Talk:P-adic number

From Wikipedia, the free encyclopedia

You have new messages (last change).
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics.
Mathematics grading: B Class Mid Importance  Field: Number theory

A better explanation of local/global principles is needed, would be "neat" to show a "picture" of the p-adic numbers (Something like Sinnot produced), also would be nice to emphasis importance in modern number theory, connections to elliptic curves, representation theory, etc..shotwell 16:12, 7 October 2006 (UTC)

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:
\left| (1/|x|_p^2) / x \right|_p = |x|_p.
The function I gave is not even continuous at 0. I now corrected the article. --SirJective 07:27, 26 Jul 2004 (UTC)

Contents

[edit] 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)
The number e, defined as the sum of the reciprocals of the factorials, is not even an element of the algebraic closure of the p-adic numbers, since the series doesn't even converge in Qp (which is already complete). As ep has p different roots in the algebraic closure, you have no canonical choice which to call e. - Holger

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

Hmmm - do you know this area of mathematics well? I only ask since you don't generally seem to edit mathematical articles. Ilan is a mathematician. Charles Matthews 19:19, 28 July 2005 (UTC)

[edit] Infinite expansions

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
But AxelBoldt says below: "...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)." Isn't that numbers with infinite expansion in both directions? -ReiVaX 21:46, 5 November 2005 (UTC)
(I've moved this discussion to the bottom of the talk page, which is where new topics are usually added) Doesn't work. The p-adic metric is defined so that sequences which expand to the left (such as 1, 11, 111, 1111 etc.) will converge in the p-adic metric, whereas in the normal metric they would diverge. But the price you pay is that sequences which expand to the right (such as 0.1, 0.11, 0.111, 0.111 etc.), which would converge in the normal metric, actually diverge in the p-adic metric. So a p-adic expansion can only have a finite number of digits to the left of the "decimal" point. Gandalf61 11:31, 6 November 2005 (UTC)

[edit] Extension vs Completion

It seems to me to not be helpful to call Qp and extension of Q, since that basically just says it has characteristic zero. As an extension, it has uncountable transcendence degree. What would make sense is to say completion, so I think I'll try to figure out how to word this. Gene Ward Smith 08:06, 6 May 2006 (UTC)

[edit] Backwards arithmetic

Copied from my talk page:

[edit] p-adic numbers notation

Gene - nice re-write on P-adic number - I liked most of your updates. However, I am not sure about your new "backwards arithmetic" section. I have always seen p-adic numbers written down as strings extended to the left. I believe that writing them as strings extended to the right is non-standard - is this your own notation ? And you have your "decimal" point in a strange place. Take -1/5 as a 5-adic number, for example - I would write it ...444.45 - how would you write it ? Personally, I thought the first part of the old "Motivation" section was a clearer introduction - introducing 10-adic expansions first at least has the advantage that newcomers are working in a base that they are familiar with.Gandalf61 09:56, 7 May 2006 (UTC)

Writing strings to the left is what Koblitz did. I find it hard to read, and don't like it. I don't know of anyone who does it to the right, but there's nothing wrong with it and the section is on trying to make the p-adics intuitive. Writing to the right makes p-adic numbers look like real numbers, and to my mind is far more digestible.

Anyway I am quite sure they are intutitive this way, because I discovered the 10-adics when I was 16 and sitting bored in an algebra class. When the teacher said carry to the left, I asked myself why not to the right? I proved to my satisfaction that this was a ring, and spent more than a week trying to show it was a field, until I discovered that in fact it wasn't, and that it depended on the base. So, I find this a very intuitive approach. I recommend it.

As for the 10-adics, I think it isn't such a good idea to fixate on them, since they aren't important. Gene Ward Smith 22:18, 7 May 2006 (UTC)

It sounds as if writing p-adic numbers to the right is your own convention. Nothing wrong with that in itself, but I think the article should stick to the standard notation. Putting your own personal notation into Wikipedia would seem to breach the No original research policy. Gandalf61 09:42, 8 May 2006 (UTC)
I don't think there is a standard notation. It's usual in computer algebra packages to write things to the right, not the left, giving p-adics as power series in a prime number. However, would "Backwards arithmetic" be a suitable topic for Wikipedia? Gene Ward Smith 01:41, 10 May 2006 (UTC)
I took a look at "no original research" and I don't think notation qualifies. Gene Ward Smith 06:17, 10 May 2006 (UTC)
As far as I can tell, the majority of authors use the "writing to the left" notation - examples are Cassels in Local Fields; Alain M. Robert in A Course in p-adic Analysis; and various sets of on-line notes such as these. The only place I can find that uses the "writing to the right" notation is here, and that does not use your unusual placement of the "decimal" point. Perhaps we should invite comments from the wider Wiki mathematics community at WikiProject Mathematics to see if there is a concensus on whether the article should use one or the other notation, or both ? Gandalf61 11:28, 12 May 2006 (UTC)
It seems to me that it's his decimal point which is non-standard, not mine. Standard in base b is that we have a0 b0 · a1 b-1 ..., and in p-adic numbers 1/p corresponds to b, so that we should get a0 p0 · a1 p1 ... .As I remarked, to-the-right is actually more common that to-the-left if you count the power series notation, where you write out the exponents and additions explicitly. Decimal notation nothing more tha a compact power series notation, and hence most logically, it seems to me, follows the same proceedure. I'll take a look at the WikiProject and see if it's a good place to stick a note. Gene Ward Smith 21:07, 13 May 2006 (UTC)
I admit I haven't worked with p-adic numbers for some time, but I've never seen the "writing to the right" notation before. (Even if you count writing out the power series as an example of "writing to right", I've seen that done backwards, as well, as in ... + 0*34 + 1*33 + 0*32 + 2*3 + 2.)
I also think that a new, non-standard, notation may be a violation of WP:NOR, if there is a standard notation. Why is a new notation not a neologism? 00:26, 16 May 2006 (UTC)
And I'm not wikistalking you, Gene, although it looks as if it would be a good use of my time and abilities. Just because I disagree with you on Real numbers, Zeration, and here, doesn't mean that I've followed you. — Arthur Rubin | (talk) 00:30, 16 May 2006 (UTC)
The "standard" notation has the feature (advantage, or disadvantage) that 234.35 has the "same" meaning as a member of Q5 or of Q. — Arthur Rubin | (talk) 00:36, 16 May 2006 (UTC)
I have mentioned this debate on the WikiProject Mathematics take page, to see whether anyone else has views on the notation used in this article - and I have re-named this talk page section to make it easier to locate. Gandalf61 08:25, 16 May 2006 (UTC)
Notation is meant to communicate. Agreement (convention) is good, and so is logic. Mathematical notation sometimes freezes its form before the underlying theory is well understood, and sometimes evolves differently at different times and places. So here we are again. We write in service of the reader. If a notation is logically far superior to its alternatives, we may consider it for purposes of explanation, but must alert the reader to what is standard and what is not. If a notation differs from overwhelmingly common usage, then it becomes that much harder to justify its use.
The notation proposed by Gene Ward Smith is not one that readers are likely to encounter elsewhere, and does not offer significant advantages over common notation. Therefore we should mention it, but not adopt it for the bulk of the article. If facts refuting my premises are brought into evidence (major advantages, frequent usage), I'll reconsider. --KSmrqT 13:12, 16 May 2006 (UTC)
I propose that we revert to the "right to left" notation in the body of the article, and mention the two variants of the "left to right" notation (with the units digit to the right or to the left of the "decimal" point) in a footnote. Gandalf61 14:32, 18 May 2006 (UTC)
I concur. — Arthur Rubin | (talk) 18:06, 18 May 2006 (UTC)
Please do not make a claim in the article that "right to left" is a standard notation. It's a notation used to introduce p-adic numbers, it's not something number theorists are in love with. In an actual math paper, if you needed a specific p-adic number, the two usual ways of representing it would be to give the number mod pn for some suitable n, or to give the number as a power series expansion (and so of course, "right to left".) The power series idea goes back to Hensel, incidentially.
If you want to see an example of how you might use and display actual p-adic computations, I recently did a lot of work on the algebraic number field article. In the section an example, I work out an example in the 23-adics. This is how you'd probably want to do it in an actual math article. Gene Ward Smith 03:42, 19 May 2006 (UTC)
Isn't that what we're doing and should be doing — introducing p-adic numbers? You're convincing me that we should use easily understandable notation. — Arthur Rubin | (talk) 05:32, 19 May 2006 (UTC)
What number theorists do is in a context where people know what p-adic numbers are; the question here is how to explain them to people who only know the real and complex numbers. From that point of view, saying p-adics carry to the right, reals carry to the left, and function fields over finite fields don't carry at all is one way of approaching it, and does bring home the idea that there is an underlying similarity. What is most likely to get people to see the analogy? Gene Ward Smith 09:01, 20 May 2006 (UTC)

[edit] Teichmüller expansions

By the way, it's by no means true that number theorists are wedded to the {0, ..., p-1} digit set for the p-adics either. The prime p has all p-1 of the p-1 roots of unity in Zp, and these, together with 0, can be used for the digits of a representation of the p-adics. Rounding off gives the Teichmüller character.Gene Ward Smith 03:56, 19 May 2006 (UTC)

Yes, there are many choices for the digit set. Yes, Teichmüller digits have advantages in some circumstances. Yes, number theorists seldom need to write out an explicit representation of a specific p-adic number. All of these issues could be discussed in a new section on notation. But I still maintain that for the purposes of an introductory section written for newcomers to the subject, the digit set {0,1,...,p-1} written right to left is the best notation, because the p-adic representation of a positive rational integer is then the same as its base p representation. If I re-write the first main section of the article with right-to-left notation, but without saying that this is a standard notation, and add a new section describing alternative notations, would you be happy with that ? Gandalf61 11:19, 20 May 2006 (UTC)
That would be fine. I'm not insisting on left-to-right, which is why I put the topic up for discussion in the first place. I think some place or other it would be nice to mention that you can carry to the left, carry to the right, or not carry at all, and all three choices produce things like numbers. Gene Ward Smith 23:20, 20 May 2006 (UTC)
Okay, I have re-written the "backwards arithmetic" section and re-named it as Introduction; added a new Notation section to describe various different notations (but without claiming that any one of them is more standard than the others); and taken out references to backwards arithmetic in other places in the article to avoid confusion. I next plan to add references for the various properties of Zp, Qp and Cp listed in the Properties section. Gandalf61 09:28, 26 May 2006 (UTC)
Looks good--at least to the extent right-to-left can ever look good. Gene Ward Smith 06:50, 27 May 2006 (UTC)

[edit] Topological approach

I like the idea. Unfortunately, it seems to be covered under "#Analytic approach", as the completion with respect to the specified metric. — Arthur Rubin | (talk) 16:12, 16 July 2006 (UTC)

[edit] ...99999 = -1?

Hi, I hope this comes out right because I usually just stick to fixing spelling mistakes!

The very first section, about how the 10-adic number ...99999 = -1 does not make things clear to me. I have seen on another occasion, elsewhere, the following idea used:

9 + 1 = 10

99 + 1 = 100

...9 + 1 = ...0

where in the last one we keep passing the "carry" to the left up to infinity, and so since x + 1 = 0, we have x = − 1, where x = ...9999

That seems clearer to me than the mentioning of "being close together if they differ by a large power of ten." --MarkHudson 15:51, 19 July 2006 (UTC)

[edit] p-adic expansion

"If p is a fixed prime number, then ...."

p does not need to be a prime number, it can be any positive integer not less than two. —The preceding unsigned comment was added by 129.13.186.1 (talk • contribs).

Yes, the base p expansion works for any integer p>1, but if p is not a prime number then the p-adic numbers are not a field, which limits their usefulness. Gandalf61 19:16, 26 November 2006 (UTC)
Retrieved from "http://en.wikipedia.org../../../p/-/a/Talk%7EP-adic_number_efbc.html"