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Virgil
Posts:
8,833
Registered:
1/6/11


Re: Systems of Numerals (not Numbers) (was: Numbersystems, bijective, padic etc)
Posted:
Sep 30, 2013 6:05 PM


In article <e6cbafe9d5f348b2853907ea1457ee03@googlegroups.com>, jonas.thornvall@gmail.com wrote:
> Den måndagen den 30:e september 2013 kl. 22:24:57 UTC+2 skrev > federat...@netzero.com: > > 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. > > Really??? > > But what about bijective ternary below, why would it need zero? > > BASE 3 BELOW > Dec = NyaNTern=StandardTern > > 1 =1 01 > 2 =2 02 > 3 =3 10 > 4 =11 3+1 11 > 5 =12 3+2 12 > 6 =13 3+3 20 > 7 =21 6+1 21 > 8 =22 6+2 22 > 9 =23 6+3 100 > 10 =31 9+1 101 > 11 =32 9+2 102 > 12 =33 9+3 110 > 13 =111 9+3+1 111 > 14 =112 9+3+2 112 > 15 =113 9+3+3 120 > 16 =121 9+6+1 121 > 17 =122 9+6+2 122 > 18 =123 9+6+3 200 > 19 =131 9+9+1 201 > 20 =132 9+9+2 202 > 21 =133 9+9+3 210 > > Why would this encoding scheme need 0?
Until you have shown us that the arithmetic of your notation is at lestas simple as that of standard base ten (or other bases like 2, 8 or 16).
How do you add, subtract, multiply, divide, take square roots, averages, etc., in your notation?
What happens when you need to subtract a number from itself?
How do you deal with integers needing both 0 and negatives?
And 



