Numerals in Positional Notation   [1]

Where does positional notation come from?

We can see the development of positional notation in early numeral systems which used a mixed-radix form (e.g., the Babylonian/Sumerian sexagesimal notation) and, loooking back further, in terms of counters categorized according to different types of goods they represented. It is straightforward to develop rules for cardianlity, enumeration and ordering of finite, mixed-radix numerals: the rules are essentially those we developed for simple composite symbols (e.g., the playing cards from lab). Given a notion of ordering on the "digit places", i.e. a notion of least and most significant digits, we see that the significance or relative contribution of the digits in a more-significant position is in terms of a multiple of the number of possible "digits" in a less-significant place (i.e., the cardianality of the digit set). In general, the significance of a more-signficiant digit is a multiple of the product of the cardinalities the sets used in less-signficiant places.

With fixed-radix positional notations, where the set of digits is uniform across all places, we get two advantages:

1. the contribution of each place value is more uniform, since the product of prior places is now just a power (exponential) of the common digit-set cardinality;
2. the set of possible values is infinite, or at least indefinite, in the sense that we can use more places as needed.

The mathematical formulae for computing the values (meanings, semantics) of numerals in mixed- and fixed-radix systems are thus as follows:

mixed radix:     [d_n ... d₀]   =   ∑_k ([d_k]⋅∏_j<k |S_j|)

fixed radix:     [d_n ... d₀]_b   =   ∑_k [d_k]⋅b^k