \ldots, n-1 \). This formula allows us to determine all \( n \) distinct roots, evenly spaced around a circle in the complex plane, showcasing a beautiful symmetry. By dividing the angles by \( n \), De Moivre's Theorem distributes the roots evenly across the complex plane, centered on the origin.

Polar Form of Complex Numbers

The polar form provides an alternative way to represent complex numbers, complementing the more familiar rectangular form \( a + bi \). In polar form, a complex number is expressed as:

• \( z = r(\cos \theta + i \sin \theta) \)
• Alternatively written as \( z = r \text{cis} \theta \)

Here, \( r \) is the modulus or absolute value of the complex number, calculated as \( \sqrt{a^2 + b^2} \), and \( \theta \) is the argument, the angle the line makes from the positive real axis.

• The modulus \( r \) represents the distance from the origin to the point \( (a, b) \) in the complex plane.
• The argument \( \theta \) is determined using inverse trigonometric functions, like \( \tan^{-1}(b/a) \).

The polar form is extremely useful for multiplication and division of complex numbers, as it simplifies complex arithmetic operations through the manipulation of moduli and the addition of angles. It also makes De Moivre's Theorem applicable, aiding in computation of complex number powers and roots.

Complex Plane

The complex plane is a two-dimensional plane used to visualize complex numbers. Think of it as an extension of the conventional XY-plane, but designed for complex numbers. It's constructed by:

• Representing the real part along the horizontal axis (real axis).
• Placing the imaginary part along the vertical axis (imaginary axis).

Each point \( (a, b) \) on this plane corresponds to a complex number \( a + bi \). By plotting these numbers, we can easily grasp their relationship and properties.

• Distance from the origin is the modulus \( r = \sqrt{a^2 + b^2} \).
• Angle from the positive real axis is the argument \( \theta \).

The complex plane is crucial when working with the polar form, as the modulus and argument directly correspond to the position of the point. This visualization helps in understanding De Moivre's Theorem and the symmetric nature of the \( n \)th roots, showing how they are distributed evenly around the circle. Each root, for example, can be viewed as points rotating an equal angle distance in the complex plane.