Edits

Advanced/Nonstandard Analysis/main.tex

@@ -32,13 +32,18 @@
 }
 
 
+% This handout is pretty difficult. Make sure you can all solve all the problems yourself,
+% and remember that each SECTION was a two-hour lesson with a smart class.
+% From experience, the following holds:
+% supremum is a better lesson than dual numbers, which is better than extensions.
+
+
 \begin{document}
 
 	\maketitle
 
-
+	\input{parts/supremum}
-	\input{parts/1 extensions}
+	\input{parts/dual}
-	\input{parts/2 dual}
+	\input{parts/extensions}
-	\input{parts/x supremum}
 
 \end{document}

Advanced/Nonstandard Analysis/parts/dual.tex

@@ -21,8 +21,7 @@ In the problems below, let $\varepsilon$ a positive infinitesimal so that $\vare
 \note{Note that $\varepsilon \neq 0$.}
 
 \definition{}
-The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par
+The set of \textit{dual numbers} consists of elements of the form $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
-that consists of elements of the form $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
 
 \problem{}
 Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
@@ -33,11 +32,7 @@ Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
 
 \definition{}
 Let $f(x)$ be an algebraic function $\mathbb{R} \to \mathbb{R}$. \par
-(that is, a function we can write using the operators $+-\times\div$, powers, and roots) \par
+(that is, a function we can write using the operators $+-\times\div$ and integer powers) \par
-\note[Note]{
-	Why this condition? These are the only operations we have in an ordered field! \\
-	Powers, roots, and division aren't directly available, but are fairly easy to define.
-}
 
 \vspace{2mm}
 

Advanced/Nonstandard Analysis/parts/extensions.tex

@@ -13,7 +13,7 @@
 % If you edit this, please give credit!
 % Quality handouts take time to make.
 
-\section{Nonarchimedian Extensions}
+\section{Extensions of $\mathbb{R}$}
 
 \definition{}
 An \textit{ordered field} consists of a set $S$, the operations $+$ and $\times$, and the relation $<$. \par
@@ -49,7 +49,16 @@ An ordered field must satisfy the following properties:
 \end{itemize}
 
 \definition{}
-An ordered field that contains $\mathbb{R}$ is called a \textit{nonarchimedian extension} of $\mathbb{R}$.
+An ordered field that contains $\mathbb{R}$ is called an \textit{extension} of $\mathbb{R}$.
+
+\definition{}
+The \textit{Archimedian property} states the following: \par
+For all positive $x, y$, there exists an $n$ so that $nx \geq y$.
+
+\theorem{}
+All extensions of $\mathbb{R}$ are nonarchemedian. \par
+Proving this is difficult.
+
 
 \vfill
 \pagebreak
@@ -71,7 +80,8 @@ Which of the following are ordered fields?
 
 
 \problem{}
-Show that each of the following is true in any ordered field.
+Show that each of the following is true in any ordered field. \par
+The list of field axioms is provided below, for convenience.
 \begin{enumerate}
 	\item if $x \neq 0$ then $(x^{-1})^{-1} = x$
 	\item $0 \times x = 0$
@@ -89,8 +99,36 @@ Show that each of the following is true in any ordered field.
 %	And thus $(x^{-1})^{-1} = x$
 %\end{solution}
 
-
 \vfill
+
+\begin{itemize}
+	\item \textbf{Properties of $+$:}
+	\begin{itemize}
+		\item Commutativity: $a + b = b + a$
+		\item Associativity: $a + (b + c) = (a + b) + c$
+		\item Identity: there exists an element $0$ so that $a + 0 = a$ for all $a \in S$
+		\item Inverse: for every $a$, there exists a $-a$ so that $a + (-a) = 0$
+	\end{itemize}
+
+	\item \textbf{Properties of $\times$:}
+	\begin{itemize}
+		\item Commutativity
+		\item Associativity
+		\item Identity (which we label $1$)
+		\item For every $a \neq 0$, there exists an inverse $a^{-1}$ so that $aa^{-1} = 1$
+		\item Distributivity: $a(b + c) = ab + ac$
+	\end{itemize}
+
+	\item \textbf{Properties of $<$:}
+	\begin{itemize}
+		\item Non-reflexive: $x < x$ is always false
+		\item Transitive: $x < y$ and $y < z$ imply $x < z$
+		\item Connected: for all $x, y \in S$, either $x < y$, $x > y$, or $x = y$.
+		\item If $x < y$ then $x + z < y + z$
+		\item If $x < y$ and $z > 0$, then $xz < yz$
+		\item $0 < 1$
+	\end{itemize}
+\end{itemize}
 \pagebreak
 
 

Advanced/Nonstandard Analysis/parts/supremum.tex

@@ -14,7 +14,7 @@
 % Quality handouts take time to make.
 
 
-\section*{Bonus: The supremum \& infimum}
+\section*{The supremum \& the infimum}
 
 \definition{}
 In this section, we'll define a \say{real number} as a decimal, infinite or finite.