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\begin{document}
\centerline{Math 112 Homework for Friday, Week 6}
\bigskip
\begin{enumerate}
	\item Recall that $\mathbb{R}$ is complete, meaning that every nonempty bounded above subset of $\mathbb{R}$ has a least upper bound.  In this problem, you will show that the corresponding fact about the existence of greatest lower bounds is a consequence.  First, we introduce some notation: given a subset $S \subseteq \mathbb{R}$, let $-S$ be the subset of $\mathbb{R}$ defined by $-S = \{ -x : x \in S \}$.
	\begin{enumerate}
		\item Let $S \subseteq \mathbb{R}$, and suppose that $M$ is the supremum of $S$.  Prove that $-M$ is the infimum of $-S$.
		\item Suppose that $T$ is a nonempty subset of $\mathbb{R}$ that is bounded below.  Prove that $\inf T$ exists in $\mathbb{R}$.  (Hint: use part (a) and the completeness of $\mathbb{R}$.)
	\end{enumerate}
	
	\item Write each complex number in the form $a + bi$, where $a, b \in \mathbb{R}$.
	\begin{enumerate}
		\item $(1 + 2i)(3 - 2i) + (2 - 5i)$
		\item $(1 + i)^{-1}$
		\item $(3 + 4i)(3 - 4i)$
		\item $\dfrac{1 - 2i}{2 + 5i}$
	\end{enumerate}
	
	\item Consider the set $\mathbb{R}^2$ with addition and multiplication defined by
	\[
	(a, b) + (c, d) = (a + c, b + d) \qquad \text{and} \qquad  (a, b)\cdot(c, d) = (ac, bd),
	\]
	respectively.  Indicate which field axioms fail, giving a concrete counter-example in each case.
	
	\item Let $x = \frac{\sqrt{3}}{2} + \frac{i}{2}$.  Draw $x, x^2, x^3, x^4, x^5, x^6,$ and $x^7$ in the complex plane.  What do you observe?
	
	\end{enumerate}

\end{document}


