Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


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


Re: Systems of Numerals (not Numbers) (was: Numbersystems, bijective, padic etc)
Posted:
Sep 30, 2013 10:04 PM


In article <d4d1395f1dae4d46a23b44311226a736@googlegroups.com>, jonas.thornvall@gmail.com wrote:
> Den tisdagen den 1:e oktober 2013 kl. 01:32:18 UTC+2 skrev Virgil: > > > Den tisdagen den 1:e oktober 2013 kl. 00:05:18 UTC+2 skrev Virgil: > > > > In article <e6cbafe9d5f348b2853907ea1457ee03@googlegroups.com>, > > > > jonas.thornvall@gmail.com wrote: > > > > > Den m?ndagen 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 > > > > > > > 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 > > > > > > 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 > > > > > > {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 > > > >  > > > I know my arithmetic will do just fine without zeros since i have > > > implemented > > > it before. > > You claim you do you do but we don't, and do not choose merely to take > > your word for it. > > For example how do you represent 0 and negatives? > > What is your alternative for decimal fractions? > > E.g., how would you represent decimal fractions, > > like 0.01, 0.001, 0.0001, and so on, without zeros? > >  > Well you could use the first digit in the decimal expansion to represent the > negative exponentiation standard form. Or just simply write out a variation > of standard form x*10^2 x*10^3 x*10^4 using bijective encoding? > Notice though that the standard form would be adopted to the base. > So for ternary > 1/3 = .1 > 2/3 = .2 > 1/9 = .(1)1 > 2/9 = .(1)2 > 1/27 = .(2)1 > 2/27 = .(2)2 > 1/81 = .(3)1 > 2/81 = .(3)2 ... > But partitioning of a single hashmark into a base has its problems like none > ending termination of string when ***trying*** to represent a certain > fraction in respective base. What about 4/9, 5/9, 7/9 and 8/9, and the other n/27 and n/81 fractions?
And in your "base 3" notations, how do you do fractions whose denominators are NOT powers of 3? What are your algorithms for adding, subtracting multiplying and dividing in your pseudodecimals?
Fir example 2/9 < 1/4 > 1/3, so how do you represent 1/4.
> So since fractions seem wastly superior i would probably mix bijective > encoding of naturals for each base with fractions instead of a lossy > representation and arithmetic. > I consider the naturals to be discrete, while fractions are part from a > continuum hold in each individual unary member forming the set of naturals. > Partitioning into bases isn't really that beautiful to me and further more > always seem to end up with lossy aproximations when doing arithmetic > operations.
Since the world has invested so much on the standard notations in math books and papers and has even hardwired them into many computers and calculators, what advantage do you see for changing all that that will be worth the billions upon billions of dollars, pounds or whatever, and the junking of all books, the adding machines, computers, etc., that are incompatible with your new notation and rewriting all those books using the old standard decimal notation? 



