Talk:Numeral system/Archive 1

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

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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

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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

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

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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.

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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

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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."

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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.

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.

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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.

• we have a page called Numeral system (that "includes" Binary_numeral_system and trinary and Roman_numerals and other ways of representing integers),
• do we have a corresponding page called ____(what?)____ (that "includes" continued fraction and Egyptian fraction and other ways of representing real numbers) ?

-- 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:

• MathWorld -- http://mathworld.wolfram.com/Base.html
• Dr Math -- http://mathforum.org/library/drmath/view/60405.html
• http://www.ceafinney.com/miscellania/bases.html citing Schwartzman S (1994). The Words of Mathematics: an etymological dictionary of mathematical terms used in English (ISBN 0-88385-511-9)
• Some copied pages from unknown books
• Between 40 and 90, I added the numbers based on the Latin numbers found in a Latin grammar http://www.gutenberg.org/files/15665/15665-h/15665-h.htm

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