% \renewcommand{\familydefault}{\sfdefault} % \renewcommand{\sfdefault}{phv} \documentclass[11pt]{report} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{multicol} \usepackage{hyperref} \usepackage{appendix} \usepackage{boxedminipage} \usepackage{color} \usepackage{fancyhdr} \usepackage{geometry} \usepackage{graphicx} \usepackage{helvet} \usepackage{times} \usepackage{fancybox} \usepackage{mathptmx} \usepackage{sectsty} \usepackage{makeidx} \usepackage[none]{hyphenat} \usepackage{setspace} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Thursday, December 29, 2016 04:53:04} %TCIDATA{LastRevised=Saturday, September 25, 2021 15:29:18} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=40 LaTeX Report.cst} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.5in,right=0.5in,top=0.6in,bottom=0.3in} \setlength{\parindent}{0in} \setlength{\textheight}{9.8in} \pagestyle{fancy} \lhead{\color{blue}\large Lecture Notes} \chead{\Large The Language of Mathematics} \rhead{\small page \thepage} \cfoot{} \lfoot{\footnotesize \copyright \; Hidegkuti, 2017} \rfoot{\footnotesize Last revised: July 21, 2018} \renewcommand{\headrulewidth}{0.8pt} \newcommand{\defstart}{ \begin{boxedminipage}{7.4in} \setlength{\fboxrule}{15pt} \setlength{\fboxsep}{15pt} \setlength{\rightskip}{15pt} \setlength{\leftskip}{15pt} \color{white} . \color{black} } \newcommand{\defcont}{ \setlength{\hangindent}{58pt} } \newcommand{\defend}{ \end{boxedminipage} \setlength{\parindent}{0pt} \setlength{\hangindent}{0pt} \setlength{\parskip}{0.06in} \setstretch{1.12} } \newcommand{\exstart}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{60pt} \setlength{\parskip}{0.04in} \setstretch{1.1} } \newcommand{\excont}{ \setlength{\parindent}{60pt} \setlength{\hangindent}{75pt} } \newcommand{\solstart}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{63pt} } \newcommand{\solcont}{ \setlength{\parindent}{63pt} \setlength{\hangindent}{63pt} } \newcommand{\solrestart}{ \setlength{\parindent}{48pt} \setlength{\hangindent}{63pt} } \newcommand{\vissza}{ \setlength{\parindent}{0pt} \setlength{\hangindent}{0pt} \setlength{\parskip}{0.1in} \setstretch{1.12} } \begin{document} %TCIMACRO{\TeXButton{parskip}{\setlength{\parskip}{0.1in}}}% %BeginExpansion \setlength{\parskip}{0.1in}% %EndExpansion %TCIMACRO{\TeXButton{\vissza}{\vissza}}% %BeginExpansion \vissza% %EndExpansion \begin{center} {\Large Part 1- What is Mathematics?} \end{center} As a graduate student, I had the annoying habit of asking my teachers and peer students what they think mathematics is. \ To my surprise, I received many different answers, and, to this day, I agree with many of them. \ In my eyes, mathematics is many things. \ In mathematics, we will be talking a lot about things being true or being not true. \ Although this probably happens in every course in every discipline, mathematical truth can be objectively established and agreed upon. \ To achieve such an objective approach, we have to develop a language that is objectively understood. \ In this sense, mathematics is also a language. %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ \textbf{Mathematics} is a collection of \emph{true statements} that are developed, expressed, and interpreted using an objective \emph{language} and rules of \emph{logic}.% %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in}\vspace{0.03in} To understand mathematics, we need to first agree on an objective language. \ Reading and writing mathematical notation correctly will be important. %TCIMACRO{% %\FRAME{dtbpF}{5.3359in}{0.3797in}{0pt}{}{}{separator1.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.3359in;height 0.3797in;depth 0pt;original-width 6.826in;original-height 0.4601in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'separator1.bmp';file-properties "XNPEU";}}}% %BeginExpansion \begin{center} \includegraphics[ natheight=0.4601in, natwidth=6.826in, height=0.3797in, width=5.3359in] {//Mac/Dropbox/lnotes/1Prealgebra/intro/graphics/separator1__1.pdf}% \end{center} %EndExpansion %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{% %\TeXButton{minipage start - 3.5in}{\begin{minipage}{3.5in} %\setlength{\rightskip}{25pt}}}% %BeginExpansion \begin{minipage}{3.5in} \setlength{\rightskip}{25pt}% %EndExpansion \vspace{0.1in} %TCIMACRO{% %\TeXButton{font family }{\setstretch{1.08} %\sffamily %\small}}% %BeginExpansion \setstretch{1.08} \sffamily \small% %EndExpansion 'Then you should\vspace{0.03in} say what you mean', the March Hare went on. \ 'I do,' Alice hastily\vspace{0.03in} replied; 'at least - at least I mean what I say - that's the same thing, you\vspace{0.03in} know.' \ 'Not the same thing a bit!' said the Hatter.\vspace{0.03in} \ Why, you might just as well say\vspace{0.03in} that 'I see what I eat' is the same thing as 'I eat what I see!'\vspace{0.16in}\newline Lewis Carroll \vspace{0.03in}\vspace{0.03in}\vspace{0.03in}\newline \textit{Alice's Adventures in Wonderland}\vspace{0.38in}% %TCIMACRO{\TeXButton{end of minipage}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{% %\TeXButton{second minipage start - 3.5in}{\begin{minipage}{3.5in} %\setlength{\leftskip}{5pt}}}% %BeginExpansion \begin{minipage}{3.5in} \setlength{\leftskip}{5pt}% %EndExpansion %TCIMACRO{% %\FRAME{itbpF}{3.3416in}{2.5529in}{0in}{}{}{alice1.jpg}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.3416in;height 2.5529in;depth 0in;original-width 4.1667in;original-height 3.1773in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'Alice1.jpg';file-properties "XNPEU";}}}% %BeginExpansion \includegraphics[ natheight=3.1773in, natwidth=4.1667in, height=2.5529in, width=3.3416in] {//Mac/Dropbox/lnotes/1Prealgebra/intro/graphics/Alice1__2.jpg}% %EndExpansion \ %TCIMACRO{\TeXButton{end of minipage}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.1in} \begin{center} {\Large Part 2 - The Language of Mathematics}\vspace{0.05in} \end{center} Mathematical statements are much like English sentences. \ As English sentences are built from different kinds of words, mathematical statements usually contain three types of components: \textbf{objects, operations, and relations}.\vspace{0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ The concept of an \textbf{object} (very much like nouns in English sentences) is usually clearly\ understood and needs no explanation. %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in}\vspace{0.03in} Examples of objects from algebra include the number $2$, the number $3$, and numbers in general. \ Objects from geometry include\ lines, points, triangles, circles, line segments, etc.\vspace{0.03in}\vspace{0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ An \textbf{operation} is an action (very much like verbs in English sentences) that can be applied to objects and usually result in new objects. %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in}\vspace{0.03in} Examples of operations from algebra include addition, subtraction, multiplication, and division. \ \ Operations from geometry include\ reflection to a line, rotation, \ or translation that can be performed on points, triangles, circles, line segments, etc.\vspace{0.03in}\vspace{0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ A \textbf{relation} is something we use to compare two objects.% %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in}\vspace{0.03in} Unlike operations, relations do not produce new things; we use relations to compare already existing objects. \ For example, the operation addition produces $7$ if applied to $2$ and $5$. \ Relations in algebra are equal, or less, or greater. \ Relations from geometry are how geometric objects can be compared to each other: similar, congruent, parallel, perpendicular. Many statements use at least one of each of these three components. \ In the statement $2+5=7$\ the numbers $2$, $5$, and $7$ are the objects, addition (denoted by $+)$ is the operation, and being equal (denoted by $=$) is the relation. It is a common misconception to think of mathematics as the study of \textit{% only} numbers. \ Numbers are only certain types of objects. \ As we progress in the study of mathematics, we will find that there are many other types of interesting objects. \ \ For example, sets are objects we will soon study. Furthermore, the study of operations and relations is also interesting and fruitful. So, what kind of true statements can be established in mathematics? \ There are three types of true statements in mathematics: \textbf{definitions}, \textbf{axioms}, and \textbf{theorems}.\vspace{0.03in}\vspace{0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ A \textbf{definition} is a labeling statement in which we agree to use an expression to refer to an object,\ operation, or relation in mathematics.% %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in} Definitions are all true statements, because they simply reflect an agreement in the terms of the language. \ To be precise, these decisions were made without consultating any of us, often decades (if not centuries) before any of us were ever born. \ An example of a definition would be if we pointed to a clear sky and said: "From now on, let's call this color blue".% \vspace{0.03in}\vspace{0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} $\,$A \textbf{theorem} is a statement that we insist on proving before believing that it is true. \ To \textbf{prove} a theorem\ means\ to derive it from previously established true statements, using logically correct steps.% %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in} If you think about that last definition a little, you will see that no theorem can exist, unless we agree on accepting a few statements to be true, without proving them. \ These are our "starting true statements".\vspace{% 0.03in} %TCIMACRO{\TeXButton{def start}{\defstart}}% %BeginExpansion \defstart% %EndExpansion %TCIMACRO{\TeXButton{def cont}{\defcont}}% %BeginExpansion \defcont% %EndExpansion \textbf{Definition:} \ An \textbf{axiom} is a statement we agree to accept to be true without proof.% %TCIMACRO{\TeXButton{def end}{\defend}}% %BeginExpansion \defend% %EndExpansion \vspace{0.03in} Axioms are usually simple, basic statements that are in agreement with our intuition. \ For example, the statement "\textit{It is possible to draw a straight line from any point to any other point.}" is an axiom. \ It has been a constant effort to keep the number of axioms to a minimum. We prove a theorem by deriving its statement from statements already established to be true.\vspace{0.03in} To be precise, when we prove our first theorem, we derive its statement from the axioms. \ When we prove our second theorem, we derive its statement from the axioms and the first theorem. \ When proving the third theorem, we can use all the axioms, and the first and second theorems. \ And so on. \ For our tenth theorem, we have all the axioms and the first nine theorems at our disposal. \ At this point, we are building a logically sound theory, a unified discipline within mathematics. \ It is one thing to suspect, to feel, or to have a hunch that something is true. \ It is entirely different from proving it, with unescapable force of logic.\vspace{0.03in} \pagebreak The ancient Greek mathematician Euclid discussed mathematics in this manner, i.e stating axioms and building a theory by deriving a sequence of theorems from the axioms. \ (He called axioms postulates.) \ Mathematicians immediately accepted and embraced this logical approach to the study of mathematics - and it is how it is done still today. \ Although Euclid has contributed to several parts of mathematics (including geometry and number theory), he completely axiomatized of what we now call classical geometry or Euclidean geometry. \ He stated five postulates, accepted them to be true and derived most basic theorems of classical geometry. \ \vspace{0.3in} %TCIMACRO{% %\FRAME{itbpF}{0.8596in}{0.6806in}{0.2006in}{}{}{tree.jpg}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.8596in;height 0.6806in;depth 0.2006in;original-width 4.4685in;original-height 3.531in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'tree.jpg';file-properties "XNPEU";}}}% %BeginExpansion \raisebox{-0.2006000000in}{ \includegraphics[ natheight=3.531in, natwidth=4.4685in, height=0.6806in, width=0.8596in] {//Mac/Dropbox/lnotes/1Prealgebra/intro/graphics/tree__3.jpg}% }% %EndExpansion \ \ \ %TCIMACRO{\TeXButton{\sffamily}{\sffamily}}% %BeginExpansion \sffamily% %EndExpansion {\LARGE Enrichment} %TCIMACRO{\TeXButton{\begin{minipage}{4.5in}}{\begin{minipage}{4.5in}}}% %BeginExpansion \begin{minipage}{4.5in}% %EndExpansion %TCIMACRO{% %\TeXButton{font family sans sherif}{\sffamily %\small}}% %BeginExpansion \sffamily \small% %EndExpansion \begin{enumerate} \item Look up Euclid's book \textit{Elements} on the internet. \ (Start at Wikipedia). \ List Euclid's five postulates. \ Explain the significance of these five statements. \item What is the parallel postulate? \ Look up the history of Euclid's parallel postulate on the internet. \ (Start with Wikipedia.) \ How would we go about proving that an axiom is really a theorem? \ List statements from geometry that are logically equivalent to the parallel postulate. \ What exactly could it mean for two axioms to be equivalent to each other? \vspace{% 0.8in} \end{enumerate} %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{\TeXButton{\begin{minipage}{3in}}{\begin{minipage}{3in}}}% %BeginExpansion \begin{minipage}{3in}% %EndExpansion %TCIMACRO{% %\FRAME{dtbpFU}{1.5177in}{2.5996in}{0pt}{\Qcb{Euclid (\symbol{126}400 BC - \symbol{126}300 BC)}}{}{euclid.jpg}{% %\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5177in;height 2.5996in;depth 0pt;original-width 4.5835in;original-height 7.9165in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'Euclid.jpg';file-properties "XNPEU";}}}% %BeginExpansion \begin{center} \includegraphics[ natheight=7.9165in, natwidth=4.5835in, height=2.5996in, width=1.5177in] {//Mac/Dropbox/lnotes/1Prealgebra/intro/graphics/Euclid__4.jpg}\\ Euclid (\symbol{126}400 BC - \symbol{126}300 BC) \end{center} %EndExpansion %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{% %\TeXButton{\rmfamily}{\rmfamily %\small}}% %BeginExpansion \rmfamily \small% %EndExpansion \vspace{1.8in}\vspace{2.4in} {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}