Talk:P-adic number
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Article says "Real numbers are obtained by allowing for infinite expansions to the right; p-adic numbers are obtained by allowing for infinite expansions to the left." What if you allow infinite expansions in both directions? -- SJK
- Doesn't work. You get numbers that cannot be calculated with. Any repeating p-imal in both directions is equal to zero; if you add e (which is in no p-adic field) and anti-e (the sum of all factorials, which is in all p-adic fields) you get a number whose square cannot be taken. -phma
A question - I'm confused by the intention of the original article writer to compare the infinite sum ∑ aipi with the algebraic definition of p-adic integer, which appears to distinguish integers not by their sum in any real sense, but by their distinctiveness as an infinite sequence. Is the "sum" analogy a good one? How is it related to the convergence under the p-adic metric? It does seem to make sense that the partial sums of the p-adic metric, where we just extend the final digit to the right, should converge; but it's not clear to me.Chas zzz brown 09:09 Dec 21, 2002 (UTC)
- Yup, the connection between sequences and sums was missing. I'll add it. Basically, if you have the sequence (1, 3, 3, 11, 11, 43, ...) in the 2-adics, you write it as the series 1 + 1*2 + 0*4 + 1*8 + 0*16 + 1*32 + ... The partial sums of this series form the original sequence. AxelBoldt 00:32 Jan 4, 2003 (UTC)
Furthermore, it's hard to see how we get a field using the algebraic description as given here; since not every p-adic integer has a multiplicative inverse (any m with p | m canjnot have an inverse). Is the given definition correct? Chas zzz brown 22:24 Dec 21, 2002 (UTC)
- The p-adic integers are an integral domain and therefore have a field of quotients. m doesn't have an inverse in the p-adic integers, but it does have an inverse in the p-adic numbers (of which you can think as infinite p-adic expansions to the left which also have finitely many digits to the right of the "decimal" point). For instance, the inverse of 12 in the 2-adics is 1*2-2 + 1*2-1 + 0 + 1*2 + 0*4 + 1*8 + ... which you can check by multiplying the latter with 12. AxelBoldt 00:32 Jan 4, 2003 (UTC)
- Muchly appreciated Axel. The fact that the indexing can include negative numbers was a missing piece of the original explanation. Chas zzz brown 08:59 Jan 6, 2003 (UTC)
Kudos to whoever wrote this article. I had heard of p-adic numbers for a while and was looking for an accesible definition. The clear exposition here encouraged me to look farther into the Wikipedia. (BTW, this comes up 6th on a Google search for p-adic numbers.) JPB 03:37 11 Jul 2003 (UTC)
I changed the function f(x) = (1/|x|p)2 into (|x|p)2 since the former is not defined at 0. (May have been a typo.) -- SirJective 12:41, 12 Aug 2003 (UTC)
- Well, that was a mistake of mine: The given function was correct, I didn't work out the derivative:
- <math>\left| (1/|x|_p^2) / x \right|_p = |x|_p<math>.
- The function I gave is not even continuous at 0. I now corrected the article. --SirJective 07:27, 26 Jul 2004 (UTC)
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cut a Square into triangles
There is a nice applictation of 2-adic numbers, Theorem states that it is impossible to cut a square into odd number of equal triangles, I thought to write about that, but I do not remeber the names of authors, seach on web gave me nothing (if you know it put it here: User talk:Tosha)
Tosha 14:22, 14 Jun 2004 (UTC)
In section "Algebraic approach" I have changed (1, 3, 3, 3, 35, 35, ...) to (1, 3, 3, 3, 3, 35, 35, ...). I think this is the right version.
I am not sure whether the following sentence (in section Properties) is correct: "Thus e is a member of all algebraic extensions of p-adic numbers." Would you give it a thought? Mikolt (How do you put in your name and the date automatically?)
- Since ep is a p-adic number, let's call it k, then e is a solution of xp = k. Since an algebraic closure (which I think is what is meant here by an "algebraic extension") of Qp must contain all roots of any polynomial whose coefficients are in Qp, then it must contain all roots of xp = k, and hence it must contain e. (and you get your name and date added automatically by typing "~~~~" at the end of your message) Gandalf61 18:42, Jun 25, 2004 (UTC)
- Yes, this is exactly what I thought. However, an extension is by no means a closure! In an extension you can just add the roots of one polynomial, plus the things you get by requiring that the field remain closed under the field-operations. And one more thing: there is the algebraic closure, not an, because up to isomorphism it is unique. I try to fix it in the article. (And thank you for the other (actually both) answer.) Mikolt 11:01, 28 Jun 2004 (UTC)
Spherical completeness
Here an end is reached, as Ωp is algebraically closed.
- Actually, this may not be the end. The completion of the algebraic closure of Qp is algebraically closed and topologically complete, but not spherically complete. (Meaning: every decreasing sequence of closed balls has a nonempty intersection.) Turns out spherical completeness is something worth having. (See "A Course in p-adic Analysis", Alain Robert) Also, there appears to be a conflict in notation. Some authors use Ωp for the spherical completion and Cp for the topological completion of the algebraic closure of Qp. This seems to make much more sense to me, because the "C" part matches with what we expect from the complex numbers, except that of course "C∞" = "Ω∞" since the complex numbers are already spherically complete. Revolver 13:18, 7 Nov 2004 (UTC)
p-adic number system
Am I completely wrong in that I believe that one also refers to 'p-adic' number system as synonym of "base-p positional notation" ?
I think we should be not too categorical about the use of "system", which some, feeling very "rigourous", categorically use as synonym of "set", while it really means "(finite(?)) family", and definetly not "semiring" (i.e. numbers themselves (be they real, complex, natural or whatsoever) do not form a system).
Everyday people (and without doubt most dictionaries) use "system" for a collection of conventions, in that sense the "numbering (I mean: number writing) system in base n" seems to me quite well defined.
In any way, a big effort is to be made in interconnecting all that is written about positional notation (decimal etc. etc.) and in making it clear what one speaks about, not in being too bourbakist about definition of a the only true one terminology, but in adding well-explained cross-references on top of each article on the field (or at least one "disambiguation" page). MFH 14:58, 8 Apr 2005 (UTC)
- I do not recall ever seeing p-adic number used to mean base-p notation. They are quite different concepts - one is a number system, the other is a numeral system - the difference is well explained in those two articles. Having said that, I can see no harm in adding a clarification note with a pointer to number system at the beginning of the p-adic number article, to avoid confusion. Gandalf61 15:34, Apr 9, 2005 (UTC)
material moved from the article:
[ilan]: Hello! You don't need any of these technicalities as motivation (I usually think of motivation as being non-technical, but that's me). First off, your p doesn't have to be a prime number, the construction works for any base, including the usual base 10. So, a 10-adic integer is simply an integer with possibly an infinite number of digits to the left. Since most people are used to doing ordinary arithmetic with an infinite number of digits to the right of the decimal point, they should be able to adjust to an infinite number of digits to the left, just arithmetic as usual.
A p-adic number is the same thing, but now p is a prime number, that is, you are doing your digit expansions and arithmetic in base p with possibly an infinite number of digits to the left. I don't see why you have to say anything more complicated that that!
- Indeed the construction works for any composite number, except in some sense the p-adic numbers describe all such completions of the integers. For example, the 10-adic integers you describe are isomorphic to the 2-adic integers cross the 5-adic integers. (In general, the n-adic integers are isomorphic to the product of the p-adic integers, where p ranges over the distinct prime factors of n.) However, I do see your point that those not acquainted with college math would find the idea of infinite decimals to the left easier to digest than numbers expanding infinitely to the left in other bases. Stuwanker 15:08, 22 Jun 2005 (UTC)
More material moved here from article
[ilan]: There is no need to limit the base to a prime number. In fact, base 10 will do just fine, and 10-adic numbers are just ordinary integers, except with possibly infinitely many digits to the left, otherwise, all rules of addition and multiplication as usual. A good exercise is to understand why the 10-adic number ...111 = -1/9. A good application of this result is the non Archimedean Zeno paradox: http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/zeno.html P.S. To the people who feel the need to remove what I write: Do some research on my person, and decide whether you are more qualified than I am to write about such subjects and therefore whether you are qualified to delete what I have written (I guess you didn't find the p-adic Zeno paradox very interesting). Otherwise, am I mistaken, or is anyone allowed to contribute here? Rx StrangeLove 22:52, 22 May 2005 (UTC)
- I've removed this same material again, mostly commentary but it looks like there's some math content as well, I'm not sure what to make of it. Maybe someone can look and see if it should be returned to the article page. Thanks! Rx StrangeLove 23:17, 22 May 2005 (UTC)