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Systems of Numerals (not Numbers) (was: Numbersystems, bijective, padic etc)
Posted:
Sep 30, 2013 4:24 PM


On Monday, September 30, 2013 10:57:49 AM UTC5, jonas.t...@gmail.com wrote: > When i've played with constructing *zeroless* numbersystems i've come a cross terms like bijective and padic, since my formalised knowledge of math terms is null.
The correct term, actually, would be systems of numerals. Numerals are not numbers, but merely symbols to denote numbers. So, technically, this is not a question of mathematics at all, but of orthography, which is a part of linguistics.
Others have deal with the main question (including a Wikipedia link), but there are a few comments to be made on zeroless orthographies.
In most cases (including the Wikipedia links, last I checked), there is a failure to note that the question has to be asked RELATIVE to the set that's being represented!
The smallest base for a numeric orthography for the natural numbers N = {0, 1, 2, 3, ... } is 2. Of necessity, any positional system has to either include a symbol for 0 or a representation of 0 formed of the other symbols. Since the base can only be positive (lest negative be represented), then 0 has to be a symbol.
For the counting numbers { 1, 2, 3, ... } the smallest base is 1. That does not require any 0. Nor does any other base. For base 10, for instance, the digits would have the values 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10.
For the integers {..., 2, 1, 0, 1, 2, ... } the smallest bases (in absolute value) are 3 or 3. Any attempt to construct a base 2 representation for the integers will either (a) require extra symbols not part of the positional orthography (i.e. a "negative sign") or (b) both a positive and negative digit, resulting inevitably in cases where 2 or more numerals represent the same number.
For the complex integers, there is are base 3i and base 3 orthographies, each of which have 9 digits. There MIGHT be a smaller digit set suitable for representing complex integers.
For the integer elements of the quaternions, octonions, Clifford algebra (and even matrix algebras), there are also positional systems of numerals, which all have the (more or less) usual methods for the 4 basic arithmetic operations. Multiplication, however, will be a bit more complicated. Division will be a lot more complicated, since it may not even exist, if the set does have the nozerodivisor property (meaning: matrix algebras and Clifford algebras other than R and C).



