Problem 34 - Precalculus

Question

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

Step-by-Step Solution

Answer

The roots are \(\sqrt{2} \text{cis} \frac{3\pi}{8}\) and \(\sqrt{2} \text{cis} \frac{11\pi}{8}\).

Step 1: Convert to Polar Form

To find the polar form of the complex number \(-\sqrt{2}+\sqrt{2}i\), first calculate the modulus, \(r\), as \(r = \sqrt{(-\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = 2\). Next, find the argument \(\theta\) using \(\tan \theta = \frac{\sqrt{2}}{-\sqrt{2}} = -1\). Thus, \(\theta = \frac{3\pi}{4}\), considering the complex number is in the second quadrant. The polar form is \(2\text{cis}\frac{3\pi}{4}\) (where \(\text{cis} \theta = \cos \theta + i\sin \theta\)).

Step 2: Apply the n-th Root Theorem

For a complex number in polar form \(r\text{cis}\theta\), the \(n\)-th roots are given by \(r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right)\) for \(k = 0, 1, ..., n-1\). Here, \(n = 2\), so we find the square roots \(2^{1/2}\text{cis}\left(\frac{3\pi}{4} + 2k\pi\right)/2\).

Step 3: Calculate the Roots

The modulus of the roots is \(\sqrt{2}\). Calculate for \(k = 0\) and \(k = 1\):1. \(k = 0\): \(\sqrt{2}\text{cis}\frac{3\pi}{8}\)2. \(k = 1\): \(\sqrt{2}\text{cis}\frac{11\pi}{8}\)

Step 4: Plot the Roots on the Complex Plane

The roots in polar form can be plotted on the complex plane as vectors originating from the origin. They will both have a radius \(\sqrt{2}\):1. For \(\sqrt{2}\text{cis}\frac{3\pi}{8}\), the direction is approximately 67.5 degrees from the positive real axis.2. For \(\sqrt{2}\text{cis}\frac{11\pi}{8}\), the direction is approximately 247.5 degrees from the positive real axis.

Key Concepts

Polar Form

The polar form of a complex number is a way to express the number using a radius and an angle, rather than the traditional real and imaginary components. This form is particularly useful for multiplication and division, as well as finding powers and roots.
If you have a complex number \(z = a + bi\), the polar form is given by \(r\text{cis} \theta\), where:

• \(r\) is the modulus, calculated as \(r = \sqrt{a^2 + b^2}\).


• \(\theta\) is the argument, found using \(\tan \theta = \frac{b}{a}\).

In our example of \(-\sqrt{2} + \sqrt{2}i\), the modulus \(r\) was computed as 2, and the argument \(\theta\) was determined to be \(\frac{3\pi}{4}\). This places the complex number in the second quadrant of the complex plane. Thus, the polar form was expressed as \(2\text{cis} \frac{3\pi}{4}\). Converting complex numbers from rectangular to polar form simplifies many operations, especially when using trigonometric identities in calculations.
Understanding polar form also aids in visualizing complex numbers geometrically, as they can be envisioned as points or arrows emanating from the origin at a specific angle and distance in the complex plane.

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. It consists of a horizontal axis, known as the real axis, and a vertical axis, called the imaginary axis.
In the complex plane, a complex number \(a + bi\) is represented by the point \((a, b)\). This visualization allows us to handle complex numbers as geometrical objects, making complex arithmetic easier to comprehend.

• The real axis divides the plane into upper and lower parts, while the imaginary axis divides it into left and right halves.
• The complex number \(-\sqrt{2} + \sqrt{2}i\) is situated in the second quadrant where the real part is negative and the imaginary part is positive.

Graphical operations on the complex plane offer insights into concepts such as rotation and scaling. For instance, multiplying a complex number by another effectively rotates and scales the point in the plane.
Plotting roots on the complex plane, as done in our example, further exemplifies how these numbers relate geometrically. Two roots found for our problem were positioned as vectors at distinct angles, illustrating their symmetric nature about the origin.

n-th Roots

Finding the \(n\)-th roots of a complex number involves identifying all distinct complex numbers that, when raised to the power \(n\), result in the original complex number. This procedure is essential for both theoretical and practical applications in complex analysis.
The n-th root of a complex number in polar form \(r\text{cis}\theta\) is expressed through the formula:

• \(r^{1/n}\text{cis}\left(\frac{\theta + 2k\pi}{n}\right)\)

where \(k = 0, 1, ..., n-1\). Each value of \(k\) gives a different root, ultimately resulting in \(n\) unique roots.
In this case, we were tasked with finding the square roots (\(n=2\)) of the complex number \(-\sqrt{2} + \sqrt{2}i\). By applying the formula:

• For \(k = 0\), the root is \(\sqrt{2}\text{cis}\frac{3\pi}{8}\).
• For \(k = 1\), it is \(\sqrt{2}\text{cis}\frac{11\pi}{8}\).

These roots correspond to points that are equidistant from the origin, split equally around the complex plane. Understanding how to locate \(n\)-th roots is vital for tackling problems in which dividing angles and distributing magnitudes around a circle are required.