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Chapter 6: Problem 42

Find all complex solutions to the equation \(x^{4}+1=0\).

Short Answer

Expert verified
\( x = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}, \text{ and } \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)

Step by step solution

01

- Rewrite the Equation

Rewrite the equation in the form that highlights its roots: \[x^{4} = -1\]
02

- Express \(-1\) in Exponential Form

Express \(-1\) in its exponential form using Euler's formula: \[-1 = e^{i\pi}\] Hence, we can write the equation as: \[x^{4} = e^{i\pi}\]
03

- Apply the General Root Finding Formula

Find the fourth roots of the equation. The general formula for the nth roots of \(e^{i\theta}\) is: \[e^{i(\theta + 2k\pi)/n}\] Here, \(k\) can take the values 0, 1, 2, and 3 (since there are four 4th roots). Thus, we get: \[x = \sqrt[4]{e^{i\pi}} = e^{i(\pi + 2k\pi)/4}\] for \(k\) in {0, 1, 2, 3}.
04

- Calculate Each Root

Evaluate the expression for each value of \(k\): For \(k = 0\), \[x_{0} = e^{i(\pi + 2\cdot0\cdot\pi)/4} = e^{i\pi/4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\] For \(k = 1\), \[x_{1} = e^{i(\pi + 2\cdot1\cdot\pi)/4} = e^{i3\pi/4} = \frac{-\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\] For \(k = 2\), \[x_{2} = e^{i(\pi + 2\cdot2\pi)/4} = e^{i5\pi/4} = \frac{-\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\] For \(k = 3\), \[x_{3} = e^{i(\pi + 2\cdot3\pi)/4} = e^{i7\pi/4} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\]
05

- Summarize the Solutions

Summarize the four complex solutions obtained: \[x = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}, \text{ and } \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\]

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Key Concepts

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

Euler's Formula
Euler's formula is a key tool in complex number calculations. It links complex exponentials to trigonometric functions. The formula states that for any real number \theta\, we have:\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \].
This relationship helps us to express complex numbers in exponential form, simplifying many mathematical problems.
For example, in the exercise, we used Euler's formula to express -1 as an exponential: \[-1 = e^{i\pi} \]. This enabled us to apply root-finding techniques more easily when solving for complex solutions.
nth Roots of a Complex Number
To find the nth roots of a complex number, we use a specific formula derived from Euler's formula. The general formula for finding the nth roots of a complex number in exponential form is:\[ e^{i(\theta + 2k\pi)/n} \] where k is an integer (0, 1, 2,...,n-1).
This formula ensures that we account for all possible roots by covering a full rotation around the complex plane.
Consider the equation in the exercise: \[-1 = e^{i\pi} \], which we transformed into: \[x^{4} = e^{i\pi} \]. Using the nth root formula with n=4, we applied \[{0,1,2,3}\] for k to find all four roots. This exhaustive approach gives us all distinct solutions of the equation.
Complex Numbers
Complex numbers are an extension of the real numbers and are very powerful in many areas of math and science. A complex number is typically written in the form \(a + bi\), where a and b are real numbers, and i is the imaginary unit, satisfying \(i^2 = -1\).
Complex numbers can be represented geometrically on the complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part.
  • They provide solutions to equations that have no real solutions.
  • They are foundational in fields like electrical engineering, quantum physics, and applied mathematics.
In our example, each complex solution for \(x^{4} = -1\) is written in Cartesian form, such as: \[\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\]. This helps us visualize and understand the nature of these roots in the complex plane.

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Most popular questions from this chapter

Find the indicated roots. Express answers in the form \(a+b i\) The cube roots of \(\sqrt{3}+i\)

For each rectangular equation, write an equivalent polar equation. $$ x=4 $$

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ (4-5 i)^{2} $$

For each polar equation, write an equivalent rectangular equation. $$ \theta=0 $$

Graph each pair of polar equations on the same screen of your calculator and use the trace feature to estimate the polar coordinates of all points of intersection of the curves. Check your calculator manual to see how to graph polar equations. $$ r=3 \sin 4 \theta, r=2 $$
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