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.

---

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.

---

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)

---

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

---

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

---

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.

---

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

---

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

---

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?

The roman cadasteral system is based on units of 120 or 240 feet [120 = actus], ultimately getting to 4800 ft. The itenitary mile is 1000 paces of 5 feet. In England, the mile was divided into 8 furlongs, each of 40 rods, the size of the rod varying according to the productivity of the country. The value adopted in a statue, wishing to state 5 miles, stated that the rod was 5 1/2 yards. Values vary from 10 to 24 feet for the rod.

There is evidence of a foot of 1.1 imperial feet, or 13.2 BI inches, which would make the corresponding mile some 4800 feet, in the roman practice. This mile and a shorter foot gives the required 5280 feet, or 1760 yds. --Wendy.krieger (talk) 08:30, 24 October 2009 (UTC)

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

Railway gauge is set by the flanges of the wheels, which gives 4' 8 1/2". The Roman road were set with tracks at five foot centres. --Wendy.krieger (talk) 08:30, 24 October 2009 (UTC)

---

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.

Sorry I'm not an expert in these matters but a number like three point thirty four (3.34) sounds like a version number in software than a proper number. I would hate to say a number like this 1.1234567. Darrenaustralia (talk) 06:18, 21 October 2008 (UTC)

It's normally three point three four. However, 48.49 would most likely be forty-eight point forty-nine, because of the pattern established by the 48. If you read off the individual digits to the left of the radix, you will to the right as well. kwami (talk) 02:31, 26 February 2009 (UTC)

confusion of base 10 with positional

The article confuses base 10 with positional systems. Someone commented on this above, but was never answered. If there is no discussion here, I'll go ahead and rewrite the article.

Chinese numerals are decimal, even though they're not positional. Likewise, Roman numerals are also decimal, though with a minor auxiliary base in 5. Hebrew gematria are decimal. There are very few written systems which are not decimal—Mayan and Babylonian are the only ones which comes to mind—though of course in spoken languages there are all kinds of bases and base combinations.

Besides decimal numeration, there are decimal fractions. All this requires is extending base 10 to fractional notation, though in practice in the modern world it nearly always implies a positional system. (Roman combined decimal numeration with duodecimal fractions, neither of which were positional.) India did not invent decimal numeration, which is the most common in the world, and AFAIK did not invent decimal fractions either; it invented the positional system and the zero that went along with it. (Mayans had a zero but not a fully positional system, due perhaps to religious considerations.) kwami (talk) 02:44, 26 February 2009 (UTC)

Most number systems from the ancient world make sense when these are read off a two-row (or alternating) abacus. The number systems are various attempts to record the count of stones, with some abbreviations. The main systems are

• Repetition of symbols (eg 70 = LXX [Roman, Greek] or sumerian - and | arrows (to form the digits). The egyptians used only the lower row, and so repetitions to 9 are not unknown. Empty columns are typically unrepresented since denominations do this, but zero is used for an empty count.
• Demotic style (ie symbol for 70, 76 as 70 6.) This can be overlayed with the alphabet (ancient or modern), to give alphabetic numbers of hebrew, greek and gothic systems.
• Digit + unit, in the style of 1h 6t 4 (for 164). Used by Mayans and by Chinese. In the mayan, the short form of the digit (sticks+stones) were attached to the unit glyph, eg '17xx' for '17score' Chinese is written as eg 1 h 6 t 4 as seperate characters.

Our number words go this way. Butterworth (The mathematical brain) notes that this is the most advanced form, since one is less likely to read two thousand and six as 20006 (ie 2000 6).

• True positional, first used in a division-system by the sumerians (of alternating 6, 10), zeros are recorded in leading and medial positions, eg 0 0 5 for 5". When used outside the sexigesimal context, units are attached, eg 0° 0' 5".

The mayan system with the column of 18 is purely a calendar system, the common count was in scores and scores of scores. Fractions were done in the greek fashion (ie ratio of numbers).

Zero, in its modern sense, was invented by the greeks around 600 AD. It followed the muslums to india, and returned by the same route. One notes that there are five different kinds of zero.

• Zero by itself. the egyptians had a symbol for an empty bag, even though no column or row zero,

• Leading zero: This is needed when the leading or most significant position carries meaning. Sumerian systems had this, because the first-referenced number is then divided to get other columns. 5 and 0 5 are different numbers, but 5 0 = 5 (where the columns stand for degree minute second)

• Trailing zero: These are useful in multiple-systems, where columns to the left depend on the right column. In fractions, these have no measure except to show precision, eg 5 0 0 is different to 5 0, but 5.0 0 is the same as 5., except to show more precise. Trailing zeros are not recorded in the sumerian system).

• Medial zeros: Zeros might be shown where the denomination is absent, even when units are used (Mayan system records, eg 1lb 0oz 2dr). Medial zeros are known in sumerian number systems. (See Neugebauer "The exact sciences in antiquity")

• Semimedial zero: None are recorded, but if as the alternating-digit systems go, let A=10, B=20, &c. A is 10, A7 is 17, A 0 is 600, and A 7 is 607. The semimedial zero is represented by the staggered column represented by dots in "A. .7"

Historical bases (being the repititions of columns in the abacus), include 10, 20, 40, 60, 80, 120. These arise from the two-row abacus, of 2/5, 4/5, 4/10, 6/10, 8/10, and 12/10. A 10/10 style in the vein of the sumerian system is also recorded, but this is not the precursor of the modern number system.

Modern decimal fractions derive from several inputs, the historical evidence is that of submultiples (where a count of 10 represents 1 in the next unit), since the earliest modern decimals have indicess written over them. However, the inspiration of this may well have been added fractions (eg 3 1/7 2/24 being 3+(1+(2/24)/7), as one sees eg 3 weeks, 1 day, 2 hours. In any case, the system is contrasted with the sumerian degree/minute/second notation.

By the way, i regularly use an alternating base (120) for calculations.

--Wendy.krieger (talk) 08:17, 26 February 2009 (UTC)

"It is the most widely used numeral base"

The lead says:

It is the most widely used numeral base

and someone added

[citation needed].'

and I thought: Don't be ridiculous... but, thinking a bit more about it, I realized that the most widely used base IN NUMERICAL CALCULATIONS is binary, in the sense that most of the additions, multipliplications, etc. that are performed on any given day are done by computers, not by humans. So while we don't really need a source, we may need a qualification; something like:

It is the most widely used numeral base in human communications

... but I don't really like that either. Any good ideas?--Noe (talk) 06:48, 21 October 2009 (UTC)

Mistake in the article

The article claims "decimal fractions were first used ... by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century". That is incorrect. He was not the 1st person to use it.

According to the Cambridge University in England, decimal fractions were 1st developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and then to Europe.

Source : Science and Civilization in China (Vol. 3) (Published by the Cambridge University Press)

Wikiwikidaddy (talk) 07:45, 30 October 2009 (UTC)

The decimal fractions as used in China did not make it to the west. Decimal numbers, with the use of zero derive from late greek, borrowed through the arabs to india and europe. The use of decimal fractions is modeled originally on the use of "second tenths", where a number had a superscript representing the order of division (eg writing 1234 over the digits 1 4 1 6 in pi= 3.1416.

The present decimal fractions are derived from the system of minutes and seconds, using 10, rather than 12 or 60 as the value of the minute.--Wendy.krieger (talk) 08:03, 30 October 2009 (UTC)

Decimal fractions are not the same thing as decimal numbers (and the use of zero). I have created another section below for that so as not to confuse the issue. As for decimal fractions (and also the section below), if you dispute the findings of the Cambridge University, you need to provide an argument backed up by clear evidence. Wikiwikidaddy (talk) 08:42, 30 October 2009 (UTC)

If no-one objects, I will update the main article with the corrections tomorrow. Wikiwikidaddy (talk) 03:50, 2 November 2009 (UTC)

Mistake in the article (2)

The article claims "The modern number system originated in India". That is incorrect.

According to the Cambridge University, the decimal system (together with the digit zero) originated in China. The most conservative estimate for the use of the decimal system dates it to no later than the 14th century BC (although it is known to have been in use long before that).

Source (1) : Science and Civilization in China (Published by the Cambridge University Press)

Source (2) : Genius of China (by Robert Temple) (This book has won numerous major literary awards including ones from the American Library Association and the New York Academy of Sciences, and was translated by UNESCO into 43 different languages ).

Wikiwikidaddy (talk) 08:42, 30 October 2009 (UTC)

If no-one objects, I will update the main article with the corrections tomorrow. Wikiwikidaddy (talk) 03:50, 2 November 2009 (UTC)

list of recorded decimal writers

We have a "list of recorded decimal writers" in this article:

• c. 3500–2500 BC Elamites of Iran possibly used early forms of decimal system.
• c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.)—see Ifrah, below
• c. 2600 BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures
• c. 1300 BC Chinese writers show familiarity with the concept: for example, 547 is written "Five hundred plus four decades plus seven of days" on an inscription of an oracle bone
• c. 400 BC Pingala—develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
• c. 250 BC Archimedes writes the Sand Reckoner, which takes decimal calculation up to 10^{80,000,000,000,000,000}
• c. 100–200 The Satkhandagama written in India—earliest use of decimal logarithms
• c. 476–550 Aryabhata—uses an alphabetic cipher system for numbers that used zero
• c. 598–670 Brahmagupta—explains the Hindu-Arabic numeral system (modern number system) which uses decimal integers, negative integers, and zero
• c. 780–850 Muḥammad ibn Mūsā al-Ḵwārizmī—first to expound on algorism outside India
• c. 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi—earliest known direct mathematical treatment of decimal fractions.
• c. 1300–1500 The Kerala School in South India—decimal floating point numbers
• 1548/49–1620 Simon Stevin—author of De Thiende ("The Tenth")
• 1561–1613 Bartholemaeus Pitiscus—(possibly) decimal point notation.
• 1550–1617 John Napier—use of decimal logarithms as a computational tool
• 1765 Johann Heinrich Lambert—discusses (with few if any proofs) patterns in decimal expansions of rational numbers and notes a connection with Fermat's little theorem in the case of prime denominators
• 1800 Karl Friedrich Gauss—uses number theory to systematically explain patterns in recurring decimal expansions of rational numbers (e.g., the relation between period length of the recurring part and the denominator, which fractions with the same denominator have recurring decimal parts which are shifts of each other, like 1/7 and 2/7) and also poses questions which remain open to this day (e.g., a special case of Artin's conjecture on primitive roots: is 10 a generator modulo p for infinitely many primes p?).
• 1925 Louis Charles Karpinski—The History of Arithmetic
• 1959 Werner Buchholz—Fingers or Fists? (The Choice of Decimal or Binary Representation)
• 1974 Hermann Schmid—Decimal Computation
• 2000 Georges Ifrah—The Universal History of Numbers: From Prehistory to the Invention of the Computer
• 2003 Mike Cowlishaw—Decimal Floating-Point: Algorism for Computers.

What is the point of this? It is not explained at all. At first, I assumed it was notable developments in the use of the Hindu-Arabic numeral system, but it also includes modern writers who discuss binary and related issues for computing. I think this section should be better motivated, focused and pared down. I've commented out most of the modern authors because I don't think that they belong here. Cheers, — sligocki (talk) 21:20, 30 October 2009 (UTC)

==================

Some of your points above are not backed up by evidence. Eg, the Indus Valley Civilization's use of measurement units that are fractions of our measurement units does not itself imply they used fractions. And more importantly, the use of fractions is not the same as the use of decimal fractions. Most advanced ancient civilisations used fractions. But there is no concrete evidence any used decimal fractions (as opposed to the common fractions) before the 14th century BC. Wikiwikidaddy (talk) 01:23, 31 October 2009 (UTC)

What are you talking about? I just copied this list from the page. I am saying that I think it is poorly constructed. Do you agree with me? Cheers, — sligocki (talk) 21:23, 1 November 2009 (UTC)

--- I see. My appologies. Thanks for the clarification. (And yes, I think you have a point) Wikiwikidaddy (talk) 03:36, 2 November 2009 (UTC)

Alright, I just got rid of it. If anyone wants to make a more appropriate list, it's copied here. Cheers, — sligocki (talk) 17:41, 3 November 2009 (UTC)

Arthur's revert

Simon Stevin's contribution to decimals is generally recognized to be the seminal one, see for example van der Waerden, or the St Andrews pair of articles which clearly relate to Stevin as a watershed. Is it reasonable to have all sorts of multicultural characters mentioned here, and leave out Stevin? Tkuvho (talk) 15:11, 21 February 2010 (UTC)

Your statement was incorrect. It appears he developed the basis for the modern notation, but he did not develop or "introduce" "the decimals we use today". — Arthur Rubin (talk) 15:19, 21 February 2010 (UTC)

Yes, his notation was different. I now see that my phrasing was misleading. At any rate he should be mentioned here. Tkuvho (talk) 15:26, 21 February 2010 (UTC)

See what I changed it to. I think the citation is still needed, but it's appropriate there. — Arthur Rubin (talk) 15:27, 21 February 2010 (UTC)

van der Waerden, B. L. (1985) A history of algebra. From al-Khwarizmi to Emmy Noether. Springer-Verlag, Berlin. Tkuvho (talk) 15:28, 21 February 2010 (UTC)

Arbitrary or Logically Imperative?

Is there actually a specific reason that in our time almost the entire world uses the decimal system, i.e. is there a logically imperative reason for its popularity or is this simply arbitrary? Is there a scientfically valid difference in using another base system?