Date: Sep 30, 2013 8:35 PM Author: JT Subject: Re: Systems of Numerals (not Numbers) (was: Numbersystems, bijective,<br> p-adic etc) 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:

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> > > In article <e6cbafe9-d5f3-48b2-8539-07ea1457ee03@googlegroups.com>,

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> > > jonas.thornvall@gmail.com wrote:

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> > > > Den m?ndagen den 30:e september 2013 kl. 22:24:57 UTC+2 skrev

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> > > > federat...@netzero.com:

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> > > > > On Monday, September 30, 2013 10:57:49 AM UTC-5, jonas.t...@gmail.com

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> > > > > wrote:

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> > > > > > When i've played with constructing *zeroless* numbersystems i've come

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> > > > > > a

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> > > > > > cross terms like bijective and p-adic, since my formalised knowledge

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> > > > > > of

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> > > > > > math terms is null.

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> > > > > The correct term, actually, would be systems of numerals. Numerals are

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> > > > > not

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> > > > > numbers, but merely symbols to denote numbers. So, technically, this is

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> > > > > not

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> > > > > a question of mathematics at all, but of orthography, which is a part

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> > > > > of

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> > > > > linguistics.

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> > > > > Others have deal with the main question (including a Wikipedia link),

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> > > > > but

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> > > > > there are a few comments to be made on zero-less orthographies.

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> > > > > In most cases (including the Wikipedia links, last I checked), there is

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> > > > > a

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> > > > > failure to note that the question has to be asked RELATIVE to the set

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> > > > > that's being represented!

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> > > > > The smallest base for a numeric orthography for the natural numbers N =

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> > > > > {0,

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> > > > > 1, 2, 3, ... } is 2. Of necessity, any positional system has to either

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> > > > > include a symbol for 0 or a representation of 0 formed of the other

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> > > > > symbols. Since the base can only be positive (lest negative be

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> > > > > represented), then 0 has to be a symbol.

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> > > > Really???

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> > > > But what about bijective ternary below, why would it need zero?

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> > > > BASE 3 BELOW

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> > > > Dec = NyaNTern=StandardTern

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> > > > 1 =1 01

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> > > > 2 =2 02

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> > > > 3 =3 10

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> > > > 4 =11 3+1 11

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> > > > 5 =12 3+2 12

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> > > > 6 =13 3+3 20

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> > > > 7 =21 6+1 21

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> > > > 8 =22 6+2 22

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> > > > 9 =23 6+3 100

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> > > > 10 =31 9+1 101

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> > > > 11 =32 9+2 102

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> > > > 12 =33 9+3 110

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> > > > 13 =111 9+3+1 111

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> > > > 14 =112 9+3+2 112

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> > > > 15 =113 9+3+3 120

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> > > > 16 =121 9+6+1 121

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> > > > 17 =122 9+6+2 122

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> > > > 18 =123 9+6+3 200

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> > > > 19 =131 9+9+1 201

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> > > > 20 =132 9+9+2 202

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> > > > 21 =133 9+9+3 210

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> > > > Why would this encoding scheme need 0?

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> > > Until you have shown us that the arithmetic of your notation is at

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> > > lestas simple as that of standard base ten (or other bases like 2, 8 or

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> > > 16).

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> > > How do you add, subtract, multiply, divide, take square roots, averages,

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> > > etc., in your notation?

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> > > What happens when you need to subtract a number from itself?

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> > > How do you deal with integers needing both 0 and negatives?

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> > > And

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> > > --

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> >

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> > I know my arithmetic will do just fine without zeros since i have implemented

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> > it before.

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>

>

> You claim you do you do but we don't, and do not choose merely to take

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> your word for it.

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>

>

> For example how do you represent 0 and negatives?

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>

>

> What is your alternative for decimal fractions?

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>

>

> E.g., how would you represent decimal fractions,

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> like 0.01, 0.001, 0.0001, and so on, without zeros?

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> --

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.

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.