Talk:Number system

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Have you any reference justifying that "number system" does mean "set of numbers"? &mdash; MFH: Talk 19:46, 8 Apr 2005 (UTC)

First 10 hits for "number system" in Google are to what here is called Numeral systems. The 11th,, agrees with this article on the meaning. MathWorld disagrees with this article. - I have modified the article to allow both meanings, but a more radical change may be appropriate.--Niels Ø 20:19, Apr 8, 2005 (UTC)


 * This article does not make a great deal of sense. If this were a mathematical term, it would have a rigorous mathematical definition, but the article doesn't mention one. It even goes on to say that "Whether the quaternions should be considered a number system is a matter of convention." which even seems to imply that there isn't a formal definition at all. At first I thought it might be talking about a ring or a field, both of which are sets and have something to do with arithmetic operations. &mdash; Furthermore, I think that most people think of things like binary when they hear "number system". I therefore think numeral system should be moved to here. &mdash; Timwi 21:24, 8 Apr 2005 (UTC)


 * Michael Hardy's recent changes, including the top matter
 * This article is not about numeral systems, which are systems of symbols for representing numbers.
 * seems to me again to obscure the point I tried to make clear: The most common meaning of "number system" is the one covered in the article Numeral system. Michael, I agree that something in italics before the article proper is the best solution, but I'm not keen on this wording. I'm not sure how to rephrase it, though. Other solutions could include a disambiguation page, or completely abandoning the rare meaning of "number system" covered here, redirecting to Numeral system. Can someone see a good solution?--Niels Ø 12:19, Apr 10, 2005 (UTC)


 * I would like to see numeral system moved here because this is by far the more common name for it. Then that article can have something on the top saying mathematicians also use the term "number system" to refer to sets on which arithmetic operations can be performed. For example, refer to ring and field. &mdash; Timwi 15:41, 10 Apr 2005 (UTC)


 * I absolutely agree!--Niels Ø 18:45, Apr 10, 2005 (UTC)

conversion from one number to another
—Preceding unsigned comment added by 202.5.143.2 (talk • contribs) 03:38, February 3, 2006 (UTC)
 * Er, you need to explain a little more. Do you want Numeral system? --AySz88 ^  -  ^  21:46, 4 February 2006 (UTC)

Natural numbers

 * 1) Zero should not be considered a natural number without some very serious explanations. Counting to three is saying "one, two, three", not "zero, one, two".
 * 2) The notation a' as an alternative to S(a) is not used and should be omitted.
 * 3) The definition "a*b = a + a + ... + a (b times)" in not the style of Peano. Use a&middot;1=a, a&middot;(b+1)=(a&middot;b)+a.
 * 4) Use &middot; rather than * for multiplication.
 * 5) The cancellation rule "x+a=y+a implies x=y" should be given.

The arithmetic characterization:
 * 1) if A and B are natural numbers, then so is A+B.
 * 2) if A and B are natural numbers, then so is AB.
 * 3) 1 is a natural number.

Generalizes to the similar characterization of polynomials:
 * 1) if A and B are polynomials, then so is A+B.
 * 2) if A and B are polynomials, then so is AB.
 * 3) 1 is a polynomial.
 * 4) X is a polynomial.

Then the short path of explanation is: natural number &rarr; polynomial &rarr; integer &rarr; algebraic number. Bo Jacoby 10:42, 1 March 2006 (UTC)


 * Ad 1: I teach math in International Baccalaureate and in the Danish high school system. In the Danish system (where I myself was educated), 0 is not a natural number. When I realised that IB required me to teach that 0 is a natural number (a member of N), I disliked it, and so I searched for ammunition on the internet in support of my own view. Alas, it turned out that serious as well as popular sources are divided roughly evenly on the matter. Wikipedia should not tell people how we think things ought to be, but how they actually are. In this case, we simply have to state that both views are common. - By the way, I think the Greeks believed counting started with two: One represents unity; proper numbers represent plurality.
 * Ad 4: I prefer &middot; for multiplication myself, and * is definitely not a good choice. However, &times; would be an acceptable choice too.
 * Apart from that, I think you could just go ahead and make the changes you want.--Niels Ø 14:11, 1 March 2006 (UTC)

Glad to see so much discussion here.
This article was a stub when I got to it. I've tried to expand it into something useful, and am glad for all the help I can get. To explain a few points.

Wikipedia already has major articles number and numeral. It also has a good article on abstract algebra and separate articles on group, ring, and field.

Someone had tried to put into the article on number a long section on the construction of the various systems of numbers. That got reverted, and did not really belong under number, and so I suggested moving it here.

Which got me started working on this article.

A couple of replies to the posts above.

Mathematicians distinguish between numbers and numerals. The google hits for "number system" are mostly "numeral systems", but we don't want to make that mistake.

This answers the question about changing from one "form" of a number to another. The mathematical properties of a "number system" are independent of how the numbers are written. Which is why we need to be sure operations are well defined. That is, we need to be sure they depend on the number itself, not on the numeral or other symbol used to express the number.

As for 0 being an element of the natural numbers -- I don't like it, either. But that's the way Peano defined it. Rick Norwood 20:04, 1 March 2006 (UTC)

Reply to Bo Jacoby.


 * The notation a' as an alternative to S(a) is not used and should be omitted.
 * The notation is here because it is in the article on Peano's Axioms. It should be in either both places or neither.


 * The definition "a*b = a + a + ... + a (b times)" in not the style of Peano. Use a·1=a, a·(b+1)=(a·b)+a.


 * This is a fairly low level article, and except for using the Peano axioms as a starting point does not claim to follow Peano. It seems to me that the first definition is easier for the layperson to understand.


 * Use · rather than * for multiplication.


 * The new multiplication sign is *. Get used to it -- I have.  The sign you suggest: ·  is both hard to see and not in universal use.


 * The cancellation rule "x+a=y+a implies x=y" should be given.


 * Proving all the properties of the various systems is beyond the scope of this article. For the cancellation rule, see group.

Thanks for your suggestions. I've offered my responses, but am always happy to go along with a consensus. Rick Norwood 20:16, 1 March 2006 (UTC)

Reply to Rick Norwood:


 * As for 0 being an element of the natural numbers -- I don't like it, either. But that's the way Peano defined it.


 * Peano's axioms work just as well with 1 substituted for 0. Or 2 substituted for zero.  I like natural numbers to have zero, but the convention of starting with 1 is widely used and not to be ignored.


 * Use · rather than * for multiplication.


 * The new multiplication sign is *. Get used to it -- I have. The sign you suggest: · is both hard to see and not in universal use.


 * "X is the new Y" is the new "I like Y and people like X have to change to Y because I say it's new"--not a terribly persuasive approach. Asterisk is no more "the new multiplication sign" than &amp;middot;.  Asterisk is ambiguous (look for the footnote) and also not in universal use--not even in programming languages, for which it was invented.  The only acceptable choices I see are "&middot;" (&amp;middot;), "&times;" (&amp;times;, duh), and "" (concatenation).


 * Thank goodness there's no one agitating for changing all the "^"s to "**".

I think Knuth or Tarjan has an article online about extensions of the sequence "+", "&times;", "^", ...; there is probably even such an article online. To cite the article by Rick Norwood in Mathematics Magazine goes close to the edge of WP:NPOV and WP:OR. I'm not too happy about those policies, either, but the alternative is worse.

Please understand that I appreciate your additions to this article very much, and these three issues are the only ones I disagree strongly with. I didn't notice the &amp;middot; controversy until I had completely &amp;middot;ified the article. And I'll have to at least footnote the 0 vs. 1 controversy. As for the citation issue, I'll try to find the article I recall if no one else does. --Dan 20:19, 8 March 2006 (UTC)

I was a little embarassed about quoting myself -- but I think the ban against original research means no original research appearing in Wikipedia for the first time. I don't think it is intended to prohibit quoting research a Wikipedian published in a reputable journal a long time ago. What sense would that make? On the other hand, I could be wrong. Rick Norwood 00:33, 9 March 2006 (UTC)

I'm not sure either just what is intended, though the sense it would make is that Wikipedians would be distanced from personal involvement in the choice of references they cite. Thats why I suggested both WP:OR and WP:NPOV. I've solved the problem here by replacing your citation with a link to the article on Tetration, which discusses the topic in detail, online. If you think a citation to your article will improve that article, I believe you should take it up there. But if there are enough references there, you might save yourself the embarassment or controversy. --Dan Hoey 02:08, 10 March 2006 (UTC)

Just because I ended up here while trying to figure out the situation on quoting yourself, the answer is here: WP:NOR. Quoting yourself is fine. —Darxus (talk) 05:17, 18 December 2010 (UTC)

Counting Numbers
I was always taught that there is a subset of Natural Numbers called "Counting Numbers" which consists of 1, 2, 3, 4, ect to infinity (positive integers), while a Natural Number is the same, but includes 0. Has this been changed or what? I have some references some where, most notably a college level mathbook from 2002 or something. Can someone explain this more in depth and/or why there was a change or mistake? 76.116.109.221 (talk) 03:56, 23 December 2007 (UTC)


 * Most people will agree about the terms "counting number" (1, 2, 3, 4, ...) and "whole number" (0, 1, 2, 3, ...). It is a religious issue whether the natural numbers are the same as counting numbers or the same as whole numbers.  Generally, number theorists side with the former, and set theorists side with the latter. Michael Slone (talk) 19:50, 23 December 2007 (UTC)

Ordering of sets
Can we have some orderings of the number systems on the page? Or a tree?

Natural numbers in Integers in Rational numbers in Constructible numbers in Algebraic numbers in Computable numbers in Real numbers in Complex numbers, for example. --Raijinili (talk) 19:22, 14 August 2008 (UTC)

Exsistance of this article?
I'm not quite sure what this adds over Number, natural numbers and other articles on specific systems and more technical articles like Set-theoretic definition of natural numbers. It seems to be largely a duplication of material which increases the editing and maintance work. Perhaphs a redirect to Number would be appropriate. --Salix alba 17:16, 23 September 2008 (UTC)

Gaussian integer
Should Gaussian integer be added as a type of Number system? This would entail adding it to the number system box at the bottom of this (and all the other number system) articles, so I feel the need to get consensus before making such a change myself. Thanks in advance for any opinions on this question!&mdash;GraemeMcRaetalk 16:10, 5 March 2009 (UTC)

What qualifies as a number system? And what is the point of this article?
I don't think "number system" is a coherent mathematical idea. As far as I can gather from the ones listed, an algebraic object is a number system if and only if it either has "numbers" in its name or is a real normed division algebra. What's so special about things satisfying these unnatural-looking criteria that they get an article all to themselves? An article entitled "List of people whose name contains more than four occurrences of the letter 'e'" would be equally useful.Saulglasman (talk) 17:29, 29 September 2009 (UTC)


 * Oddly tempted to create said list...Clifsportland (talk) 22:21, 6 December 2010 (UTC)

Article Sections.
There is a section heading titled “Logical construction of number systems” that does not seem to be described by its subsections. Is this an old heading of a section that has been removed? If so, the sections should be renamed accordingly. Clifsportland (talk) 22:20, 6 December 2010 (UTC)