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Chapter 6: Problem 25

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

Short Answer

Expert verified
The subset of the complex plane consisting of complex numbers z such that \(z^3\) is a real number is the set of complex numbers with arguments \(\theta = \frac{n\pi}{3}\), where n is an integer (with no restrictions on the magnitude r).

Step by step solution

01

Express the complex number in polar form

For a complex number z, we can express it in polar form as follows: \(z = re^{i\theta}\), where r is the magnitude of z and \(\theta\) is the argument of z.
02

Compute the cube

Compute the cube of the complex number in polar form: \(z^3 = (re^{i\theta})^3 = r^3e^{i3\theta}\).
03

Find conditions for real solutions

For a complex number to be real, its imaginary part must be zero. For this to happen in a polar representation, the exponent of \(e\) must be a multiple of \(\pi\) so that the imaginary part vanishes. So, we must have: \(3\theta = n\pi\), where n is an integer.
04

Solve for the argument of z

If \(3\theta = n\pi\), then \(\theta = \frac{n\pi}{3}\).
05

Interpret the solution

For the complex number z to have its cube real, its argument must be \(\theta = \frac{n\pi}{3}\), where n is an integer. Therefore, the subset of the complex plane consisting of complex numbers z such that \(z^3\) is a real number is the set of complex numbers with arguments \(\theta = \frac{n\pi}{3}\) where n is an integer (with no restrictions on the magnitude r).

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Most popular questions from this chapter

Suppose \(z\) is a complex number. Show that \(\frac{z-\bar{z}}{2 i}\) equals the imaginary part of \(z\).

Suppose \(a \neq 0\) and \(b^{2}<4 a c .\) Verify by direct calculation that $$ \begin{array}{l} a x^{2}+b x+c= \\ a\left(x-\frac{-b+\sqrt{4 a c-b^{2}} i}{2 a}\right)\left(x-\frac{-b-\sqrt{4 a c-b^{2}} i}{2 a}\right) \end{array} $$.

Suppose \(z\) is a complex number. Show that \(\frac{z+\bar{z}}{2}\) equals the real part of \(z\).

Show that \(\overline{w-z}=\bar{w}-\bar{z}\) for all complex numbers \(\mathcal{w}\) and \(z\).

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (1+3 i)-(6-5 i) $$
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