Problem 89 - Precalculus with Limits

Question

Finding the \(n\) th Roots of a Complex Number \(\operatorname{In}\) Exercises \(81-96,(\) a) use the formula on page 446 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fourth roots of $$16$$

Step-by-Step Solution

Answer

The fourth roots of 16 are: \(2, 2i, -2, -2i\).

Step 1: Identifying the complex number

The given complex number is \(16 + 0i\). The real part is \(16\) and the imaginary part is \(0\).

Step 2: Converting complex number to polar form

The magnitude \(r\) of the complex number is obtained by the formula \(\sqrt{a^2 + b^2}\) and the argument (angle) \(\Theta\) is obtained by \(\arctan(\frac{b}{a})\). Here, \(a = 16\) and \(b = 0\). Hence, \(r = \sqrt{16^2 + 0^2} = 16\) and \(\Theta = \arctan(\frac{0}{16}) = 0\). So, in polar form, the complex number is represented as \(16\operatorname{cis}(0)\).

Step 3: Finding the fourth roots

The \(n\)th roots of a complex number in polar form \(r\operatorname{cis}(\Theta)\) are given by \(\sqrt[n]{r}\operatorname{cis}(\frac{\Theta + 2k\pi}{n})\) where \(k\) varies from \(0\) to \(n-1\). As \(n = 4\) here, the four roots are: \(\sqrt[4]{16}\operatorname{cis}(\frac{0+2*0*\pi}{4}) = 2\), \(\sqrt[4]{16}\operatorname{cis}(\frac{0+2*1*\pi}{4}) = 2\operatorname{cis}\frac{\pi}{2}\), \(\sqrt[4]{16}\operatorname{cis}(\frac{0+2*2*\pi}{4}) = 2\operatorname{cis}\pi\), \(\sqrt[4]{16}\operatorname{cis}(\frac{0+2*3*\pi}{4}) = 2\operatorname{cis}\frac{3\pi}{2}\).

Step 4: Converting the roots to standard form

The standard form of a complex number is \(a+bi\). Using the euler's formula \(cis(\Theta) = \cos(\Theta) + i\sin(\Theta)\), the four roots in standard form are: \(2, 2i, -2, -2i\).

Key Concepts

Complex Number Polar Form

The polar form of a complex number is a way of expressing the number in terms of its magnitude and angle relative to the real axis. This format is especially beneficial when dealing with multiplication, division, or finding roots of complex numbers.

In general, a complex number can be written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The polar form, however, expresses this number as \( r(\cos(\Theta) + i\sin(\Theta)) \), where \( r \) represents the magnitude \(\sqrt{a^2 + b^2}\) and \( \Theta \) represents the argument \(\arctan(\frac{b}{a})\), the angle made with the positive real axis.

The formula to convert a complex number to polar form can be remembered as: \[ r\operatorname{cis}(\Theta) \]Where \( \operatorname{cis}(\Theta) \) is shorthand for \( \cos(\Theta) + i\sin(\Theta) \). So, for a given real number like 16, it can be represented as \( 16\operatorname{cis}(0) \), since it is located on the real axis and makes a \( 0 \)-degree angle with it.

Graphical Representation of Complex Roots

Complex roots can be visualized on a two-dimensional plane known as the complex plane or Argand plane. Here, the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.

To illustrate the roots of a complex number graphically, each root is plotted as a point in this plane. For instance, the fourth roots of the number 16 are equally spaced along a circle with a radius of 2. Starting from the real axis and moving counter-clockwise, these roots are positioned at 0, \( \frac{\text{π}}{2} \), \( π \), and \( \frac{3π}{2} \) radians, corresponding to the complex numbers \( 2 \), \( 2i \), \( -2 \), and \( -2i \) respectively.

Such a graphical interpretation helps in visualizing the symmetrical nature of complex roots about the real and imaginary axes, and it shows that these roots are evenly spaced around a circle when you're looking for the \( n \)th roots of a number.

Standard Form of Complex Numbers

When we describe complex numbers, we often use the standard form, which is the most familiar way to represent them for many math students. The standard form is written as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary component.

After finding the roots of a complex number, converting these roots from polar form to standard form makes them easier to understand and perform additional operations on. For instance, the polar roots \( 2\operatorname{cis}(0) \), \( 2\operatorname{cis} \frac{π}{2} \), \( 2\operatorname{cis}π \), and \( 2\operatorname{cis} \frac{3π}{2} \) can be converted to \( 2 \), \( 2i \), \( -2 \), and \( - 2i \) respectively in standard form by applying Euler's formula: \( e^{i\Theta} = \cos(\Theta) + i\sin(\Theta) \). This step is crucial in simplifying the final results and making them applicable in a variety of mathematical and engineering contexts.