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:

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

Where each 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 is chosen, the value of a number $s_2 s_1 s_0 . s_{-1} s_{-2} s_{-3}$ would be calculated like:

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

In our now-standard, Arabic-derived decimal system, the base is equal to 10. Other very common (and useful) systems are binary, hexadecimal, and octal, having , , and respectively ².

Each is a member of an alphabet of symbols which must have 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 th binary digit of $\pi$ , but not for decimal- one still must calculate all of the preceding decimal digits of $\pi$ to get the th (see this paper).

Footnotes

...Most ¹: but not all- see Roman numerals for an example of a baseless number system.
... respectively ²: 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.

"digital number system" is owned by akrowne.

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