Talk:Decimal

terminology

This article does not make it clear whether it is about the decimal aspect of the current world system, or the positional aspect. It says that our system is the one of two decimal positional systems. But then it compares non decimal systems to the system only discussing their base. What is truly needed is a grid of articlesh that looks like this, having an introductary article for decimal systems, binary systems, dodecimal, binary, vigesimal and sexigesimal systems, as well as an introduction for each of the ways of denoting the powers, positional, different symbols.. &c. However, as a start, the following might make sense:

an article on systems that use a different symbol to show how many, but use positions for powers -can be based on this article after a title change, and moving some stuff around -will also contain mention of common binary notation, and its dervitives (hex &c) -will also discuss all the different notations for the decimal system of this type, arab, western, gujarati, &c -base sixty fractions

an article about systems that have a different symbol for each amount in each order such as greek, hebrew, older arabic one -(abjad systems???)

an article about systems that have a different symbol for each power of the base, but write it multiple times in order to show amount -decimal ones: Roman, Egyptian, that other greek one that looks like hang man -sexigesimal ones: Sumerian, babylonian

an article about systems that use positions to show order and use accumalation of the symbols to show amound -sexigesimal: later babylonian -vigesimal: maya (the above two both use alternating symbol sets for the two factors of their base, so are not really pure)

Once this framework is done, there are probably lots of main articles that can be pointed to. —Preceding unsigned comment added by Alexwebjitsu (talk • contribs) 03:43, 18 February 2008 (UTC)

Initial comments

I've made two changes. First, there was a statement that "It is the most widely used numeral system, perhaps because a human usually has four fingers and a thumb on each hand, giving a total of ten digits on both hands." The proper preposition is "over", not "on". "On" creates an ambiguity as to whether it means that each hand has a total of 10 fingers, or together have 10 fingers.

Also, there was a statement that + means plus and - means minus. When it comes to sign, that is WRONG. The signs are positive and negative, not plus and minus.

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Decimal is the number system humans use because of the fact that we have ten fingers.

I heard that some cultures prefered to use the hexadecimal system because they didn't count their fingers on their hands. But instead, they counted with one hand using one thumb to touch on the finger tips and the bends at their finger joints. (There are 16 points on each human hand, hence a hexidecimal system.) However, the decimal system became so wide spread internationally that it dominates now.

I heard about this over twenty years ago from my high school teacher. I don't know his source of this information. I am wondering if any wikipedians out there can confirm this.

If the counting finger-joints technique were more prevailing than counting fingers, human society could have adopted the hexadecimal system which is much better compatible with binary computers nowadays.

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The ancient Mayan civilization used base 20 in their numbering system. Their numeric symbols denote values from 0 to 19. (source: http://www.eecis.udel.edu/~mills/maya.htm)

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Avoid fallacies in arguments. Just because the people that use decimal do so because they have 10 fingers doesn't mean that all humans use decimal. Nor does it invalidate any of these base 16 or base 20 systems. The article should point out that not all people use decimal (and I will edit it). --drj

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I don't think there are any societies that used base 16 though. The highschool teachers story seems suspect. Base 20 is of course fingers and toes. But where does base 12 come from? --AxelBoldt

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12 presumably comes from months of the year. Many calendars have 12 months in a year (not just because it is nearly the number of lunar months in a year). Imagine you are an early geek into factors and astronomy. Observe: 360 days in a year, aha! that factorises easily with nice factors like 12, 60, 24, etc. The base 16 claim seems very dubious to me. Fingers and toes didn't occur to me though it is plausible. --drj.

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I think the 12, 24, 60 business came from the Babylonians/Persians? Somewhere that direction and long before Greece. --rmhermen I said "geek" not "greek"! Bablylonians/Mesopotamia is the generally agreed source I believe. --drj

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Are roman numerals a number system? What is the base?

In Wikipedia, this is now called a numeral system -R. S. Shaw.

It's a number system, but not a positional one, so it doesn't have a base. --AxelBoldt

So perhaps the article on number systems should mention it?

In Wikipedia, this is now called a numeral system -R. S. Shaw.

Yes it should. --AxelBoldt

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In the US weighing system, one pound = 16 ounces. In Chinese weighing system, one catty = 16 taels. Though they are not number systems, but at least it give some hints why the number 16 is involved in measurements universally. In any systems that use division, any power of 2 is a good candidate for convenience sake. For example, a gallon = 4 quarts = 8 pints = 128 fluid ounces = 1024 fluid drams etc.

One pound is also 16 ounces in the Imperial system from which the US one is derived, although (1) the Imperial pint is now 20 fluid ounces rather than 16, obscuring the (former) relationship between "pound" and "pint", and (2) for some reason the two "fluid ounces" are slightly different (the fluid ounce used to be that amount of pure water which weighed 1oz. at 70°F, just as today one definition of "kilogram" is the weight of 1 litre of pure water at 4°C). Thus a gallon of water weighs (about) 8Lb. in the US and 10lb. in Britain. -- 217.171.129.68 (talk) 22:35, 29 March 2008 (UTC)

Looks like human are attracted to the power of 2 and astronmonical periods and our fingers and toes.

A old British pound = 20 shillings

one old shilling = 12 pences

"Pence" is already plural (one of the plurals of "penny", the other of course being "pennies"), hence there's no such word as "pences". Incidentally, one penny nowadays (1p — in the few years after 1972, called "one new penny" to distinguish it from the previous penny) == 2.4 (old, pre-1972) pence (2.4d -- d for "denarius", anthough as noted there were 12 to the next unit, not 10 as the name implies). — 217.171.129.68 (talk) 22:35, 29 March 2008 (UTC)

20 and 12 can still be explained, but 1 mile = 1760 yards??? how did they come up with that number?

Have you heard the story about how the butt size of the Roman horses decided the rail guage in the current US railroad system?

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In decimal counting, the Fibonacci numbers repeat the sequence of the last digit over a period of 60. Every other numeral system with base less than 14, repeats in less than half of this (often 24).

Base   Period of last digit of Fibonnacci Numbers
  2      3
  3      8
  4      6
  5     20
  6     24  (last two digits too)
  7     16
  8     12
  9     24
 10     60  (unusually big)
 11     10
 12     24  (last two digits too)
 13     28
 14     48

Karl Palmen

12 and 60

I think it is quite established that the use of 12 and 60 in many cultures on several continents comes from the fact that 12 and 60 have 'relatively' many divisors, {1,2,3,4,6,12} for 12, and {5,10,15,20,30} in addition for 60. This is very useful when it comes to fractional quantities etc, especially before the introduction of p-adic numeral systems with fractional part.

I think the number of months is rather a consequence than a reason for the choice of 12: observe that a (natural, i.e. lunar) month is rather 4 weeks than 30 days, and 52 / 4 = 13, and not 12, (a (solar) year having 52 weeks plus about 1.25 days.) MFH 12:52, 8 Apr 2005 (UTC)

A base other than 10 and 20 may be used in a measurement system for the division's sake, but it is rare to use such a base for a language's counting system. Duodecimal systems are used only in several North Nigerian languages and in the Chepang language of Nepal. The latter seems to be from the Nepalese way of counting using fingers (see the figure).

The Babylonian sexagesimal system clearly had an internal decimal system, and 60 was used instead of 100 for ease of division. It is more appropriate to call it a mixed-radix system of bases 10 and 6.

By the way, a year has about 12.368 months, not 13 — divide the solar year (365.2422 days) by the lunar month (29.53059 days). - TAKASUGI Shinji 10:42, 2005 Apr 12 (UTC)

Abacus (Stone-board)

One should note that Multiplication and Division take part in different parts of the brain (see Butterworth "The Mathematical Brain". and by separate processes. Multiplication is closer to animate numbers (such as other animals recognise), division more distant.

While a single method of counting is reckoned (a count of batches), there are many different division systems. The simple representation of numbers can be shown on a stone-board, where N stones in one column becomes 1 stone in the right. The most common form of number is the alternating system, where one replaces M stones for a single stone in the row above, and D stones in the top row for a single stone in the bottom row, one to the right. M is usually a number of count (eg 5, 10), where D is usually a division number (2, 4, 6, 8, 12). The chinese abacus is D=2 over M=5.

A precusros for an alternating system is a system of stand-alone fractions in D. This is known of the sumerians, and of the romans, but not elsewhere. However, the roman uncia is probably not the source of the germanic 100 of vi score.

One sees that many different systems become apparant eg 20 = 4*5 or 2*10, 40=4*10, 60=6*10, 120=12*10 are all historically known. The inherent 'decimal' system is seen, except that some have '5' at that point. To some extent, these arise by "making things bigger". The prehistory of 60 is 3*20. [O. Neugebauer - the exact sciences in ancient times]

Fractions are more complex. One has the Greek system (also mayan), where one makes ratio, eg 1944 parts where 2000 make the English foot, or the Roman weight-fractions (an uncia of weight, length and time, and number = 1/2, whence ounce and inch). One sees from their measurement systems, an ace (1) is variously a foot, a pound and a grain, and that these are invariably decimally counted (centar = 100 lb, millier = 1000 lb, mile = 100 paces), but divided approximately duodecimally (eg uncia = 1/12).

The sexagesimal numbers form the sumerian division system, these being to avoid division. For example, the most significant column is on the right, and subsequent places are divisions of the first. Zeros occur where they add meaning (eg leading, medially), but not finally, so 3 and 0 3 are different (3 and 1/20 respectively), but 3 0 is the same as 3 (ie as we write 3 vs 3.0). The count of numbers in the common system is the motely collection of decimal, sexagesimal etc (eg 192 = 100 60 30 2) See eg O Neugebaur.

That the system is a system for divisions is seen by the contents of the reckoners that come to us: tables of multiples of x by 1..20, 40, and tables of recriprocals (in ascending order logrithmically over sixty). The method of calculation was to determine the recriprocal and then look up these in the reckoners. (x = 44:26.40 exists, because this is 1/81.] We further note the existance of papers of the style of 'the problem of seven brothers' exist, giving 1/7 lieing between 0:8 34 16 and : 8 34 18, supporting the notion that it is indeed a system for fractions.

Sixty spread, along with the astronomy it used, both eastwards to India etc and westwards to Europe etc.

We see this fantasy with the duodecimal system. Historically, 12 is a division number, and that dozens and grosses were "super-divisions", ie measures, that on division, will reveal a unit peice. We have a grocer as one who deals in grosses, and sells off dozens and units.

The use of 10-like numbers (8, 12, 14, 16), is more to do with the recently devised method of using tables (the first tables, along with the first modern 0, appeare in late greece, and spread by the muslims to india and europe.

Wendy.krieger (talk) 06:35, 2 June 2008 (UTC)

synonym for decimal

I think we should start using the unambiguous word aal instead of decimal, because every base is decimal in its own base. Actually every time we say "base 10" we should say "base A" instead.

In this way we would always represent the base with the first digit that is not used for that base eg.:

base 2 ( = 10 in base 2) -> digits 0,1

base 3 ( = 10 in base 3) -> digits 0,1,2

base 9 ( = 10 in base 9) -> digits 0,1,2,3,4,5,6,7,8

base A ( = 10 in base A) -> digits 0,1,2,3,4,5,6,7,8,9

base B ( = 10 in base B) -> digits 0,1,2,3,4,5,6,7,8,9,A

I've been thinking about this for years, so I hope you all will agree with me on this.

--Ortonormale 00:47, 2005 May 11 (UTC)

It may be good to remember that the root "deci" means ten and "a" is a letter that is not associated with the base ten number system (why add another thing to confuse people?). Besides, saying "aal" would be more ambiguous than saying "Decimal". It sounds the same as "all" -Michael

very funny. but

• what comes after the zal base?
• I disagree, not every but only the aal system is decimal. "decimal" does not mean "digit 1 followed by digit 0"! you contradict yourself! — MFH: Talk 21:12, 11 May 2005 (UTC)

Yes, you would be right. I mean that since the discovery of base conversion, numerals have acquired new meanings while losing the direct link to their etymology. We could say that a new abstract level has been introduced between the original etymological meaning and the new virtual meaning. For example: 11 in base 2 represents the same quantity represented by 3 in base 8. Luckily or unluckily (according to your point of view) we have not a different set of names for each numeral in each base, therefore we have two possibilities:

• we can spell each numeral in every base but the base A (really sad)
  
• we can extend the same language structure we already use for base A to all bases up to Z.

In this latter case, we could simply say "eleven" to read the numeral 11 in whichever base. The same concept would apply to "ten", "decimal" and "digit".
Obviously, we would have ambiguities when not specifying the actual base, but this already happens when writing.
Nothing really fun so far. The funny part comes when we want to read numbers like

• APPLE that is: "aytypee thousand pee hundred eltyee"
• CRAZY that is: "ceetyar thousand ay hundred zeewy"
• SPELLING that is: "espee million ee hundred eltyel thousand i hundred entyjee"
• DECIMAL that is: "dee million ee hundred ceetyi thousand em hundred aytyel"

Again: obviously (as you have noticed) we would have an obstacle to complexity increase trying to use bases that are greater than Z, but this already happens when writing. It is a common problem for non positional numbering systems, but a simple solution consists in grouping. So for example we could use the base 2xG (or simply 2G) in which each digit is represented by a group of 2 digits in base G, like

• 20 08,
  • simply spelled "two zero blank zero eight" or
  • spelled-read "twenty zerotyeight" or
  • read "twentytyzerotyeight"
• A5 47 FF 00,
  • simply spelled "ay five blank four seven blank ef ef blank zero zero"
  • spelled-read "aytyfive fourtyseven eftyef zerotyzero"
  • read "aytyfive thousandty fourtyseven hundredty eftyeftyzeroty"

--Ortonormale 00:22, 2005 May 19 (UTC)

I find it totally meaningless. Don't confuse numbers and notations. Ten is ten, the number of circles in oooooooooo whichever base you use. Likewise, decimal always means base ten. What 10 means depends on the base, but decimal is default. No one would call (10)_{2 ten. It's two. - TAKASUGI Shinji 06:25, 2005 May 19 (UTC)}

Old English, Gothic, etc had words for reckoning by counts of ten (short count, or teentywise), vs the reckoning by base 120 (long count, twelftywise). Where several bases are in use concurrently, one might use neutral names, or names that have a common scheme, to describe the various notations. Where some are technical in nature (eg hexadecimal), the name would be described in the main base.

I use 'radix or base notation', for the expression of fractions &c by an implied added fraction, such as 'decimal fractions'. On the other hand, if i want to deliberately infer the denominator is 10 (rather than 60 or 120), i might call it a decimal. The point separating the fraction from the whole, is the 'radix', or root point

10 is ultimately derived from te.hund = two hands. If ye envisage a population that is heptadactic, you might correctly infer that two hands makes 14. The notion that 100 = 10 * 10 is not always the case. As long as there is a sequence of progression, the columns can tick over after different values. Historically, the column left of the units might be different to the other columns (eg mayan long count of days, has 20,20,...,20,18,20.

It is known in English metrology (see R E Zupko: A Dictionary of English Units to the Eighteenth Century: entry hundred), that writing something like C or 100 or hundred, might need to be qualified, eg 300, where C = v^xx xii , that is, 336.

Wendy.krieger (talk) 07:03, 5 June 2008 (UTC)

Finger and base 10

The article claims that we use decimal numbering because humans have 10 fingers. I find this claim highly suspect: 10 fingers is sufficent to count in base 11 (just as one finger is sufficent to count in base 2). Does someone have a good citation for this? --Gmaxwell 20:33, 22 May 2005 (UTC)

I don't have the citation you request, but I think it's a very sensible claim even without documentation. A few things I find relevant:

• Base 10 in number words is older than base 10 in a positional number system.
• Try teaching a child (age 4-7) to count-in-11's using the fingers of two hands; then (when you have despaired) try teaching counting-in-10's instead. Or try teaching counting-in 6'5 and counting-in-5's using one hand only.

Right. 10 fingers. 0 (no fingers) 1,2,3,4,5,6,7,8,9,10 ... Which is all of the single digit symbols in base 11. Really. Base-11 is more obvious for hand counting than base-10, as long as you have a concept of zero. --Gmaxwell 22:35, 23 May 2005 (UTC)

• The chinese abacus has 5 beads on each wire to represent values 0-4 (the 5th being used only temporarily in calculations). The japanese abacus is similar but has done away with the extra beads, at the expense of making its use slightly harder to learn.

Right, I know how to use a chinese abacus. I'm not following how it helps this argument. ... Thanks for replying though... I honestly didn't expect a reply anytime soon! --Gmaxwell 22:35, 23 May 2005 (UTC)

Zero is an artificial mathematical symbol, unnatural for human perception. "Decimal" does not *necesarily* imply that 10 is spelled using two symbols. When you start counting fingers, 10 ranks the same as each of the 1 to 9 numerals. So, why ten and not eleven? Try to quickly show the number 30 by flashing your fingers. It's as natural as... 123. Now, try with 33. Still think that base eleven suits your fingers? Luciand 15:50, 29 December 2005 (UTC)

This article may need work

I am not really happy with the current TOC:

Contents

    * 1 Decimal notation
          o 1.1 Alternative notations
          o 1.2 Decimal fractions
          o 1.3 Other rational numbers
          o 1.4 Real numbers
    * 2 History
          o 2.1 Decimal writers
    * 3 See also
    * 4 External links

The article is about decimal notation, so it does not make sense (to me) to have a section titled "decimal notation". And even if it is there, I don't see why one should have a subsection called "Alternative notation". That should be its own section, preferably at the bottom, as it is a related topic to decimal notation, but not the focus of the article. Comments? Oleg Alexandrov (talk) 00:54, 22 February 2006 (UTC)

"perhaps" because of ten fingers?

Is there any other theory at all for explaining the decimal system?

From a mathematical point of view, I see no argument that could be made for ten - two (or powers of two) is special, of course, since it's the smallest possible base (powers of two are just a neat way of cramming several binary digits into one handy symbol), three would give you balanced ternary, and I believe you can formalise the fact that 12 has a large number of factors.

Of course, it's possible that there might be a psychological aspect that makes 10 a natural choice, or that it was just an accident of history, but in the absence of support for either of those theories, maybe we should state this a bit more strongly?

RandomP 18:51, 13 May 2006 (UTC)

The words for five and hand are related in many languages, especially in New Guinean languages. And ten or twenty is called as a person in some languages. That strongly suggests our ancestors counted their fingers. How easy division will be is pointless - counting is much older than division. Languages of base-6 (Ndom), base-8 (Yuki), base-15 (Huli), and base-24 (Kakoli) have been reported.

Source:

• The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania
• What We Can Learn From the Counting Systems Research of Dr. Glendon Lean

- TAKASUGI Shinji 14:33, 15 May 2006 (UTC)

Thanks. Certainly interesting to know, but note that my question was whether there is any other theory for the use of ten, other than that that happens to be the number of non-thumb fingers.

RandomP 15:21, 15 May 2006 (UTC)

Gee, counting "non-thumb fingers" would lead us to use base 8 among most members of my species. Anyway, the mention of bases 6, 15, and 24 suggest the human predisposition to finger use is not absolutely overwhelming. -R. S. Shaw 18:55, 15 May 2006 (UTC)

Oops, sorry. I meant to say "fingers including thumbs", but got confused. Nothing to do with my extra pinkies, I assure you. RandomP 23:03, 15 May 2006 (UTC)

New Guinea, the most linguistically diverse area in the world, have various bases such as 4, 5, 6, 10, 15, 20, and 24. Body-part tally systems are also common. Eurasia is almost unified under decimal, with scattered vigesimal systems in its outer rim - Celtic languages, Basque, Caucasian languages, Dravidian languages, Burushaski, Ainu, etc. That suggests decimal spread and overwhelmed other bases in Eurasia. It seems to me China was the origin of decimal, because it has had a strict decimal system from the beginning while many other languages have special words for teens and decades. - TAKASUGI Shinji 00:18, 16 May 2006 (UTC)

I'm almost positive that the decimal system is the most widely used numeral system in the world for any reason but our fingers. Can anyone get any evidence to back this up? —Preceding unsigned comment added by 24.45.212.65 (talk) 07:18, 10 September 2008 (UTC)

A system based on fours and eights would correspond not to the fingers, but the spaces between the figures. The Indogermanic word for /nine/ and /new/ both come from a common stem, being the new number (of a group of four). One notes also that there are languages that change the style of counting at a multiple of four (English, two-left, three-ten), in french, between 16 and 17, and in fininsh between eight and nine (one before ten).

The sumerian counting system is evidently based on three scores (in very ancient times), but went through a phase where the sixths had fractional names (ie symbols for 1/6, ..., 5/6), before becoming a system of repeated divisions. One notes with Oppenheimer that the most famous system is a division system, to avoid having to do division. The numbers were not used in the usual multiplication sense (that 1.0 meant 60), but there is a distinction between 1, (eg 1 degree) vs 0.1 (1 minute), and 0.0.1 (1 second). Zero is known in this sense (the actual symbol also means sentence-period or full-stop).

For larger numbers, they used the usual motley collection of mixed decimal and sexagesimal (and some twelfty), so a number we write as 192, would be consistently be shown as 3A2 (ie 3.12) in the tables, in the attached matter might be shown as CIxxxii, that is, 100+60+3*10+2*1, ie 'one hundred and sixty-thirty-two', cf french, un cent, quatre-vingt-douze (100, 4-score and 12). --Wendy.krieger (talk) 08:10, 11 September 2008 (UTC)

Failed V0.7 nomination

I failed the article for two reasons:

1. The "probably due to ten fingers" and "probably due to 20 fingers and toes" make sense, but they need to be cited... unfortunately we're not allowed to make inferences.
2. More importantly, the Grouping of digits section needs text. A link to another article doesn't cut it.

After that is done, feel free to renominate it at any time. Titoxd^{(?!? - cool stuff)} 18:40, 27 April 2007 (UTC)

Pronunciation of decimal numbers

Is it possible to add how to pronounce these numbers properly? I mean, is 3,34 three point thirty four like in French or three point three four like in German? I have heard both opinions.