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In the introduction, the 10-adic "absolute value" is defined only for powers of 10, but it needs to be defined for any rational number for the definition of the 10-adic metric that follows to make any sense. Unless I'm missing something? — Preceding unsigned comment added by 93.25.93.82 (talk) 10:52, 28 March 2020 (UTC)[reply]

You are right: r has not been defined. Now it is. - Nomen4Omen (talk) 15:20, 28 March 2020 (UTC)[reply]

why is it that wiki math articles have the worst intros, from the POV of readability by a general reader this article's intro is not pitched for a general reader; it, to use a technical term from higher math, sucks these are better but still not good I am just so pissed off that you math people in general - not just this article - do such a bad job shame on you https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/ https://divisbyzero.com/2008/11/24/what-are-p-adic-numbers/

Thanks for the suggestions. I've added a link to the Quanta magazine article to the External links section. I'll look at possible ways to improve the intro, but many math topics are difficult to explain even for specialists. Also please add "--~~~~ to the end of your comments. This adds you signature.--agr (talk) 00:25, 9 April 2021 (UTC)[reply]

I removed the section "Introduction" with the edit summary "rm the section as WP:OR, confusing, and/or out of scope: As 10 is not prime, the "10-adic absolute value" that is considered here is not an absolute value, and the strange resulting properties do not help to understand the subject of the article". This has been reverted.

So I have tagged the section, and explain here whic this section must be removed. This section is aimed to introduce the subject of the article which is p-adic numbers. In the literature, these numbers are defined only when p is prime. So, introducing the subject with p = 10 is already confusing, as 10 is not prime. Moreover, the fact that 10 is not prime induces complications that makes the introduction harder to understand than the subject of the article, and leads to nonsensical assertions, such the definition of an "absolute value" that is explicitly said to not be an absolute value (moreover this caveat is in a footnote, and this makes the section even more confusing for readers who do not spent time to open footnotes).

I intended to remove this section again, but it appeared to me that the whole article has similar issues. Especially, I have not found any workable definition of p-adic integers and p-adic numbers. So, I'll add the lacking information before removing the present mess. D.Lazard (talk) 12:15, 30 May 2021 (UTC)[reply]

The new section defines the coefficients by

«every ${\displaystyle a_{i}}$ is a rational number ${\displaystyle a_{i}={\tfrac {n_{i}}{d_{i}}},}$ such that the denominator ${\displaystyle d_{i}}$ is not divisible by p».

This is certainly a possibility, but done without any reference ― and completely new to me. In all books I know of, the p-adic coefficients are already integers although not all necessarily in the interval ${\displaystyle [0,p-1].}$ Knuth e.g. likes very much the coefficients from the interval ${\displaystyle [-1,0,+1]}$ for ${\displaystyle p=3,}$ an approach which he calls balanced ternary and which is a Teichmüller representation using the ${\displaystyle (p-1)}$-st roots of unity as digits. All such representations are certainly p-adic series, as well.
Other books start with the p-adic valuation or absolute value without going the side step with these coefficients.
Besides that, there is derived an equivalence relation which IMHO is far less important than the equivalence relation which is established by the sequences converging to 0, which means that equivalent sequences (or series) have the same p-adic limit. ―Nomen4Omen (talk) 20:06, 6 June 2021 (UTC)[reply]

I do not intend to use p-adic series for the representation of p-adic numbers. I'll use only the normalized p-adic series. However general p-adic series are needed for defining operations on p-adic numbers, specially for division of p-adic numbers.

Presently, the article does not contain any understandable definition of p-adic numbers. Definitions are given only in the sixth section, needs advanced concepts such as inverse limits, or general notions of topology (completion). Moreover the equivalence of these definitions is not mentioned. Also many properties that make p-adic numbers fundamental for number theory are completely lacking: they form an extension field of the rationals that is locally compact; the term of discrete valuation is completely lacking, except in a quotation hidden in a footnote; etc.

In summary, something is "far more important" than anything else, which is to provide a definition of p-adic numbers and a description of their main properties that is accessible to the largest possible audience. I am working on this, and this is for this reason that I have added the section "p-adic series". D.Lazard (talk) 08:56, 7 June 2021 (UTC)[reply]

1. Very good that you do not intend to use p-adic series for the representation of p-adic numbers.
2. But certainly, your p-adic series is NOT needed for defining operations on p-adic numbers, even not for division of p-adic numbers.
3. You still do not give a reference for your «p-adic series».

Pls do not misunderstand me: I do not want to defend the previous version. I fully agree with your goal: to provide a definition of p-adic numbers and a description of their main properties that is accessible to the largest possible audience. ―Nomen4Omen (talk) 15:43, 7 June 2021 (UTC)[reply]

I would like to emphasize my remark 2: There exist people (I'm not among them) who find p-adic operations of rational numbers, especially division, better than calculations with numerator and denominator !! And they do that absolutely without your unnormalized p-adic series. So again: What shall your «unnormalized p-adic series» be good for ?? ―Nomen4Omen (talk) 16:52, 7 June 2021 (UTC)[reply]

Now it is about a week that I asked you: What is your «unnormalized p-adic series» good for ?

"In this talk", let me call it the „unnormalized p-adic series of D.Lazard type“.

In your edit dated 12 June 2021 08:45 you have added: «every rational number can be considered as a p-adic series» (as I stimulated you in this talk's section Valuation first​) ― so what is the gain in introducing these unnormalized p-adic series of D.Lazard type ? Since I would like to assume that you have something in mind with them, it would only be fair to your WP-readers that you tell them what it is. Just to make further speculations superfluous and the discussion simpler. ―Nomen4Omen (talk) 09:43, 13 June 2021 (UTC)[reply]

General p-adic series are unavoidables as soon as one compute with p-adic numbers, since the normalized p-adic series are not closed under series operations (addition, subtraction, multiplication, division, etc.). The series operations applied to normalized p-adic series provide general p-adic series. So the question is not "What is «unnormalized p-adic series» good for", it is "Is there a reasonable way to define and describe operations on p-adic numbers, without using unnormalized p-adic series and without entering into the technical details of carry management? D.Lazard (talk) 15:26, 13 June 2021 (UTC)[reply]

I thought that everybody knows that the standard (in your words the normalized) p-adic series ARE closed under addition, subtraction, multiplication, division. I am sure that you know this as well: Isn't ${\displaystyle \mathbb {Q} _{p}}$ a field

• containing all rational numbers and all numbers, you are talking about (the unnormalized p-adic series of D.Lazard type), and
• being closed under these operations ??

From your post I have to assume that you don't like carry management. But also with your unnormalized p-adic series of D.Lazard type it is impossible to avoid it. Take e.g. ${\displaystyle \textstyle p=3,a={\frac {1}{2}},b=1}$ and add, then you have ${\displaystyle \textstyle a+b={\frac {3}{2}}}$ and you have to manage some carry. And I'm sure that you can't avoid it when you want to show some equivalence:

${\displaystyle \sum _{i=k}^{n}a_{i}p^{i}-\sum _{i=k}^{n}b_{i}p^{i}=\sum _{i=k}^{n}(a_{i}-b_{i})p^{i}}$

A very simple thing for you could be to pinpoint to an operation (addition, subtraction, multiplication, division) and to related operands which are not closed. ―Nomen4Omen (talk) 16:07, 13 June 2021 (UTC)[reply]

${\displaystyle \textstyle \sum _{i=0}^{\infty }5^{i}}$ and ${\displaystyle \textstyle \sum _{i=0}^{\infty }3\cdot 5^{i}}$ are normalized 5-adic series. Their difference as series is

${\displaystyle \sum _{i=0}^{\infty }5^{i}-\sum _{i=0}^{\infty }3\cdot 5^{i}=\sum _{i=0}^{\infty }(1-3)5^{i}=\sum _{i=0}^{\infty }(-2).5^{i}.}$

This is a 5-adic series, but not a normalized one. One must normalize it for getting the difference as a 5-adic numbers. D.Lazard (talk) 17:18, 13 June 2021 (UTC)[reply]

By the way, I do not like so much the introduction of p-adic numbers through p-adic series. But the representation of p-adic numbers as p-adic series is implicit in their positional notation, and this positional notation is used in many introducing textbook. As Wikipedia must be based on a solid mathematical foundation, p-adic series must appear soon in the article. It is possible to avoid series, by defining a p-adic integer as a sequence ${\displaystyle a_{1},a_{2},\ldots ,}$ such that, for every i, ${\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} ,}$ and ${\displaystyle a_{i}}$ is the image of ${\displaystyle a_{i+1}}$ under the canonical map ${\displaystyle \mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} }$ (this is the definition of the p-adic numbers as an inverse limit). I'll detail this approach in a section entitled "Modular properties". This is important as this is widely used for fast integer and rational computation (exact linear algebra), and polynomial factorization. D.Lazard (talk) 17:18, 13 June 2021 (UTC)[reply]

In many introducing textbooks, p-adic numbers are identified with p-adic series of your normalized type. And it is possible to define arithmetics (addition, subtraction, multiplication, division) on these series of your normalized type and to show that this arithmetic coincides for rational numbers with the standard arithmetic ― very much similar to the teachers for decimal arithmetic at school. These definitions and proofs are on solid mathematical foundations and there is no need to talk about inverse limits nor about modular properties, but ―yes― carry management is needed. [Several times I have tried to pinpoint you to the very strange Quote notation: these people even replace (numerator,denominator)-arithmetics by p-adic series ― and try to convince the whole world that theirs is better.] Btw, you did not yet really show how solid your mathematical foundations are: for me and up to now, the usefulness of the unnormalized p-adic series of D.Lazard type is hot air including the arithmetics on them. I can agree that some positional p-adic notation appears soon in the article and I have decided to wait, although your latest post does not help me very much.

Btw, I threw out your remark on the size of the base, because you are far away from showing an example where a problem about the size of the digits shows up.

―Nomen4Omen (talk) 18:34, 13 June 2021 (UTC)[reply]

@D.Lazard: As it appears you have finished working on this article. So I insert some amendments which make clear that not everything is clear, especially this section „p-adic series”. At the beginning you inserted 20:45, 9 June 2021‎ "formal series". So I convert this to standard notation. [Additionally I have to write "evaluated at point X=p".] As you possibly see: Your set of p-adic series is not the Formal power series as the standard user is insinuated to assume. And it is absolutely not obvious and has to be proved explicitly that your equivalence relation comprises all rational numbers. ―Nomen4Omen (talk) 09:32, 20 June 2021 (UTC)[reply]

It is wrong to say that using formal series for defining p-adic numbers is WP:OR, as many textbooks use formal series for defining p-adic numbers. The definition/representation of p-adic numbers as infinite sequences of digits use formal series implicitly and cannot be made formally correct without formal series.

The definition of p-adic series is explicitly restricted to this article, it has been introduced for convenience of writing. It would be WP:OR only if this was presented as a standard definition.

By the way, I have edited Formal power series for including the general definition of a power series. This is not WP:OR, as the completion of a local ring is often defined by using formal series with a term in each degree of the associated graded ring. This is not the case in WP, where the more technical use of inverse limit has been preferred. D.Lazard (talk) 11:26, 20 June 2021 (UTC)[reply]

@D.Lazard: You saw my remark and reverted it. But this is NOT AT ALL an answer to my question: How do you find this order (after a mathematical operation) without any evaluation ? ―Nomen4Omen (talk) 13:23, 20 June 2021 (UTC)[reply]

Besides the unresolved issue about the „p-adic series”​ above, I think it is not a good sequence of explaining some math analysis when the series do run up to infinity without having defined the conditions for convergence. Many books introduce the possible valuations (absolute values) of the rational numbers first, then define a topology, then define limits and completion within, and then (in the case of p-adics) show that all rational numbers (and not only those with p-denominators) are among these convergent series and can if needs be written as infinite p-adic series. –Nomen4Omen (talk) 16:25, 9 June 2021 (UTC)[reply]

Up to now, I have not yet considered any math analysis. Everything that I have written is pure algebra. I have just edited the article for making clear that the p-adic series are formal series, and thus that no convergence has to be considered. In any case circular reasoning must be avoided, and it seems that this is what you are asking for: one cannot define a p-adic number as a limit when the space in which the limit occurs is not yet defined.

About the terminology, I know that "p-adic series" is not a standard term. This is the reason for introducing them by "in this article". However, the concept is not original research, as my "normalized p-adic series" is exactly the same concept as the "infinite p-adic expansion" widely considered in the article, with the difference that writing the terms of the series as digits makes impossible to work with large prime numbers.

Also p-adic numbers were introduced for number theory, and most of their applications are in number theory and algebra. So, they must be described in a style adapted for number theorists and algebraists. This was not the case before my edits. D.Lazard (talk) 21:21, 9 June 2021 (UTC)[reply]

If you identify math analysis with real analysis you may be right. But if analysis starts with infinitesimal and infinity then you are wrong, then it is analysis already when you talk about formal series which run to infinity. And you certainly know that the difference between pure algebra and other math does not help you so much.
Your recourse to formal series is a problem especially when you admit coefficients with denominators ${\displaystyle d_{i}\not =1}$: as far as I can see the identification of these series with their normalizations can work only if you take non-formal (≈ analytic ??) limits, e.g. when expanding 1/2 for p=3.
You say: «So, the p-adic expansion of a rational number is a normalized p-adic series.» Why that? Isn't the (I admit: unnormalized) 3-adic series ${\displaystyle \textstyle \sum _{i=0}^{0}{\tfrac {1}{2}}\,3^{0},}$ a valid 3-adic expansion of the rational number 1/2 ??
You say: «The p-adic valuation, or p-adic order of a nonzero p-adic series is the lowest integer i such that ${\displaystyle a_{i}\neq 0.}$» What if the numerator ${\displaystyle n_{i}}$ is divisible by p ??
You do not elaborate true algorithms for addition, subtraction, multiplication, division.

As far as I can see: especially your section «p-adic series» is WP:OR and even a defective one. ―Nomen4Omen (talk) 08:42, 10 June 2021 (UTC)[reply]

Clearly some tweaks are still needed. I have fixed your objection about the numerator multiple of p. Also, I have clarified that the p-adic expansion of a rational number has all its coefficients integers in the interval [0, p − 1).

An example of computation of a p-adic expansion of a rational number is still lacking. Also, the algorithms are only sketched, but a section detailing them is needed and lacking (even in the old version). D.Lazard (talk) 14:39, 10 June 2021 (UTC)[reply]

I saw your repair wrt. numerator. Indeed, great insight !!

I'm kind of helpless wrt. your proposed division step consisting of writing ${\displaystyle r=a\,p^{k}+s}$ for r=1/2 and p=3. What do you propose: a=1 and s=−1/2 or a=0 and s=1/2 ?? Both satisfy ${\displaystyle 0\leq a<3}$, but appear not to satisfy ${\displaystyle v_{3}(s)>0}$. Or a=2 and s=−3/2 ? Then, what would be the subsequent division step ?? And the 3-adic expansion of 1/2 ?? It is   ${\displaystyle {\overline {1}}2}$, but how do you get there ? Since these algorithms are all well known and already contained in WP, I'm asking why you do not take them ― and present your phantasy instead ?

I'm extremely eager to learn about your detailing the «only sketched» algorithms of addition, subtraction, multiplication, division (I do not see a sketch anywhere).

As it appears in total: you do not have a book where you are taking your texts from.

As far as I can see the old version recurs to the well-known standard algorithms of finite partial sums and goes to the valid(!) limits. But I agree: this is not really well elaborated, if you do not take the link to the very strange Quote notation. And it has the additional (in my opinion lesser) problem of 10 not being a prime. Finally, it is indeed a question whether starting from decimal standard has to be given up in principle. ―Nomen4Omen (talk) 18:33, 10 June 2021 (UTC)[reply]

If ${\displaystyle p=3}$ and ${\displaystyle r=1/2,}$ one has ${\displaystyle v_{3}(r)=0,}$ and thus one must have ${\displaystyle v_{3}(r)>0}$ (you forgot this condition in your post). As ${\displaystyle 1=2\cdot 2+(-1)\cdot 3}$ (Bézout's identity for 2 and 3), one has ${\displaystyle 1/2=2-3/2.}$ So ${\displaystyle s=-3/2}$ and ${\displaystyle a=2.}$ At the next step, one has ${\displaystyle -1/2=1-3/2.}$ So, ${\displaystyle {\frac {1}{2}}=2+3\cdot 1-{\frac {9}{2}},}$ and the process can be repeated infinitely.

In the end, the end is not visible. So I'll let you time for your elaborations. We both know that the matter is/the matters are settled in principle ― maybe already since 1897.
Remark: I had to correct some process to be repeated indefinitely above.
Best regards. ―Nomen4Omen (talk) 14:16, 11 June 2021 (UTC)[reply]

Sorry, I still can't resist: Isn't a finite

formal series of the form ${\displaystyle \sum _{i=k}^{k}a_{i}p^{i},}$ where every nonzero ${\displaystyle a_{i}}$ is a rational number ${\displaystyle a_{i}={\tfrac {n_{i}}{d_{i}}},}$ such that none of ${\displaystyle n_{i}}$ and ${\displaystyle d_{i}}$ is divisible by p

already a kind of your «p-adic series», and a simpler one ??
Pls note that it has the additional advantage that its arithmetics is taught in school !! ―Nomen4Omen (talk) 14:42, 11 June 2021 (UTC)[reply]

1. Bind r to the formula.
2. Bind s to its role in the iteration.
3. Remove misplaced "with the order of digits reversed".
4. Make the value of s explicit.
5. Not the order of the digits is reversed, but their production wrt. significance.
6. Make explicit the "integer part".
7. The sign in ${\displaystyle -{\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}}$ has to be omitted.
8. Show the order of the digits in the standard notation.

―Nomen4Omen (talk) 18:33, 23 June 2021 (UTC)[reply]

For example for P-adic_number#p-adic_integers is not evident how to use ${\displaystyle x_{i}}$. Suppose p=2 and x as the first natural numbers...

   x   | x1 | x2 | x3 | x4 
-------+----+----+----+----
     1 |  ? |  ? |  ? |  ?
     2 |  ? |  ? |  ? |  ?
     3 |  ? |  ? |  ? |  ?
     ...
     8 |  ? |  ? |  ? |  ?
     ...

Please show a list of valid values as example.

This Gupta's link has a, perhaps, didactic definition for p-adic integers, that is not used but is valid:

${\displaystyle a_{0}+a_{1}\cdot p+a_{2}\cdot p^{2}+...+a_{n}\cdot p^{n}+...}$

where p is a prime and the ${\displaystyle a_{i}}$ are integers from {0, 1, . . . , p− 1}. So, seems the positional notation.

The reader can understand that, for p=2, the valid ${\displaystyle a_{i}}$ are in {0,1}

• ${\displaystyle 1=(1,~0\cdot 2,~0\cdot 4,~\ldots )=(1,0,0,...)}$
• ${\displaystyle 2=(0,~1\cdot 2,~0\cdot 4,~\ldots )=(0,1,0,...)}$
• ${\displaystyle 3=(1,~1\cdot 2,~0\cdot 4,~\ldots )=(1,1,0,...)}$
• ... the simple (but in reversed order) classic binary representation. Is it?

Now, returning to the article's definition of p-adic_integers, how to obtain a valid ${\displaystyle x_{i}}$? It is non-obvious in the article. And, if the Gupta's definition is valid, how to relate the two representations? they are equivalent? Krauss (talk) 14:38, 23 July 2023 (UTC)[reply]

I agree that the article is awfully written. I have rewritten the lead for having a clear (and as elementary as possible) definition of p-adic numbers and integers.

As it is, the article contains a method for computing the coefficients of the normalized p-adic series of a rational number, but, to be found, it requires a careful reading of the article.

Also, it is only for positive integers that p-adic representation resembles to positional notation. For example, for every p, the p-adic representation of –1 is $${\displaystyle -1=(p-1)+(p-1)\cdot p+(p-1)\cdot p^{2}+\ldots +(p-1)\cdot p^{i}+\ldots .}$$ D.Lazard (talk) 13:31, 24 July 2023 (UTC)[reply]

That example still resembles positional notation, to any programmer familiar with two's complement. —David Eppstein (talk) 18:02, 24 July 2023 (UTC)[reply]

I took a shot at a more accessible first sentence, but folks should feel free to adapt or rewrite it for clarity/style. Can we find any existing materials aimed at laypeople which have a clear 1–2 sentence summary? –jacobolus (t) 20:21, 26 July 2023 (UTC)[reply]

The order of the sections was confusing. In particular, the p-adic expansion of a rational number into a p-adic series was described before the definition of a p-adic series. So, I have written a section "p-adic series". This is only after publishing it that I saw that a section existed already with this title, in a style that resemble to mine. In fact, it is myself who partially wrote the section two year ago.

So, these two sections need to be merged, and I will do this soon. However, the old section contains some mistakes. Mainly that, with the chosen definition of p-adic series, the sum of two p-adic series is not always a p-adic series. D.Lazard (talk) 16:48, 16 August 2023 (UTC)[reply]

 Done. The subsequent sections still require to be updated and upgraded. D.Lazard (talk) 15:00, 19 August 2023 (UTC)[reply]

I'm disappointed by the removal of the previous lead image (shown at right) in this August edit: [1]. I created the image, so I'm biased! But I think there's a lot of value in having a lead image that is compelling and, dare I say, pretty. And I want to emphasize that the image isn't some flight-of-fancy original research on my part. It's just the solenoidal embedding of the p-adic integers. The description for the full-size image cites Chistyakov (1996), but you can find similar images in the dynamical systems literature even before that.

I'm open to improving the image to have more explanatory value, although at the outset, I want to warn that there's only so much explanation that is possible in a lead image. The image description page has plenty of room for explanatory text, but the image itself doesn't, nor does the caption. It's asking too much of the lead image to fully explain the topic to a general-audience reader with no background.

That said, I think the most straightforward improvement would be to label the selected group elements. I could embed, in the image, labels like:

• 0 = 0_3
• 9 = 100_3
• 3 = 10_3
• 1 = 1_3
• −1/8 = …001001_3
• −1/2 = …111_3
• 1/4 = …2021_3
• −1/4 = …0202_3
• 1/2 = …1112_3
• 1/8 = …221222_3
• −1 = …222_3
• −3 = …2220_3
• −9 = …22200_3

Such labels would have the nice benefit of reinforcing the lead paragraph's discussion of digit expansions. They would make the whole image larger, so to avoid losing detail, I would want to increase the WP:THUMBSIZE by 50% or so.

@D.Lazard: Before I go to the trouble of putting such an SVG together, what do you think? Of course, other editors' opinions are also welcome. Melchoir (talk) 00:34, 2 October 2023 (UTC)[reply]