Talk:Numeral system

Older discussion
Archive 1

Nenets source
No source cited on the Nenets base 9, and can't find any other info. -LinguisticsStud The burden is on the uploader. No cite, no write. Here's the text I deleted:


 * "The Nenets language once used a base-9 system (nonary), but has since shifted to decimal under the influence of Russian. The word yúq originally meant 9, but took the value 10 on account of Russian influence; so in current Nenets the word for 9 is xasu-yúq, lit. 'Nenets yúq ', whereas 10 is simply yúq, but in Eastern dialects also lúca-yúq, lit. 'Russian yúq ' ."


 * Whoever uploaded it is welcome to reupload it with a cite. See Wiki guidelines on the policy. Cbdorsett 09:57, 29 January 2007 (UTC)

Number system- Not Numeral
Knuth does a nice history of "Number Systems" in the Art of Computer Programming.

Text after text describe radix based and/or positional "number systems".

Here are a few I've glanced at recently.

- “Number Systems and the Foundations of Analysis”; Elliott Mendelson; Academic Press, Inc; 1973

- "Encyclopedia of Computer Science"; Ralston, Reilly, Hemmendinger; Nature Publishing Group; 2000

- "The Art of Computer Programming - Semi Numerical Algorithms"; Knuth; Addison Wesley; 1981 (Oh.  Look.  The Wikipedia article on "numeral systems" cited this one, including the term "number systems" in the citation.)

If Wikipedia has a basis in credible literature for renaming "Number Systems" to "Numeral Systems", it should be prominently displayed in the article.

Plain and simple, a number system provides meaning for a numeral. Without the number system, the numeral is just a string of characters. The number system maps the numeral to the number which it represents. The actual type of the number system is (numeral -> number). Calling it a "numeral system" is no more descriptive then calling it a number system. That doesn't matter any more; Not even a little. The convention is established and Wikipedia doesn't get to change it. It's a "number system".

Someone needs to rewrite this article, eliminating the term "numeral system".

And by the way, in the page on "number systems", reals, integers, complex, imaginaries etc are sets not systems.


 * I don't think Wikipedia is changing a standard convention; it's just adhering to one. So you cite some examples that are exceptions.  The words "number" and "numeral" have standard meanings; Wikipedia is following them.
 * You are wrong to say the reals, integers, etc. are simply sets; they are sets with structures; some (reals, complexes) are fields; some (integers, quaternions) are rings but not fields; in the case of the reals, the ordering, rather than just the algebraic operations, is often taken to be part of the structure. The reals are just one number system, but there are many different numeral systems by which one can identify particular reals. Michael Hardy 23:24, 6 November 2005 (UTC)


 * ... and I am suspicious of the claim about Mendelson's book. Can the anonymous poster describe its topic with enough specificity to make it clear that it's about numeral systems and not things like the reals, rationals, etc.? Michael Hardy 23:27, 6 November 2005 (UTC)


 * ... and now I see: right at the top of the article, very conspicuously, it says: "Occasionally the term "number system" is used for this concept, but...". I think that is full and sufficient answer to this anonymous poster's concern. Michael Hardy 23:48, 6 November 2005 (UTC)

Why not "numeric systems"?148.252.128.44 (talk) 10:17, 5 September 2017 (UTC)

Split the page
This page describes two different things and this causes problems on pages that link to it, like Arabic numerals. This article should be split into: What say you? Zocky 22:46, 7 December 2005 (UTC)
 * An article about systems employed to represent numbers, probably Numeral system, i.e. here.
 * An article about sets of glyphs used to this, probably Numerals, which redirects here.


 * Not necessary yet. I don't see sufficient content on glyphs that can be extracted from a description of how they are used. Davilla 22:05, 8 February 2006 (UTC)

Deleted material
This was parenthetical so I'm assuming the contributor wasn't sure of him/herself. I've tried to look up the history of neuf with no luck. Is there a French equivalent to the OED? Unless both senses can be traced back centuries, I think it's better to leave it out.
 * In French, the word neuf still means both 9 and 'new'.

BS. The twelve months are a more recent invention than the number system.
 * [Base-12 systems were popular] because the year has twelve months.

First of all, it would be much more interesting to note the number systems used naturally, before discussing the computing age. Second, this "summary" adds some other bases that aren't discussed in the article. Davilla 22:14, 8 February 2006 (UTC)
 * The bases that were used in the past or used today are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16, 20, 60.


 * "Neuf" (new) comes from Latin novus and "Neuf" (recent) from Latin novem. They come from similar-sounding (but different) Indo-European roots. AnonMoos 22:38, 8 February 2006 (UTC)

agree —Preceding unsigned comment added by 188.86.17.140 (talk) 15:57, 29 March 2010 (UTC)

Chinese, Japanese, Korean should be grouped under same system in the numeral template
Chinese, Japanese and Korean numerals are all based on Chinese numerals AND on the Chinese numeral system. The base ordering is also entirely Chinese (ordered by myriads 10^4 instead of thousands 10^3), the official written form of all three languages are also in Chinese characters (一, 二, 三, 四, 五, 六, 七, 八, 九, 十). It is ridiculous to separate the three if Hindu-Arabic are grouped under one; the differences between Chinese, Japanese and Korean numerals are merely linguistic (much like the difference between French and English counting of Arabic numerals). Japanese and Korean numerals are not genuine numeral differences and should not be classified separately in the Numeral Template on the right of each numeral-related page; instead they should be grouped together under a "Sinitic" category. Naus 03:23, 26 March 2006 (UTC)

Similarly, I object slightly to appending “Japanese” with Chinese in the table. I had changed it from Kanji because that is a Japanese word that literally means “Chinese characters.” The Japanese and Koreans are well aware that the characters are entirely of Chinese origin. MJ (t • c) 15:39, 15 November 2007 (UTC)


 * I agree absolutely. J IM ptalk·cont 00:30, 23 June 2008 (UTC)

History - where are the refs?
I think there's some bold statements in the history section. I think the stuff about modular arithmetic is pretty weak.


 * "In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing."

I think this is supposed to be a reference to the Chinese remainder theorem or some interpretation therein. I see the following issues:

1. I'm pretty sure the specific examples about rice and armies was a theoretical idea used by another historian (need to find ref) to explain why the CRT might have been developed by Sun Zi. Sun Zi just quotes the problem with numbers, and I don't think makes reference to armies or provisions.

2. I think the statement that addition is harder in modular arithmetic is at worst wrong and at best confusing.


 * "The binary system (base 2), propagated in the 17th century by Gottfried Leibniz who had heard about it from China, came in common use in the 20th century because of computer applications."

I'm aware of Leibniz and binary numbers, but I had not heard the connection to China. I would assume the connection is somehow to the I Ching. Does anyone have a reference for this?

Thanks,--M a s 19:47, 5 May 2006 (UTC)

Degrees into minutes???
"and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds)"

Maybe I'm wrong here but that makes absolutely no sense. Since when has minutes been a measure of angle?


 * I don't know since when, but several hundred years, I'd guess. Many electronic calculators (especially slightly older models) can do conversion between angles in degrees with decimals, and angles in degrees, minutes, and seconds with decimals. E.g., 17.375° = 17°22'30", and 73.220644° = 73°13'14.32".--Niels Ø 14:02, 6 May 2006 (UTC)


 * The minutes and seconds here are not the same minutes and seconds used in time.
 * See degree (angle) for details. --68.0.124.33 (talk) 02:50, 9 March 2009 (UTC)


 * Minutes is just a word used to describe a division of degrees. It doesnt actually mean time. Its a homonym, just like pound describes both a type of british currency and a unit of weight.T4k (talk) 00:53, 15 April 2009 (UTC)

Sumerian notation?
Why no separate page for Sumerian numeral notation, given its foundational importance? I'd do it myself, but my books covering it are 2000 miles away and I have a hard time trusting webby sources... Anyone? JDG 05:28, 11 May 2006 (UTC)

I moved some of the History section to a new page titled "History of writing numbers" where I added 4 paragraphs on the historical origin of numbers in prehistoric Sumer. If you know where I can find gif files for archaic numerals and cuneiform numerials, please tell me and I will add them to History of writing numbers. Bob Best, 1:20am, 19 June 2006.

The term "sub-base" is slightly ambiguous
The explanation of base-five, within the 'Bases used' topic, uses the term "sub-base", which I am not familiar with and isn't defined in Wiktionary. —Preceding unsigned comment added by 124.168.230.203 (talk) 13:03, 16 June 2008 (UTC)

Base 8 in Proto-Indo-Europeans
The evidence given for the use of base 8 amongst proto-indo-europeans doesn't make sense. A base 8 system only requires 8 digits, 0-7, therefore both 8 and 9 would be new digits for proto-indo-europeans. Application of Okham's razor would result in the conclusion they used base 9 based on this evidence and reasoning. If someone has access to the reference given for this snippet, could they look it up and check this? Make sure it is base 8, and check to see if any other evidence is given for base 8 (and why 9 is "new" but 8 isn't). Ooooooooo 00:29, 1 December 2006 (UTC)

Nenets --

Really need a source on the Nenets base 9 thing. Can't find it anywhere else.


 * I question the assertion "Zero to seven are the only possible digits." I believe most early numbering systems started with 1, thus the original digits in a base 8 system would be 1-8.  This would also mesh with the "9 = new" postulate.  None of this is in my area of expertise, but this statement immediately struck me wrong.  I've changed it for now, but would be interested in hearing other viewpoints or if there are any relevant citations.

--EMU CPA (talk) 18:54, 29 May 2008 (UTC)


 * It says somewhere else in the article that Arabic is the only system that uses 0T4k (talk) 00:57, 15 April 2009 (UTC)

Back to numbers
I feel the article should be moved to numeral, since the topic is clearly the numerals themselves. That there is more than one numeral system is rather self-evident and could be explained more intuitively than making a disambiguation where one isn't actually required.

Peter Isotalo 20:45, 6 February 2007 (UTC) jesus loves you man

Important or not?
At the end of the article (in 'Properties of numerical systems with integer bases') a theorem is stated and proved. However the reader is not informed whether or not this theorem has any relevance or importance whatsoever. If some authors here think that this theorem can be interesting to some readers, then some kind of motivation for this should be stated in the article. Bob.v.R 11:11, 22 June 2007 (UTC)
 * No respons until now. Bob.v.R (talk) 18:33, 11 January 2008 (UTC)

Examples on how to do basic arithmetics?
I have contributed to the similar page in Swedish and I have some examples on how to do arithmetics in other bases (in this case, base 14). Maybe this is something worh including on this page aswell?

Paxinum 19:46, 13 July 2007 (UTC)

The five and twenty system
is (or at least was) used in Welsh counting. Numbers 1 to 10 are used, and then counted as 1-and-10, 2-and-10, 3-and-10, 4-and-10, 15, 1-and-15... The next major numbers are 20, 40, 60, 80 and 100 (no specific words for 30, 50, 70 or 90).

unique representation
The article contains the sentence: ...the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits... What about 0.999...=1. Don't we have 2 different representations for the same number 1 here.--Shahab (talk) 10:22, 22 November 2007 (UTC)

But 0.9999.... is not a finite representation, 1.0 is. Every whole number can be represented in (at least, at most?) 2 ways, one with infinitely many 9999.. and one with 0. Paxinum (talk) 11:50, 27 November 2007 (UTC)

Twelve, the common British unit of measurement
There are twelve inches to a foot. Prior to 1971, in British currency, there were 12 pennies to a shilling.

"Twelve is a common British unit of measurement."

Or so the article claimed. It's a number, not a unit of measurement nor is it really all that common.

"English words for numbers are also 'base-12' in that there is a unique word for the numbers one through twelve, with 'thirteen' being the first word that was formed by combining numbers (three and ten)."

The above was from the same paragraph. This is just not true. The word eleven and twelve are related to one and two no less than thirteen is to three. The difference is just how the relationship was formed.

eleven O.E. endleofan, lit. "one left" (over ten), from P.Gmc. *ainlif- (cf. Goth. ain-lif), a compound of *ain "one" + PIE *leikw- "leave, remain" (cf. Gk. leipein "to leave behind;" see relinquish). Viking survivors who escaped an Anglo-Saxon victory were daroþa laf "the leavings of spears," while hamora laf "the leavings of hammers" was an O.E. kenning for "swords" (both from "The Battle of Brunanburgh"). Eng. twelve reflects the same formation; outside Gmc. the only instance of this formation is in Lith., which uses it all the way to 19 (vienio-lika "eleven," dvy-lika "twelve," try-lika "thirteen," keturio-lika "fourteen," etc.) Phrase eleventh hour is from Matthew xx:1-16.

twelve O.E. twelf, lit. "two left" (over ten), from P.Gmc. *twa-lif-, a compound of the root of two + *lif-, root of the verb leave (see 'eleven'). Cf. O.S. twelif, O.N. tolf, O.Fris. twelef, M.Du. twalef, Du. twaalf, O.H.G. zwelif, Ger. zwölf, Goth. twalif. Outside Gmc., an analogous formation is Lith. drylika, with second element -lika "left over." FIREFLY: Give me a number from 1 to 10. CHICOLINI: eleven! FIREFLY: Right!

I've removed the sentences. J IM ptalk·cont 00:26, 23 June 2008 (UTC)

Twenty System
This system still exsists in Georgian and it is not mentioned in the article. For example 40 (ormotsi) means "two times twenty"; 50 (ormotsdaati) means "two times twenty and ten"; 60 (samotsi) is "three times 20"... 130 (as otsadaati) means "one hundred twenty and ten" --გიგა (talk) 01:53, 22 July 2008 (UTC)

Base 10?
Every base is technically a "base 10" since the expression jumps up to "10" when a number equals its base. This would be more accurate if it were listed as "base decimal" —Preceding unsigned comment added by 66.92.38.183 (talk) 19:51, 6 August 2008 (UTC)
 * "Base ten" should be unambiguous enough, or maybe you would prefer "base 1010". Mild Bill Hiccup (talk) 21:31, 6 August 2008 (UTC)

Unary vs Positional
Isnt Unary just a variation on Positinal with 1 as the base? T4k (talk) 01:01, 15 April 2009 (UTC)


 * Nope, it doesn't have the same abstraction, see Talk:Unary. Basically if we tried to work with a base-1 positional notation all numbers would be the same 0 = 00 = 000. Cheers, — sligocki (talk) 22:40, 28 October 2009 (UTC)


 * It is, as a bijective (1-adic) system, see Bijective_numeration. — Preceding unsigned comment added by 77.169.168.165 (talk) 18:13, 2 December 2012 (UTC)

Moksha numerals
This article was supposed to be merged into this one after an AfD in October 2008, but it didn't happen. Does anyone want to do the merge? Fences &amp;  Windows  00:58, 30 July 2009 (UTC)

Move content to positional numeral system
Unless we want to merge that article here, we ought to move most of the second and third sections to positional numeral system. Cheers, — sligocki (talk) 22:43, 28 October 2009 (UTC)

Alright, I did the move. Cheers, — sligocki (talk) 00:47, 11 November 2009 (UTC)

Hereditary base-n notation
There is a variant of the base-n notation (but not positional notation if I understand well) called hereditary base-n notation, where the exponents also have to be expressed in base-n until no coefficient is larger than n, e.g., 33 = 2^5 + 1 = 2^(2^2 + 1) + 1 in hereditary base-2. Since this is not really a positional numbering system, I think it can't be moved into a more specialized page, but wonder where it would fit into this article. Maybe in the first big section? I hope someone can insert this conveniently in this page. &mdash; MFH:Talk 20:36, 19 February 2017 (UTC)

Writing only?
The article starts:
 * A numeral system (or system of numeration) is a writing system for expressing numbers...

Now, take tally sticks. Do they show a numeral system? (Yes I think so.) Is that writing? (No I'm not sure you can call it that.) The article on tally sticks says:
 * ...used to record and document numbers

(Actually, once later in the article this is referred to as "writing".) Maybe our article could start:
 * A numeral system (or system of numeration) is a system used for recording numbers...

though that particular phrasing may be too inclusive (e.g., saving an spreadsheet on a computer may be a way of recording numbers). I'm not at all sure about this - do you have a better solution, or is there not a problem at all?--Nø (talk) 12:08, 13 October 2018 (UTC)
 * Why not replacing simply "writing system" by "method"? D.Lazard (talk) 13:14, 13 October 2018 (UTC)
 * Perhaps, but I believe that, too, is too broad. Spoken numbers, like "twentyfive thusand seven hundred and fortysix", are AFAIK generally not considered examples of numeral systems. But the distinction, the delineation of what is and what isn't, seems rather arbitrary to me:
 * Writing (but not written numberwords) on paper, parchment, bark, blackboard, whiteboard, or the like, using a colouring agent like ink, chalk, or laser printer toner: Yes.
 * Etching, scratching or cutting marks into wood, stone, paraffin or the like: Yes.
 * Speech, written speech, signing: No.
 * Bit patterns on magnetic surfaces, optical disks, flash memory etc.: Yes and No.
 * When I write "Yes and No" for bit patterns, it's because the principles for mapping numbers into bit patterns are considered numeral systems (binary, BCD, or whatever), but the way bits are represented physically (voltages, or whatever) is not. This seems to me to be different from the e.g Hindu-Arabic numerals, where the number system is a package comprised of the abstract decimal system and a rather specfific set of digit symbols and a writing direction.
 * I think (and I have written before) that there is a tendency to confuse abstract number systems (like Hexagesimal or Grey code) and cultural, physical expressions like the Babylonian numerals. We are not here to invent new terminology, but I think this confusion is built into the term "Numeral system".
 * But maybe I'm making things more complicated than they are; maybe I'm just wrong about this delineation?
 * --Nø (talk) 10:35, 14 October 2018 (UTC)
 * One must distinguish between a numeral system, which is a systematic method for denoting/representing/manipulating numbers, and its realization/implementation on physical devices. These realizations/implementation include the glyphs that are used for the digits (some Arabic speaking countries use the Hindu-Arabic numeral system with other glyphs than 0, 1, ...).
 * The case of spoken numbers and written number words is slightly different: as a "positional notation" makes no sense for speech, it has been replaced by words for positions with a nonzero digit. Also the number words have been introduced a long time before the Hindu-Arabic numeral system, which may explain the existence of specific words for some small numbers. Nevertheless, when writing "twentyfive thusand seven hundred and fortysix", you have used implicitly the Hindu-Arabic numeral system for memorizing the syntax that should be used, and I have used it to understand which is the number that you have written.
 * Therefore, I strongly suggest to replace "writing system" by "method" or "systematic method". As this would need some modifications in the sentences that follow, I prefer wait to a consensus instead being WP:BRD and doing the change immediately. D.Lazard (talk) 13:25, 14 October 2018 (UTC)

Why does this article omit base 16?
Why does this article omit the hexadecimal (base/radix 16) system?—Finell 01:59, 20 June 2019 (UTC)
 * It's there, along with binary and octal, in the paragraph on computer number systems. Just plain Bill (talk) 04:13, 20 June 2019 (UTC)

"Numeric symbology" listed at Redirects for discussion
A discussion is taking place to address the redirect Numeric symbology. The discussion will occur at Redirects for discussion/Log/2021 November 13 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Hildeoc (talk) 22:23, 13 November 2021 (UTC)

Quipu in the see also tab.
Why is the Quipu explicitly mentioned rather than having only general topics? The existence of quipu being in the list would mean that every numeral system should be in the see also section, unless it was specifically mentioned due to it’s uniqueness. It realistically should be either deleted from the see also tab or every number system should be added. Legendarycool (talk) 22:31, 30 May 2024 (UTC)


 * Agree; done (along with a few other trimmings) Nø (talk) 06:54, 31 May 2024 (UTC)