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Chapter 9: Problem 27

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

Short Answer

Expert verified
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 by step solution

01

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 (胃) 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 (胃), 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: $$ 1+i = \sqrt{2}(\cos{\frac{\pi}{4}} + i \sin{\frac{\pi}{4}}) $$
02

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 \(k=1\): $$ z_1 = \sqrt[2]{\sqrt{2}}\left(\cos{\frac{\frac{\pi}{4} + 2(1)\pi}{2}} + i \sin{\frac{\frac{\pi}{4} + 2(1)\pi}{2}}\right) = \sqrt[4]{2}\left(\cos{\frac{9\pi}{8}} + i \sin{\frac{9\pi}{8}}\right) $$
03

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 $$ z_1 = \sqrt[4]{2}\left(\cos{\frac{9\pi}{8}} + i \sin{\frac{9\pi}{8}}\right) $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攁 real part and an imaginary part鈥攅xpressed 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.

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Most popular questions from this chapter

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=5, \theta=30^{\circ}$$

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=12 \text { newtons, } \theta_{\mathbf{u}}=130^{\circ} ; \mathbf{v}=20 \text { newtons } \theta_{\mathbf{v}}=250^{\circ}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\frac{\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}}{\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}}$$

Solve \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0 .\) IHint: Consider \(\left.x^{6}-1 \text { and } x-1 \text { and see Exercise } 47 .\right]\)

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$\frac{1+i}{1-i}$$
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