Problem 27 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$1+i ; \quad n=2$$

Step-by-Step Solution

Answer

The roots of the complex number \(1+i\) are given by \(\sqrt[4]{2}\left(\cos{\frac{\pi}{8}} + i \sin{\frac{\pi}{8}}\right)\) and \(\sqrt[4]{2}\left(\cos{\frac{9\pi}{8}} + i \sin{\frac{9\pi}{8}}\right)\) respectively, where the roots are presented in polar form.

Step 1: 1. Convert complex number to polar form

To convert the complex number \(1+i\) to polar form, we need to find the magnitude (r) and angle (θ) of the complex number. \ For the magnitude (r), use the formula: $$ r = \sqrt{a^2 + b^2} $$ \ Where a and b are the real and imaginary parts of the complex number respectively. In this case, \(a=1\) and \(b=1\): $$ r = \sqrt{1^2 + 1^2} = \sqrt{2} $$ \ To find the angle (θ), use the formula: $$ \theta = \arctan{\frac{b}{a}} $$ \ In this case, \(\theta = \arctan{\frac{1}{1}} = \arctan{1} = \frac{\pi}{4}\). \ Thus, the polar form of the given complex number is: $$ 1+i = \sqrt{2}(\cos{\frac{\pi}{4}} + i \sin{\frac{\pi}{4}}) $$

Step 2: 2. Apply De Moivre's theorem to find the roots

To find the nth roots of the complex number in polar form, use De Moivre's theorem: $$ z_k = r^{\frac{1}{n}}\left(\cos{\frac{\theta + 2k\pi}{n}} + i \sin{\frac{\theta + 2k\pi}{n}}\right) $$ \ Where \(z_k\) are the roots, n is the number of roots to be found, and k ranges from 0 to \((n-1)\). In this case, \(n=2\), \(r=\sqrt{2}\), and \(\theta=\frac{\pi}{4}\). \ Now, we will compute the roots for \(k=0\) and \(k=1\). For \(k=0\): $$ z_0 = \sqrt[2]{\sqrt{2}}\left(\cos{\frac{\frac{\pi}{4} + 2(0)\pi}{2}} + i \sin{\frac{\frac{\pi}{4} + 2(0)\pi}{2}}\right) = \sqrt[4]{2}\left(\cos{\frac{\pi}{8}} + i \sin{\frac{\pi}{8}}\right) $$ For \(k=1\): $$ z_1 = \sqrt[2]{\sqrt{2}}\left(\cos{\frac{\frac{\pi}{4} + 2(1)\pi}{2}} + i \sin{\frac{\frac{\pi}{4} + 2(1)\pi}{2}}\right) = \sqrt[4]{2}\left(\cos{\frac{9\pi}{8}} + i \sin{\frac{9\pi}{8}}\right) $$

Step 3: 3. Write the roots in polar form

The two roots of the complex number \(1+i\) in polar form are: $$ z_0 = \sqrt[4]{2}\left(\cos{\frac{\pi}{8}} + i \sin{\frac{\pi}{8}}\right) $$ and $$ z_1 = \sqrt[4]{2}\left(\cos{\frac{9\pi}{8}} + i \sin{\frac{9\pi}{8}}\right) $$

Key Concepts

Complex Numbers

Complex numbers form a mathematical field that expands the concept of what we define as 'number'. Inherently, they provide solutions to equations that no real number can solve, the most famous being \( x^2 + 1 = 0 \).

A complex number is composed of two parts—a real part and an imaginary part—expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, defined as \( i^2 = -1 \). This vastly enriches our numeric landscape, allowing us to solve and graph equations previously out of reach.

For instance, the complex number \( 1 + i \) represents a point \( (1, 1) \) on the complex plane, also called the Argand plane, where the x-axis represents the real part, and the y-axis represents the imaginary part of the complex number.

De Moivre's Theorem

De Moivre's Theorem is a powerful mathematical tool often used with complex numbers, especially when they are presented in polar coordinates. The theorem connects complex numbers and trigonometry, stating that for any real number \( x \) and any integer \( n \) we have:
\[ (\cos x + i \sin x)^n = \cos(nx) + i \sin(nx) \]
This becomes incredibly useful when calculating powers and roots of complex numbers, as well as when working with trigonometric identities. In essence, De Moivre's Theorem simplifies the processes of raising complex numbers to powers or extracting roots by expressing the operations as a manipulation of the complex number's magnitude and angle.

Polar Coordinates

Polar coordinates offer an alternative method to describe the position of points in a plane using an angle and a distance from the origin. Unlike the traditional Cartesian coordinates, which use a grid of x and y, polar coordinates are represented by \( r \) (the distance from the origin) and \( \theta \) (the angle from the positive x-axis).

In polar coordinates, complex numbers take the form \( r(\cos \theta + i\sin \theta) \), which is also called the polar form of a complex number. This is particularly useful for multiplication and division of complex numbers, as well as converting between rectangular and polar forms.

Using polar form also streamlines solving for roots of complex numbers. As seen in the exercise, once we have \( 1 + i \) in polar form, we can apply De Moivre's Theorem to find its nth roots in an efficient manner, leading to elegant solutions that highlight the symmetry and structure inherent in complex numbers.