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Chapter 6: Problem 44

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a positive number.

Short Answer

Expert verified
The subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a positive number can be described as: \(z = re^{i\frac{2k\pi}{3}}\), where \(r > 0\) (magnitude greater than zero) and \(k\) is an integer. This subset forms equidistant rays of complex numbers originating from the origin, with an angle of 120 degrees (\(\frac{2\pi}{3}\) radians) between them.

Step by step solution

01

Understand Complex Numbers and Complex Plane

A complex number is a number in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, which equals \(\sqrt{-1}\). The complex plane is a two-dimensional plane, with the x-axis representing the real part of the complex number, and the y-axis representing the imaginary part.
02

Identify Polar Form of Complex Numbers and Roots

The polar form of a complex number is given by \(z = re^{i\theta}\), where \(r\) is the magnitude of \(z\) and \(\theta\) is the angle that the line connecting the origin to the complex number makes with the positive x-axis. De Moivre's Theorem states that a complex number raised to a power, say \(n\), can be computed as: \(z^{n} = r^{n}e^{in\theta}\). In our case, we are interested in cubing a complex number, so \(n = 3\).
03

Analyze the Condition Given by the Problem

We need to find the subset of complex numbers whose cube is a positive number. That is: \(z^3 = r^3e^{i3\theta} > 0\). It's important to note that a positive number in the complex plane has a magnitude greater than zero and lies entirely on the positive real-axis, which means that the imaginary part is zero. That means the angle \(\theta\) will be a multiple of \(2\pi\) after the complex number is cubed.
04

Find the Subset of Complex Numbers

We know that \(z^3 = r^3e^{i3\theta} > 0\) and that the imaginary part should be zero. So, we can set up the following equation: \(3\theta = 2k\pi\), where k is an integer. Now, we can solve for \(\theta\): \(\theta = \frac{2k\pi}{3}\). We can finally describe the subset of the complex plane. The complex numbers \(z\) will have the polar form: \(z = re^{i\frac{2k\pi}{3}}\), where \(r > 0\) (magnitude greater than zero) and \(k\) is an integer. This subset contains points at every 120 degree turn (\(\frac{2\pi}{3}\) radians) around the complex plane, forming equidistant rays of complex numbers originating from the origin.

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Most popular questions from this chapter

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=9, \theta=-\frac{\pi}{3} $$

What is the range of the function \(\cos ^{2}(3 x) ?\)

Convert the rectangular coordinates given for each point to polar coordinates \(r\) and \(\theta .\) Use radians, and always choose the angle to be in the interval \((-\pi, \pi)\). $$ (4,-4) $$

Convert the rectangular coordinates given for each point to polar coordinates \(r\) and \(\theta .\) Use radians, and always choose the angle to be in the interval \((-\pi, \pi)\). $$ (-4,1) $$

Show that $$ \cos (2 \theta) \leq \cos ^{2} \theta $$ for every angle \(\theta\).
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