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\begin{document}
\section*{Math 331 Fall 2026, Lecture 1: Number systems and binary operations}

\textbf{website:}
\texttt{\href{https://www.cip.ifi.lmu.de/~grinberg/t/26fa}{\texttt{https://www.cip.ifi.lmu.de/\symbol{126}%
grinberg/t/26fa/}}}

\setcounter{section}{0}

\subsection{General}

Welcome to Math 331: the first part of a two-quarter sequence on abstract
algebra. The website of this course will be
\[
\text{\texttt{\href{https://www.cip.ifi.lmu.de/~grinberg/t/26fa}{\texttt{https://www.cip.ifi.lmu.de/\symbol{126}%
grinberg/t/26fa/}}\ \ };}%
\]
this is where you can find typeset lecture notes and references. Homework will
be submitted (and graded) on
\href{https://drexel.instructure.com/courses/9597}{Canvas}. Homework is to be
done by yourself, with no use of AI or outside help (collaboration with your
fellow students is allowed, as is use of
\href{https://drexel.edu/coas/academics/departments-centers/mathematics/math-resource-center/}{the
MRC}). If you collaborate, please acknowledge your coauthors. All homework
should be typeset.

\section{What is this about?}

Abstract algebra is called abstract for a reason: it abstracts some features
of objects you know (numbers, matrices, maps, etc.), i.e., isolates them from
their context and studies them as independent things, ready to be applied to
other contexts. And it is called algebra because it is still fundamentally
about algebraic operations, albeit in a more general sense. Indeed, roughly
speaking, it is about the same four operations that you have been working with
for most of your mathematical life:

\begin{itemize}
\item \textbf{Middle-to-high-school algebra} is about adding, subtracting,
multiplying and dividing numbers.

\item \textbf{Linear algebra} is about adding, subtracting, multiplying and
dividing matrices.\footnote{The word \textquotedblleft
dividing\textquotedblright\ is commonly frowned upon, in favor of
\textquotedblleft inverting\textquotedblright, but the idea is the same. A
more serious problem is that matrices cannot be multiplied when their
dimensions don't fit together; but let's think about square matrices here.
Also, matrices can be scaled.}

\item \textbf{Abstract algebra} is about adding, subtracting, multiplying and
dividing anything that can be added, subtracted, multiplied and divided. The
\textquotedblleft anything\textquotedblright\ here hides quite a lot of
fineprint (what exactly makes an operation an \textquotedblleft
addition\textquotedblright?), and this is exactly what we will be defining in
the coming lectures.
\end{itemize}

\subsection{A quick overview of number systems}

So we will be studying \textquotedblleft generalized number
systems\textquotedblright, even though many of them don't look like numbers at
all. But before we come to them, let us recall some of the \textquotedblleft
number systems\textquotedblright\ (sets of numbers) we know, in a more-or-less
historical order:

\begin{itemize}
\item The set $\mathbb{Z}_{>0}=\left\{  1,2,3,\ldots\right\}  $ of all
positive integers originates in prehistory.
(\href{https://en.wikipedia.org/wiki/Proto-cuneiform}{Some of the earliest
writing was numbers}.)

\item The set $\mathbb{Q}_{>0}$ of all positive rational numbers was known to
the Babylonians.

\item The set $\mathbb{R}_{>0}$ of all positive real numbers was first
\textquotedblleft discovered\textquotedblright\ by the Greeks (and possibly
the Babylonians before them). Arguably, it was not properly (rigorously)
defined until the 19th Century by
\href{https://mathshistory.st-andrews.ac.uk/HistTopics/Real_numbers_2/}{Dedekind,
Cantor, Heine, M\'{e}ray, Weierstrass and others} (who thus founded rigorous
real analysis).

\item The number $0$ and the negative numbers took a surprisingly long time to
reach mankind's imagination; I refer to the Wikipedia
(\href{https://en.wikipedia.org/wiki/0}{zero} and
\href{https://en.wikipedia.org/wiki/Negative_number}{negatives}) for what we
know about their history. Yet, from a modern point of view, they are one of
the most natural things (certainly $0$, the beginning of all counting). So now
we have the number systems $\mathbb{Z}$ (the set of all integers),
$\mathbb{Q}$ (the set of all rational numbers) and $\mathbb{R}$ (the set of
all real numbers).

\item The complex numbers were introduced by Cardano in the 1500s; their set
is nowadays called $\mathbb{C}$.

\item W. R. Hamilton invented what is now known as
\href{https://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions}{the
Hamiltonian quaternions} in 1843. Their set is denoted by $\mathbb{H}$. Unlike
the prior number systems, it is noncommutative -- that is, two quaternions $a$
and $b$ can satisfy $ab\neq ba$.

\item Then there are two analogues of the complex numbers: the dual numbers
$\mathbb{D}=\left\{  a+b\varepsilon\ \mid\ a,b\in\mathbb{R}\right\}  $ with
$\varepsilon^{2}=0$, and the split complex numbers $\mathbb{S}=\left\{
a+bj\ \mid\ a,b\in\mathbb{R}\right\}  $ with $j^{2}=1$. We will not say much
about them in this course, but we will see them as examples of rings in the
second part of the sequence (Math 332).
\end{itemize}

There are more \textquotedblleft number systems\textquotedblright. And there
are some things that are quite similar to numbers yet do not commonly get the
honor of that name: matrices, for example. Square matrices can be added,
subtracted, multiplied and oftentimes inverted; why aren't they called numbers?

What is a number anyway? Where do number systems end? Who decides what a
number is?

\textbf{The modern point of view} is that this is not a good question.
\textquotedblleft Number\textquotedblright\ is a vague notion not worth
defining. The rigorously definable concepts are the specific kinds of numbers,
such as integers, rational numbers, real numbers, etc.. (Interestingly, the
Greeks did not think of real numbers as numbers but as geometric measurements;
so the word has been a floating signifier throughout its history.)

But still, there is a real question to be asked here: What do the above number
systems have in common, and what other objects share these commonalities?

This is what abstract algebra is about.

So what makes a number, or, rather, what makes a number system? All the above
number systems have operations $+$ (addition), $-$ (subtraction), $\cdot$
(multiplication) and $/$ (division), even though the operations $-$ and $/$
are not always defined in some of the systems (for example, $a-b$ and $a/b$
are not always defined in $\mathbb{Z}_{>0}$, and $n/0$ is never defined). So
it makes sense to focus on the operations $+$ and $\cdot$, and view $-$ and
$/$ as their partially-defined \textquotedblleft undo
operations\textquotedblright. Let us see what we know about $+$ and $\cdot$ on
some of our number systems:

\begin{enumerate}
\item \textbf{Associativity of addition:} We have $\left(  a+b\right)
+c=a+\left(  b+c\right)  $ for all numbers $a,b,c$.

\item \textbf{Associativity of multiplication:} We have $\left(  ab\right)
c=a\left(  bc\right)  $ for all numbers $a,b,c$. (When the operation symbol is
not written, always read a $\cdot$, as usual in algebra.)

\item \textbf{Commutativity of addition:} We have $a+b=b+a$ for all numbers
$a,b$.

\item \textbf{Commutativity of multiplication:} We have $ab=ba$ for all
numbers $a,b$. (This holds for all the above number systems, except for
$\mathbb{H}$; it also does not hold for matrices.)

\item \textbf{Identity element of addition:} There exists a number -- called
$0$ -- that satisfies $a+0=0+a=a$ for each number $a$. (This holds for all
\textquotedblleft modern\textquotedblright\ number systems, but not for
$\mathbb{Z}_{>0}$, $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$.)

\item \textbf{Identity element of multiplication:} There exists a number --
called $1$ -- that satisfies $a1=1a=a$ for each number $a$.

\item \textbf{Annihilation:} We have $a0=0a=0$ for each number $a$. (Here and
in the following, \textquotedblleft$0$\textquotedblright\ and
\textquotedblleft$1$\textquotedblright\ mean the $0$ and the $1$ from the
previous two properties.)

\item \textbf{Distributivity I:} We have $a\left(  b+c\right)  =ab+ac$ for all
numbers $a,b,c$.

\item \textbf{Distributivity II:} We have $\left(  a+b\right)  c=ac+bc$ for
all numbers $a,b,c$.

\item \textbf{Existence of additive inverses:} For each number $a$, there
exists a number $a^{\prime}$ such that $a+a^{\prime}=a^{\prime}+a=0$. (This
does not hold for $\mathbb{Z}_{>0}$, $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$.
The number $a^{\prime}$ is denoted by $-a$ and called the \textbf{negative} or
the \textbf{additive inverse} of $a$. This allows us to define subtraction by
$a-b=a+b^{\prime}$.)

\item \textbf{Existence of multiplicative inverses:} For each number $a\neq0$,
there exists a number $a^{\prime}$ such that $aa^{\prime}=a^{\prime}a=1$.
(This holds for $\mathbb{Q}$, for $\mathbb{R}$, for $\mathbb{C}$ and for
$\mathbb{H}$, but not for $\mathbb{Z}$, for $\mathbb{D}$, for $\mathbb{S}$ and
for many other number systems. This number $a^{\prime}$ is denoted by $a^{-1}$
or $\dfrac{1}{a}$ and called the \textbf{reciprocal} or the
\textbf{multiplicative inverse} of $a$. This allows us to define division by
$\dfrac{a}{b}=ab^{\prime}$.)

\item \textbf{Absence of zero-divisors:} If $a$ and $b$ are two nonzero
numbers, then $ab\neq0$. (This holds for all the above number systems except
for $\mathbb{D}$ and $\mathbb{S}$. It also fails for matrices.)

\item \textbf{Ordering:} There is a total order on the number system (i.e.,
numbers can be compared by size) that \textquotedblleft behaves
well\textquotedblright\ (this means, e.g., that $a\leq b$ implies $a+c\leq
b+c$ and $ac\leq bc$ for every $c\geq0$ and so on). (This is true for
$\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{Q}$ but not for $\mathbb{C}$.)
\end{enumerate}

As you can already see, these properties (called \textbf{axioms}) are not
universal; some number systems satisfy more of them than others do. And there
are many more axioms that we can think of (for example, any real number has a
cube root, but most number systems do not satisfy this). The idea is now to
define \textquotedblleft number systems\textquotedblright\ as sets with two
operations $+$ and $\cdot$ that satisfy some of the above axioms. More
precisely, for each selection of axioms, we want to know both

\begin{enumerate}
\item examples of number systems that satisfy these axioms, and

\item general properties that all such number systems have (i.e., theorems
that can be deduced from the axioms).
\end{enumerate}

These are the daily bread of an algebraist.

In this quarter-long course, we will only cover \textquotedblleft number
systems\textquotedblright\ with one operation ($+$ or $\cdot$ or whatever else
we call it). There is already a lot of interesting things to say about them.
Then, systems with two operations $+$ and $\cdot$ will be covered in Math 332.


\end{document}