Talk:Numeral system/Archive 1

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I think everyone would agree that roman numerals are a numbering system, right? What is the name for that kind of numbering system? I'm tempted to call it a purely additive system, but there may be another name. We sould expand this article so that numbering systems like the roman one are included in the definition (even if they aren't used any more).

No. Roman numerals are a different numeral system rather than a different number system.. Eclecticology

I've seen it called a 'tally system', like other systems in which 'tally marks' are made to count or tally up a total. --MichaelTinkler

This kind of system, often with four vertical strokes followed by a fifth cross stroke, seems to be a rudimentary base five number system. Eclecticology

Someone wrote:

N=12: Duodecimal? (used by the Romans in many places)

When did the Romans use duodecimal? --Zundark

The Romans used 12 especially when doing fraction calculation. I hope someone else can write more about this. Perhaps fractions should not be considered a full blown number system. In the Chepang language of Nepal, numbers are duodecimal. Perhaps it is a better quote then the Roman example. I'll change it.
I see what you mean now: uncia / quincunx / septunx / deunx. But fractions don't really count as a number system, as you say. --Zundark

never heard of its use myself, Zundark. --MichaelTinkler

I don't know who used it, but it is suggestive that we have a word for a dozen but not a word for thirteen. Also, the number names "eleven" and "twelve" should really be "oneteen" and "twoteen" if base 10 had been in use consistently at the time. Whoever made the names for small numbers used a mixture of base 12 and base 10. --AxelBoldt

Yes,eleven and twelve are trace of base 12 usage. Other traces of base 12 usages in human languages include dozen = 12 items, gross = 12 dozens, 1 shilling = 12 pences etc.

But the word "eleven" is related to the word "one", and similarly "twelve" is related to "two". This is not obvious in modern Germanic languages, but it's quite clear in Gothic "ainlif" and "twalif". So they are still decimal, but expressed in a different way. --Zundark

A base eight system was devised by a people (the name of which I will insert here once I have tracked it down)

"... certain people in the Nawkachta area had once belonged to a now-disbanded tribe, which had the odd custom of severing the pinky fingers of a child to symbolize maturity ... their number system was based on eight."
"... the Yuki of California think their system based on eight is the most logical for a similar reason. The Yuki's base eight system is based on the number of interfinger spaces. Knuckles are used in yet other cultures."
In the tekst I read: "If A is even and A/2 is odd, all integer powers of the number (A/2)+1 will contain (A/2)+1 as their last digit" But lets apply this to "6", and we get:

-> A = 6(even) -> A/2 = 3(odd) -> 4/2+1 = 4 -> 4^2 = 16 -> which does not have 4/2+1 (=4) in any digit!!!!! Maybe I am stupid, or didn't read tekst correctly? explanation please! RHD.


Contents

Roman numeral system

I have heard the Roman number system called a "subtractive" system since the instance of putting the smaller symbol on the left indicates that it is to be subtracted from the immediate symbol on its right.

Desirable properties of numeral systems

I have added what I think to be 3 main points of numeration

  • A useful set of numbers represented
  • uniqueness of representation (nice, but not required, e.g. 1.00000000..=0.9999999999..)
  • compatibility with arithmetic, so that the numerals can be used in algorithms.

I am considering a 4th:

  • uniqueness of number represented

since one interpretation of a fixed or floating point representation is that one numeral can represent any value in a range of values. For instance, a numeral such as 1.02 in the "currency" numeration system represents any value from 1.015 to 1.025. The question is whether this is too abstract for wikipedia - there are plenty of places elsewhere in wikipedia where numeral is confused with number. Comments? --AndrewKepert 07:21, 22 Sep 2003 (UTC)


fractions

But fractions don't really count as a number system, as you say. --Zundark

Well, what are they then ?

Do we have a WikiPage that lists all the strange and wonderful ways of representing fractions ? -- DavidCary 02:47, 9 Apr 2004 (UTC)

1/t = [0; 1 1 1 1 1 1 1 1 ...] (continued fraction) = 0.618 034 448 ... = 1/2 + 1/9 + 1/200 + ... (egyptian fraction) = (sqrt(5)-1) / 2 =~= 5/8

The articles continued fraction and Egyptian fraction exist. Michael Hardy 03:11, 9 Apr 2004 (UTC)

Cool, I didn't know that.

-- DavidCary 01:41, 21 May 2004 (UTC)

  • "This article is about different methods of expressing numbers with symbols."

Fractions are numbers and a summary of unit fractions as used by Egyptians is absolutely appropriate content for this page. I would consider continued fractions to be a bit on the mathematical side, and fixed point numbers to be technical, so maybe just a mention would suffice. Davilla 22:27, 8 February 2006 (UTC)

electronic components

Today I read: "Electronic components (first vacuum tubes, then transistors) may have only 2 possible states: concat(1) and closed (0). ... Note however that a computer does not treat all of its data as integers. Thus, some of it may be treated as texts and program data. Real numbers (numbers that can be not whole) are usually written down in the floating point notation, that has different rules of arithmetic."

Huh ? What is "concat" ? I thought it was a Unix command-line utility. The whole point of the Von Neumann revolution was realizing that programs, data, text, etc. can all be represented by the same numbers and stored in the same memory locations. The computer *does* store every bit of its data as, well, bits. (And if you think of bits as the integers "0" and "1", then texts and program data *are* treated as integers.) Nearly all bits are stored as one of the 2 possible ways to magnetize of a patch of magnetic media. -- DavidCary 02:47, 9 Apr 2004 (UTC)

"stone-age binary"

I deleted from the article the sentence

(In that context, it is often referred to as "stone-age binary", on the justification that the absence of the chosen symbol functions like a different symbol (perhaps explicitly called "blank"), making the system in another sense a two-symbol one.)

Google tells me

  • "stone-age binary": 2 hits (both on the same site)
  • "unary": 218,000 hits.

2 times by the same writer doesn't count as "often".

--DavidCary 21:05, 23 Jul 2004 (UTC)

Positional systems with funny digit sets

By this I mean things like balanced ternary and decimal without a zero. There are stranger possibilities too, like ternary with digits {0, 2, -5} (% stands for -5 here):

Decimal             0     1     2     3     4     5     6     7     8     9    10    11
{0,2,-5}-ternary    0    2%     2   2%0  2%0%   2%2    20 2%0%%    22  2%00  2%2%  2%02

Is it worth having some unified discussion of these? 4pq1injbok 03:11, 2 August 2005 (UTC)

One possibility is to present the section on "Positional numeration systems in detail" in three subsections: The first with heading "The canonical base-k numeration system" (more-or-less as already written), a second subsection "Alternative base-k numeration systems", and a third subsection "Positional systems other than base-k". The second would include a very brief discussion with links to (1) the already existing bijective numeration article ("decimal without a zero" is very special case), (2) a yet-to-be created aricle on the (j,k)-numeration system (of which balanced ternary is the (1,1) special case), and maybe (3) a yet-to-be-created article that discusses the very general case of digit-sets D that contain 0 and possibly a mix of positive & negative integers, focusing on the general result that's roughly as follows:
a set D of integers (positive and/or negative) that includes 0, serves as the digit-set of a complete & unambiguous base-k (k ≥ 2) numeration system iff (a) D is a complete residue system (mod k) and (b) no length-n digit-string represents a nonzero multiple of kn-1.
The third subsection could link to articles on more exotic systems, such as the Fibonacci representations, Ostrowski numeration, and maybe even representations in complex bases. --r.e.s. (Talk) 23:24:11, 2005-08-02 (UTC)
That's a reasonable presentation. It looks like you're more qualified than me to write it, given that I didn't know the names you give for most of these systems nor the general result you quote. I gather that, in your result, a system must be able to represent all integers to be complete, so even the canonical base-k system fails (any string of the digit k-1 violates (b)). Systems in which we only require uniqueness and completeness of the natural numbers are of interest too, I'd say. Do you know of any results applying to them?
Some other interesting positional systems for the third section:
  • the golden ratio base, in which all integers (indeed, all elements of Z[√5]) have terminating representations, unique if we rule out the sequence 11 (is this a Fibonacci representation?). I don't know if there are other algebraic numbers which give such nice results. It might also be worth mentioning something like 'base pi', since this seems to be a common idea, even though no simple set of restrictions on strings will leave the system complete and even close to unique for R.
  • I think Knuth found 'factorial base' useful somewhere.
Lastly, I guess decimal without a zero should be merged into bijective numeration. 4pq1injbok 01:17, 3 August 2005 (UTC)
I've lined-through some incorrect wording in what I said above -- the theorem I paraphrased is for "mixed" digit-sets that include negative, zero, and positive digits (all three must be in the set). "Complete" means there is a representation for every number intended to be represented (e.g. the elements of a semiring). Both the canonical and the bijective base-k systems are complete for the nonnegative integers, and the alternative base-k systems that are the subject of the theorem are complete for the set of all integers. I agree that the additional systems you mention would also fit nicely in the third section. If no one does the rewrite, eventually I might give it try (assuming no dissenters), but not soon.--r.e.s. (Talk) 03:51:52, 2005-08-03 (UTC)

move fictional numbering system out

D'ni numbers are fictional, designed in the 1990s solely for certain computer games. Although a metaphysical argument can be made that all numeral systems are fictional, that is, created by humans, there is a reasonable basis for making the distinction between numeral systems that were actually used by real human cultures, and numeral systems created for artistic effect in individual works of literature. If nobody objects, and if I have any energy left this weekend, I'll create a new page to put it in. Cbdorsett 16:48, 6 September 2005 (UTC)

The etymologically correct base names

"etymologically correct" does not have to mean "etymolgically pure" which is the sense used here. For example, wouldn't "hepticosamal" (instead of viginti-septimal) be a "correct" blend of latin and greek just as hexadecimal is? ConstableBrew 02:21, 18 May 2006 (UTC)


I would like to know the source of the list of "etymologically correct" base names, or the criterion for "etymological correctness". Specifically, I see that denary is considered preferable over decimal, but no -ary term is given as alternative for octal. I wonder why that is so.

I've changed that now. – Adhemar 10:27, 13 May 2006 (UTC)

I am quite interested in these names, but I find that several sources give different names. I will post here a list of what I have found, from different sources. I'm not sure if this is authorative enought to include. The distinction is made between

  • base names per se (ending in -ary)
  • the name of a relationship between n elements (ending in -al)
  • the commonly used base names, a mix of the previous two.

Sources include:

Number

Relation

Base

Mixed

0

–

nullary

nullary

1

–

unary

unary

φ

–

phinary

–

2

binal
or dual

binary

binary

e

(natural)

–

–

3

tertial

ternary

or trinary

ternary

4

quartal

quaternary

quaternary

5

quintal

quinary

or quinquenary

quinary

6

sextal

senary
or sexenary

senary
or sexenary or sextal

7

septimal

septenary

septenary

8

octal
or octonal or octimal [1]

octonary

octal

9

nonal

nonary

nonary
or nonal

10

decimal

denary

decimal

11

undecimal

undenary

undecimal

12

duodecimal
or dozenal

duodenary

duodecimal
or dozenal

13

tredecimal
or tridecimal

tredenary
or tridenary

tredecimal
or tridecimal

14

quattuordecimal

quattuordenary

quattuordecimal

15

quindecimal

quindenary

quindecimal

16

hexadecimal
or sedecimal or sexdecimal or sexadecimal [2]

senidenary
or sedenary or sexdenary or sexadenary

hexadecimal
or senidecimal or sedecimal or sexdecimal or sexadecimal

17

septendecimal

septendenary

septendecimal

18

octodecimal

octodenary
or decenoctonary

octodecimal

19

novendecimal

or nonadecimal

novendenary
or nonadenary or decennonary

nonadecimal
or nonadecimal

20

vigesimal

vigenary

vigesimal

21

viginti-unal

viginti-unary

viginti-unal

22

viginti-dual

viginti-binary

viginti-dual

23

viginti-tertial

viginti-ternary

viginti-ternary

24

viginti-quartal

viginti-quaternary

viginti-quaternary

25

viginti-quintal

viginti-quinary

viginti-quinary

26

viginti-sextal

viginti-senary

viginti-senary

27

viginti-septimal

viginti-septenary

viginti-septenary

28

viginti-octal

viginti-octonary

viginti-octal

29

viginti-nonal

viginti-nonary

viginti-nonary

30

trigesimal

tricenary

trigesimal

32

triginta-dual
(or duotrigesimal)

triginta-binary

(or duotricenary)

triginta-dual
(or duotrigesimal)

40

quadragesimal

quadragenary

quadragesimal

42

quadraginta-dual

quadraginta-binary

quadraginta-dual

50

quinquagesimal

quinquagenary

quinquagesimal

60

sexagesimal

sexagenary

sexagesimal

64

sexaginta-quartal

sexaginta-quaternary

sexaginta-quaternary

70

septuagesimal

septuagenary

septuagesimal

80

octogesimal

octogenary

octogesimal

85

octoginta-quintal

octoginta-quinary

octoginta-quinary

90

nonagesimal

nonagenary

nonagesimal

99

nonaginta-nonal

nonaginta-nonary

nonaginta-nonary

100

centesimal

centenary

centesimal

101

centum-unal

centum-unary

centum-unary

128

centum-viginti-octal

centum-viginti-octonary

centum-viginti-octal

150

centum-quinquagesimal

centum-quinquagenary

centum-quinquagesimal

160

centum-sexagesimal

centum-sexagenary

centum-sexagesimal

200

ducentesimal

ducentenary

ducentesimal

250

ducenti-quinquagesimal

ducenti-quinquagenary

ducenti-quinquagesimal

256

ducenti-quinquaginta-sextal

ducenti-quinquaginta-senary

ducenti-quinquaginta-sextal

300

trecentesimal

trecentenary

trecentesimal

400

quadringentesimal

quadringentenary

quadringentesimal

500

quingentesimal

quingentenary

quingentesimal

512

quingenti-duodecimal

quingenti-duodenary

quingenti-duodecimal

600

sescentesimal

sescentenary

sescentesimal

700

septingentesimal

septingentenary

septingentesimal

800

octingentesimal

octingentenary

octingentesimal

900

nongentesimal

nongentenary

nongentesimal

1000

millesimal

millenary

millesimal

1024

mille-viginti-quartal

mille-viginti-quaternary

mille-viginti-quartal

2000

duo-millesimal

duo-millenary

duo-millesimal

100000

centum-millesimal

centum-millenary

centum-millesimal

1000000

decies-centena-millesimal

decies-centena-millenary

decies-centena-millesimal

Notes: [1] octaval is never used. [2] hexadecimal is the common computer-science terminology, but it is unsatisfactory because it is a combination of the Greek hexa and the Latin decim. The proper Latin should be sedecim or sexdecim, yielding either sedecimal or sexadecimal. Schwartzman writes, according to MathWorld: "Since hexadecimal is a rather long word, it is sometimes abbreviated hex. The word hexadecimal is unusual because Greek and Latin elements are combined; the expected purely Latin form would be sexadecimal, but then computer hackers would be tempted to shorten the word to sex."

-- Adhemar, 4 May 2006