Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Replies: 23   Last Post: Oct 1, 2013 3:22 PM

 Messages: [ Previous | Next ]
 JT Posts: 1,448 Registered: 4/7/12
Re: Systems of Numerals (not Numbers) (was: Numbersystems, bijective,

Posted: Sep 30, 2013 5:14 PM

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 UTC-5, jonas.t...@gmail.com wrote:
>

> > When i've played with constructing *zeroless* numbersystems i've come a cross terms like bijective and p-adic, 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 zero-less 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?
And is this a bijective or p-adic encoding scheme, or maybe bijective and p-adic. Noone have yet told me.

>
>
> 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 no-zero-divisor property (meaning: matrix algebras and Clifford algebras other than R and C).

Date Subject Author
9/30/13 JT
9/30/13 JT
9/30/13 JT
9/30/13 FredJeffries@gmail.com
9/30/13 JT
9/30/13 Brian Q. Hutchings
9/30/13 JT
9/30/13 JT
9/30/13 Rock Brentwood
9/30/13 JT
9/30/13 Michael F. Stemper
9/30/13 JT
9/30/13 JT
9/30/13 JT
9/30/13 Virgil
9/30/13 JT
9/30/13 Virgil
9/30/13 JT
9/30/13 Virgil
10/1/13 Brian Q. Hutchings
10/1/13 Brian Q. Hutchings
10/1/13 Michael F. Stemper
10/1/13 Brian Q. Hutchings
10/1/13 Karl-Olav Nyberg