\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{radix notation}

\hypertarget{radix_notation}{}\section*{{Radix notation}}\label{radix_notation}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{foundational_issues}{Foundational issues}\dotfill \pageref*{foundational_issues} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

At least in [[classical mathematics]], every [[real number]] may be written in \emph{base $n$} for any [[natural number]] $n \geq 2$ (binary for $n = 2$, decimal for $n = 10$, etc), which can be generalized to fractional $n \gt 1$ (especially $n = \mathrm{e} \approx 2.718$) and beyond.

For [[integers]], this idea goes back to the Old Babylonians using base $60$ (around 2000 BCE), perfected in the Gupta Empire using base $10$ (around 400), and popularized in Europe by [[Fibonacci]] (in 1202). For arbitrary real numbers, these were first used by [[Abu'l-Hasan al-Uqlidisi]] (around 952) and popularized by [[Simon Stevin]] (in 1585). This notation (in base $10$) is now ubiquitous, and (despite the technical difficulties of doing so rigorously) serves as a de facto definition of the [[real numbers]] in elementary mathematics.

Radix notation in base $2$ is the basis of [[floating point arithmetic]], the fast but imprecise method of calculation with real numbers usually used in modern [[computing]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[natural number]] $n \geq 2$, let $[n]$ be the set of natural numbers (including [[zero]]) strictly less than $n$: $[n] = \{0, 1, 2, \ldots, n - 2, n - 1\}$; in this context, an element of $[n]$ is called a \textbf{digit} (in base $n$). Let $\mathbb{Z}$ be the set of [[integers]]: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$; in this context, an element of $\mathbb{Z}$ is called a \textbf{place value}. Then a [[function]] from $\mathbb{Z}$ to $[n]$ is a doubly-infinite [[sequence]] of digits, one digit for each place value. Such a sequence $a = (a_i)_{i\colon{\mathbb{Z}}}$ defines a doubly-infinite [[series]] of [[rational numbers]]:

\begin{displaymath}
\sum_{-\infty \lt i \lt \infty} a_i n^{-i} .
\end{displaymath}
This series converges (to a [[real number]]) if and only if there is a place value $N$ such that $a_i = 0$ for all $i \lt N$. That is, we can write the series as

\begin{displaymath}
\sum_{i = N}^\infty a_i n^{-i} .
\end{displaymath}
In this way, a sequence of digits (finite on one end, infinite on the other) represents a nonnegative [[real number]].

As a representation of real numbers, this is almost unique. Specifically, two distinct sequences $a$ and $b$ of digits represent the same real number (meaning that the sums of the series are equal) if and only if there is some place value $m$ such that

\begin{itemize}%
\item $a_i = b_i$ for every place value $i \lt m$,
\item $a_m = b_m + 1$ (or the reverse), and
\item $a_i = n - 1$ for every $i \gt m$ while $b_i = 0$ for every $i \gt m$.

\end{itemize}
In this case, the sequence with $0$s is generally considered standard. The real numbers arising in this way are precisely the positive $n$-[[adic rational]] numbers, that is the [[rational numbers]] that are positive integer multiples of $n^{-m}$ for some integer $m$ (the same $m$ as above).

In [[classical mathematics]], every nonnegative real number may be represented in this way. Given such a number $x$ and a place value $i$, let $a_i$ be the [[remainder]] modulo $n$ of the [[floor]] of $x n^i$: that is, $a_i \coloneqq \lfloor{x n^i}\rfloor \mod n$; then $x = \sum_i a_i n^{-i}$. (For $n$-adic rational $x$, this will produce the representation with $0$s; to automatically produce the other representation, use $a_i = (\lceil{x n^i} - 1\rceil) \mod n$.) The number $0$ is represented only by a sequence of all $0$s.

Given $n$ unique symbols for the $n$ digits, the real number represented by a sequence of digits may be written by, beginning at the smallest place value whose digit is nonzero, writing the symbols in order, with a dot (or comma), called the \textbf{radix point}, between place value $0$ and place value $1$ (and padding zeroes after the radix point if the first nonzero digit has not yet been reached). Of course, the sequence is still infinite and cannot be written down, but we may write any finite portion. The $n$-adic rationals can be represented exactly by leaving out the infinitely repeating digit $0$; arbitrary [[rational numbers]] may be represented exactly by a bar over a list of digits that repeats infinitely. (It is a theorem that every non-$n$-adic rational number can be uniquely represented in this way; in fact, only rational numbers can be so represented.) A subscript may be used to indicate the base $n$ (with a default base $10$ in practice). Finally, a negative real number is written by writing its [[absolute value]] after a minus sign.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

Given any number $n$ (not necessarily a natural number), still called the \emph{base}, and any set $D$ of numbers, whose elements are called \emph{digits}, we may consider finite or infinite sequences of digits, indexed by integers (still called \emph{place values}). Such a sequence should include a smallest place value $N$ as part of its data (since $0$ may not be one of the digits), so the place values are all (or perhaps only some) of the integers $i \geq N$. Then the infinite series

\begin{displaymath}
\sum_{i=N}^\infty a_i n^{-i}
\end{displaymath}
still converges to a real number.

If $n \geq 2$ is a natural number and $D$ is $[n]$, then we recover the representation above of all nonnegative real numbers, unique except for the positive $n$-adic rationals. More generally, if $n \gt 1$ is any real number and $D$ is $[\lceil{n}\rceil]$, then we get a representation of all nonnegative real numbers, which (for non-integer $n$) is never unique (although in practice one uses the last sequence in [[lexicographic ordering]]).

If $n \geq 2$ is a natural number and $D$ is $[n] + 1 = \{1, 2, 3, \ldots, n - 1, n\}$, then we get a representation of all positive real numbers (but not $0$). If $n = 1$ and $D$ is again $[n] + 1 = \{1\}$, then every infinite sequence gives a divergent series, but the finite sequences represent all positive integers, uniquely up to a [[shift]] of place value; this is called \textbf{tally notation}. In the other direction, if $n \geq 3$ is a natural number and $D$ is a set of $n$ consecutive integers that includes both $-1$ and $1$, then we get a representation of all real numbers (positive, negative, and zero), which is unique except for a few rational numbers (in the case of $n = 3$ and so $D = \{-1, 0, 1\}$, those which are half of a power of $1/3$); this is called \textbf{balanced} radix notation.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Here are some representations of well-known real numbers in well-known radixes:

\hypertarget{foundational_issues}{}\subsection*{{Foundational issues}}\label{foundational_issues}

In [[constructive mathematics]], the existence of a radix expansion for every real number is equivalent to the (analytic) [[lesser limited principle of omniscience]].

[[Fred Richman]] considered a number system (a noncancellable [[rig]]) of nonnegative decimal sequences in which $0.\overline{9} \lt 1$; the usual rig of nonnegative real numbers is a subrig; see \hyperlink{Richman1999}{Richman 1999}. Although Richman is a prominent constructivist, the development was not (and probably cannot be made) constructive.

Usenet legend [[Alexander Abian]], before branching into speculative physics, advocated polemically that the real numbers should be defined as finite or infinite sequences of decimal digits; see \hyperlink{Abian1981}{Abian 1981}. While not favoured by most mathematicians due to the inelegance of the definitions and proofs, this is the form in which the real numbers are often presented to elementary students.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Alexander Abian]]. 1981. Calculus must consist of the study of real numbers in their decimal representation and not of the study of an abstract complete ordered field or nonstandard real numbers. International Journal of Mathematical Education in Science and Technology 12(4). \href{https://doi.org/10.1080/0020739810120417}{Doi:10.1080/0020739810120417}.


\item [[Fred Richman]]. 1999. Is 0.999\ldots{} = 1?. \href{http://math.fau.edu/Richman/html/999.htm}{Web}.



\end{itemize}
[[!redirects radix notation]] [[!redirects positional notation]] [[!redirects place-value notation]] [[!redirects place value notation]]

[[!redirects radix expansion]] [[!redirects expansion in base]]

[[!redirects decimal notation]] [[!redirects decimal expansion]]



\end{document}