\documentclass[11pt]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[nomarginpar]{geometry} \usepackage{color} \usepackage{amsfonts} \usepackage{amsmath} \usepackage{fancyhdr} \usepackage{graphicx} \usepackage{hyperref} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=Latex.dll} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{LastRevised=Monday, July 16, 2018 07:33:42} %TCIDATA{} \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}{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=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \large Lecture Notes} \chead{\color{black} \Large The Set of All Natural Numbers} \rhead{ \footnotesize page \ \thepage} \cfoot{} \lfoot{\footnotesize \copyright $\;$ Hidegkuti, 2018} \rfoot{\footnotesize Last revised: July 16, 2018} \textwidth 7.6in \textheight 9.8in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \setlength{\parskip}{0.09in} \begin{document} The \textbf{set of all natural numbers}, (sometimes also called the set of all counting numbers), denoted by $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $, is the set\FRAME{dtbpF}{3.2949in}{0.3779in}{0pt}{}{}{pic1natural.bmp}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.2949in;height 0.3779in;depth 0pt;original-width 8.7605in;original-height 0.9686in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic1natural.bmp';file-properties "XNPEU";}}It is easy to imagine that the set of all natural numbers was the first set of numbers at which human beings looked. \ The four basic operations can be defined and performed on natural numbers as follows. \textbf{Addition}, denoted by $+$, is defined as we usually think of addition: the addition of two natural numbers is obtaining the total amount of those quantities combined. \ In the statement $3+7=10$, we say that $3$ and $7$ are \textbf{addend}s and $10$ is called the \textbf{sum} of $3$ and $% 7$.\FRAME{dtbpF}{2.0972in}{1.0066in}{0pt}{}{}{pic2natural.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.0972in;height 1.0066in;depth 0pt;original-width 3.2188in;original-height 1.5316in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic2natural.bmp';file-properties "XNPEU";}}If we add two natural numbers, the sum is always a natural number. This is called \textbf{closure}. The set of all natural numbers is closed under addition: \ If $n$ and $m$ are natural numbers, then $n+m$ is also a natural number. \textbf{Subtraction}, denoted by $-$, is a mathematical operation that represents the operation of removing objects from a collection. \ In the statement $18-5=13$, we say that $18$ is the \textbf{minuend} and $5$ is the \textbf{subtrahend}, and $13$ is called the \textbf{difference} of $18$ and $% 5$.\FRAME{dtbpF}{2.527in}{0.8077in}{0pt}{}{}{pic3natural.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.527in;height 0.8077in;depth 0pt;original-width 4.2601in;original-height 1.3439in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic3natural.bmp';file-properties "XNPEU";}}If we subtract a natural number from another natural number, the difference may or may not exits within $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $. For example, the subtraction $8-5$ results in a natural number but the subtraction $3-11$ does not. In other words, the set of all natural numbers is NOT closed under subtraction. \textbf{Multiplication}, denoted by $\cdot ,$ or by $\times $, or by nothing at all between two objects, is defined as we usually think of multiplication% \begin{equation*} 3\cdot 7=21\text{ \ \ \ or \ \ \ }{3\times 7}{=21}\text{ \ \ \ or \ \ }% \left( 3\right) 7=21\text{ \ \ \ \ \ or \ \ \ \ }3\left( 7\right) =21 \end{equation*}% Please note that in the last two equations, the \textbf{parentheses do not indicate multiplication}. The parentheses helps us interpret $3$ and $7$ as two separate numbers and not the number $37$. Once we see two numbers with no operation sign between them, that NOTHING indicates multiplication. \ In the statement $3\cdot 7=21$, we say that $21$ is the \textbf{product} of $3$ and $7$. We also say that $3$ and $7$ are \textbf{divisors} or factors of $% 21 $. \FRAME{dtbpF}{2.156in}{0.8406in}{0pt}{}{}{pic4natural.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.156in;height 0.8406in;depth 0pt;original-width 3.8649in;original-height 1.4892in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic4natural.bmp';file-properties "XNPEU";}}If we multiply two natural numbers, the product is always a natural number. In other words, the set of all natural numbers is closed under multiplication: \ if $n$ and $m$ are natural numbers, then $nm$ is also a natural number. \textbf{Division}, denoted by $\div $ or by $/$ is defined as we usually think of division. \ One example is: if we have $20$ dots and we circle together every four dots, how many packages of four do we obtain? \ The answer is clearly five, because five packages of four will account for $20$ dots. \begin{equation*} {20\div 4}{=5}\text{ \ \ \ or \ \ \ }\dfrac{20}{4}{=}{5} \end{equation*}% \FRAME{dtbpF}{2.6195in}{1.0352in}{0pt}{}{}{pic5natural.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.6195in;height 1.0352in;depth 0pt;original-width 5.4163in;original-height 2.1248in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic5natural.bmp';file-properties "XNPEU";}}In the statement $20\div 4=5$, we say that $20$ is the \textbf{% dividend}, $4$ is the \textbf{divisor}, and $5$ is called the \textbf{% quotient} of $20$ and $4$.\FRAME{dtbpF}{2.4068in}{0.8216in}{0pt}{}{}{% pic6natural.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.4068in;height 0.8216in;depth 0pt;original-width 3.9894in;original-height 1.3439in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'pic6natural.bmp';file-properties "XNPEU";}}If we divide a natural number by another natural number, the quotient may or may not exist within $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $. For example, the division $12\div 3$ results in a natural number, but the division $20\div 7$ does not. In other words, the set of all natural numbers is NOT closed under division.\bigskip \bigskip %TCIMACRO{\TeXButton{\begin{minipage}{1.2in}}{\begin{minipage}{1.2in}}}% %BeginExpansion \begin{minipage}{1.2in}% %EndExpansion \FRAME{itbpF}{0.7394in}{0.6849in}{0in}{}{}{question.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.7394in;height 0.6849in;depth 0in;original-width 1.0533in;original-height 0.9729in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../7_tools/pictures/question.bmp';file-properties "XNPEU";}}% %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion %TCIMACRO{\TeXButton{\begin{minipage}{5.9in}}{\begin{minipage}{5.9in}}}% %BeginExpansion \begin{minipage}{5.9in}% %EndExpansion Discussion: \ \begin{enumerate} \item Consider the multiplication $4\cdot 9=36$ and the division $30\div 6=5$% . \ While in the division all of $30$, $6$, and $5$ have different names, both $4$ and $9$ are simply called factors in the multiplication. \ Can you explain why this will not cause any problems? \item Under which of the four basic operations (addition, subtraction, multiplication, \newline division) is $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $ closed? \end{enumerate} %TCIMACRO{\TeXButton{\end{minipage}}{\end{minipage}}}% %BeginExpansion \end{minipage}% %EndExpansion \FRAME{itbpF}{0.8484in}{0.672in}{0.1807in}{}{}{tree.jpg}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 0.8484in;height 0.672in;depth 0.1807in;original-width 4.4685in;original-height 3.531in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../7_tools/pictures/tree.jpg';file-properties "XNPEU";}}% %TCIMACRO{\TeXButton{\sffamily}{\sffamily} }% %BeginExpansion \sffamily %EndExpansion \ \ \ {\LARGE Enrichment} %TCIMACRO{% %\TeXButton{sans sherif}{\sffamily %\small} }% %BeginExpansion \sffamily \small %EndExpansion \ If you haven't already done so, read \href{http://www.teaching.martahidegkuti.com/shared/lnotes/1_prealgebra/intro/intro.pdf% }{our lecture notes on axioms}, because you need to understand the concepts of axioms and theorems for this exercise. \begin{enumerate} \item The set of all natural numbers, $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $ was not defined rigorously. \ Instead, we gave an intuitive description of $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $. \ However, mathematicians insisted on axiomatizing $% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion $. \ This means that they established a set of axioms from which many theorems can be derived. \ The resulting collection of true statements is the same as what we get starting with the intuitive definition. \ Research the Peano axioms. \ Feel free to start at Wikipedia. \ What are the Peano axioms? \ Note: when you look at the Peano axioms, you may notice that according to Wikipedia, $0$ is a natural number. \ In our class, $0$ was not defined as a natural number. \ Do not let yourself be annoyed or confused by the difference. \ Both conventions are very common. \end{enumerate} %TCIMACRO{\TeXButton{\rmfamily}{\rmfamily}}% %BeginExpansion \rmfamily% %EndExpansion \bigskip \bigskip %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion \href{http://www.teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html% }{For more documents like this, visit our page at\ http://www.teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}