Digital System

Most ¹ written number systems are built upon the concept of a digital system (or 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 place is called a 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 point ($.$ ) are worth whole units, and places to the right are worth fractional units. It is the base that tells us how much of a fraction 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 binary, hexadecimal, and octal, having $b=2$ , $b=16$ , and $b=8$ respectively ².

Each $s_i$ is a member of an alphabet of symbols which must have $b$ members. Intuitively this makes sense: when we try to represent 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.

Remark

Curiously, though one would think that the choice of base leads to merely a different way of rendering the same information, 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 binary digit of $\pi$ , but not for decimal- one still must calculate all of the $n-1$ preceding decimal digits of $\pi$ to get the $n$ th (see this paper).

Footnotes

...Most ¹

but not all- see Roman numerals for an example of a baseless number system.

... respectively²

These are generic systems which are capable of representing any number. By contrast, our system of written time is a curious hybrid of bases (60, 60, and then 10 from there on) and has a fixed number of whole places and a different number of symbols (24) in the highest place, making it capable only of representing the same discrete, finite set of values over and over again.