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%%% Primary Title: digital number system
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%%% Owner: akrowne
%%% Author(s): akrowne
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\section{Digital System}

Most \footnote{ but not all-- see \htmladdnormallink{Roman numerals}{http://planetmath.org/encyclopedia/RomanNumerals.html} for an example of a baseless \htmladdnormallink{number}{http://planetmath.org/encyclopedia/Number.html} system.} written number systems are built upon the concept of a \emph{digital system} (or \emph{positional system}) for their functioning and conveying of quantitative meaning. In these systems, meaning is derived from two things: symbols and positions. A particular symbol in a specific \htmladdnormallink{place}{http://planetmath.org/encyclopedia/Place2.html} is called a \emph{digit}.

The representation of a value in a digital system follows the schema:

$$ \ldots s_2 s_1 s_0 . s_{-1} s_{-2} s_{-3} \ldots $$

Where each $s_i$ is some symbol that has a quantitative value (a digit). Places to the left of the \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} ($.$) are worth whole \htmladdnormallink{units}{http://planetmath.org/encyclopedia/Unital.html}, and places to the \htmladdnormallink{right}{http://planetmath.org/encyclopedia/Acute.html} are worth fractional units. It is the \emph{base} that tells us how much of a \htmladdnormallink{fraction}{http://planetmath.org/encyclopedia/Fraction.html} or how many whole units. Once a base $b$ is chosen, the value of a number $s_2 s_1 s_0 . s_{-1} s_{-2} s_{-3}$ would be calculated like:

$$ s_2 s_1 s_0 . s_{-1} s_{-2} s_{-3} = s_2 \cdot b^{2} + s_1 \cdot b^{1} + s_0 \cdot b^{0} + s_{-1} \cdot b^{-1} + s_{-2} \cdot b^{-2} + s_{-3} \cdot b^{-3} $$

In our now-standard, Arabic-derived decimal system, the base $b$ is equal to 10. Other very common (and useful) systems are \htmladdnormallink{binary}{http://planetmath.org/encyclopedia/Arity.html}, \htmladdnormallink{hexadecimal}{http://planetmath.org/encyclopedia/Hexadecadic.html}, and octal, having $b=2$, $b=16$, and $b=8$ respectively \footnote{ These are \htmladdnormallink{generic}{http://planetmath.org/encyclopedia/SomethingRelatedToMartinsAxiomAndTheContinuumHypothesis.html} systems which are capable of representing any number. By contrast, our system of written time is a curious hybrid of \htmladdnormallink{bases}{http://planetmath.org/encyclopedia/Bases.html} (60, 60, and then 10 from there on) and has a \htmladdnormallink{fixed}{http://planetmath.org/encyclopedia/Fixed3.html} number of whole places and a different number of symbols (24) in the highest place, making it capable only of representing the same \htmladdnormallink{discrete}{http://planetmath.org/encyclopedia/Discrete2.html}, \htmladdnormallink{finite set}{http://planetmath.org/encyclopedia/Finite.html} of values over and over again.}.

Each $s_i$ is a member of an \htmladdnormallink{alphabet}{http://planetmath.org/encyclopedia/Alphabet.html} of symbols which must have $b$ members. Intuitively this makes sense: when we try to \htmladdnormallink{represent}{http://planetmath.org/encyclopedia/RepresentableFunctor.html} the number which follows ``9'' in the decimal system, we know it must be ``10'', since there is no symbol after ``9.'' Hence, position as well as symbol conveys the meaning, and base tells us how much a unit in each position is worth.

\section{Remark}

Curiously, though one would think that the choice of base leads to merely a different way of rendering the same \htmladdnormallink{information}{http://planetmath.org/encyclopedia/MutualInformation.html}, there are instances where things are variously provable or proven in some bases, but not others. For instance, there exists a non-recursive formula for the $n$th \textbf{binary} digit of $\pi$, but not for decimal-- one still must \htmladdnormallink{calculate}{http://planetmath.org/encyclopedia/SearchProblem.html} all of the $n-1$ preceding decimal digits of $\pi$ to get the $n$th (see \htmladdnormallink{this paper}{http://www.nersc.gov/~dhbailey/dhbpapers/digits.pdf}).

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