Complex Numbers
Author
Junha Park
License
Creative Commons CC BY 4.0
Abstract
Simple problem set involving complex numbers.
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\title{Complex Numbers}
\author{Junha Park}
\date{\today}
\begin{document}
\maketitle
\section{Warm-Up Set}
\begin{enumerate}
\item Let $z^4$ = 1.
(a) What are the four complex roots of $z$?
(b) Draw the roots of $z$ on the complex plane.
\item Now let $z^5$ = 1.
(a) What are the five complex roots of $z$?
(b) Draw the roots of $z$ on the complex plane.
\item Let $w$ be a complex number such that $|w|$ = 3. Find the largest possible value of $|i + 1 - w|$.
\end{enumerate}
\section{Practice}
\begin{enumerate}
\item The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are the same, what is $b$ equal to?
\item There is a complex number z with imaginary part 164 and a positive integer n such that
$\frac {z}{z + n} = 4i$.
Find n.
\item Let $P(z)=x^3+ax^2+bx+c$, where $a$, $b$, and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $|a+b+c|$.
\item Let $z=a+bi$ be the complex number with $\vert z \vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized, and let $z^4$ = $c+di$. Find $c+d$.
\item The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m<n$. Find $n$.
\item (Challenge Problem) Complex numbers $a$, $b$, and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r$, and $|a|^2 + |b|^2 + |c|^2 = 250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.
\end{enumerate}
\end{document}