Talk:Arithmetic

Numeralization

Don't you think "numeralization of the null concept" should be changed to "invention of zero"? If not, I think at least "numeralization of the null concept (invention of zero)" should be written instead. This would be much clearer. A.Z.

operations section covers 4 of 6

The intro paragraph lists: addition, subtraction, multiplication, division, exponentiation, and extraction of roots. Then the section on Arithmetic operations, gives a brief overview of the first four (explaining how they are inverses, etc.), but then the section stops and doesn't finish with the pair of exponentiation, and roots. I think completing the summary would be fitting for this overview article (at the same brief level) without readers having to go to a more specialized article. DKEdwards (talk) 18:55, 23 January 2022 (UTC)

Peano formalizing Arithmetic

The claim that "Peano formalized arithmetic with his Peano axioms," seems misleading given that Peano was building off Dedekind's axiomization. Ted BJ (talk) 01:56, 16 August 2022 (UTC)

I cannnot understand how a man (Peano) can be built off an axiomatization. Moreover, The only Dedekind's axiomization that I know of is Dedekind's construction of real numbers, which cannot be confused with Peano's axiomatization of natural numbers. So, there is nothing misleading here, and I'll remove the tag {{disputed inline}}. D.Lazard (talk) 12:33, 16 August 2022 (UTC)

First of all, by "Peano was building off Dedekind's axiomization" I meant that Peano built his axiomatization off Dedekind's axiomization. This is a very common idiom, for example see here: https://www.google.com/books/edition/Protest_on_the_Page/0JMvBwAAQBAJ?hl=en&gbpv=1&dq=%22was+building+off+earlier%22&pg=PA106&printsec=frontcover

Secondly, Dedekind provided an axiomatization of arithmetic in his 1888 paper "Was sind und was sollen die Zahlen?" aka "What are numbers and what should they be?". See here https://mathcs.clarku.edu/~djoyce/numbers/dedekind.pdf and here http://aleph0.clarku.edu/~djoyce/numbers/peano.pdf. This axiomatization was the basis of for Peano's later axiomatization, so claiming that "Peano formalized arithmetic with his Peano axioms" is misleading, because it leaves out Dedekind's contributions. I will put the tag back. Ted BJ (talk) 15:06, 22 August 2022 (UTC)

Changes to the article

I was thinking about implementing changes to this article with the hope of moving it in the direction of GA status. There is still a lot to do since the article has various problems in its current form. Its sourcing needs a lot of work. It further lacks various key topics, like a proper explanation of the difference between different types of numbers and numbering systems. The history section contains very little about developments after the Middle Ages. The arithmetic operations of exponentiation and logarithm are not properly discussed. There also should be more on the foundations and axiomatizations of arithmetic.

I was thinking about doing more in-depth research and preparing a draft to address and implement the ideas pointed out here. It will take me a while to go through the sources. Feedback on these ideas and other suggestions are welcome. Phlsph7 (talk) 09:04, 21 October 2023 (UTC)

Good idea. Nevertheless, instead of a global draft, it would be better to work one section after the other. This would make be the discussion easier. Also, you may be bold and make your change directly in the article. If everybody agree with your change, this will be OK. Otherwise, you will probably be reverted per WP:BRD, and this will allow discussing only the points of disagreement. D.Lazard (talk) 11:38, 21 October 2023 (UTC)

@D.Lazard: That's a good idea about adding the changes section-wise instead of all in one go. I'm not sure how feasible it is to implement the changes incrementally and directly in the article without a draft. The problem I see is that quite a few substantial changes would be needed to prepare the article for a GA nomination. I have to do a proper literature review anyways before I make any non-trivial changes and I usually take notes and make drafts as I go along. I intend to keep you in the loop to ensure that I don't stray too far from the expected direction. Phlsph7 (talk) 16:25, 21 October 2023 (UTC)

Feel free to make a draft if you prefer.

I'd recommend predominantly focusing on pen-and-paper positional base-ten arithmetic, which is what the word "arithmetic" typically refers to, what I would expect most readers to be looking for when they arrive here, and which is already a pretty broad scope about which plenty can be said to fill a long article, especially if you add discussion of arithmetic pedagogy/curriculum, the use of arithmetic in society, the role of changing technology, and so on, then mostly leaving broader questions about other number systems, other kinds of calculation methods (counting boards, slide rules, computer algorithms, ...), number theory, formal axiomatizations, etc. to other articles with a more directly relevant scope.

I'd personally recommend moving the history section much further down the page. It would be great to fill out History of arithmetic in much greater detail (ideally this could be 5000+ words; cf. the Russian version ru:История_арифметики [machine translation]), with just a summary (no more than maybe 2000 words) at Arithmetic § History. It would be especially nice for someone to do some proper research into medieval Islamic material (e.g. Abu'l-Hasan al-Uqlidisi), which is not very well covered anywhere on Wikipedia.

–jacobolus (t) 12:08, 21 October 2023 (UTC)

@Jacobolus: Your explanation of how to handle the history section is a good example of how to approach this kind of overview article by following WP:SUMMARYSTYLE. You are right that for topics like this one, the history section is usually better placed at the end.

I agree that it's important to keep the reader's expectations in mind when writing this type of article, for example, regarding a more detailed explanation of positional base-ten arithmetic. However, one of the GA criteria is that the article covers all the major aspects of the topic. To fulfill it, I think we have to discuss arithmetic in its widest sense and not just what people familiar with elementary arithmetic from school expect. One danger especially common among math articles is to make discussion of difficult topics overly technical by filling sections with formal definitions and jargon. We'll have to see how it goes in relation to discussions of topics like alternative number systems and the foundations of arithmetic. I think we can't just skip these topics so we'll have to struggle to make them accessible. As for length, I usually aim at a readable prose size of 40-50kB as per WP:SIZERULE but this is difficult to plan in advance. Phlsph7 (talk) 16:26, 21 October 2023 (UTC)

have to discuss arithmetic in its widest sense – The word "arithmetic" is sometimes used very broadly to include all of number theory, e.g. see the title of Gauss's Disquisitiones Arithmeticae or Serre's Cours d'arithmétique. Broadening the scope like that is not useful for readers and not manageable for a single article here unless it gets turned into a high-level summary overview, which is frankly not that helpful for this kind of case, because it necessarily significantly detracts from the attention available for the topic of "arithmetic" meaning mathematical calculations of the type used in school or everyday life, which, as I said, I think is already more than wide enough a scope to fill out an arbitrarily long article.

A "good" Wikipedia article just has to cover its own self-defined scope, it doesn't have to cover every sense of the title ever used by anyone (that's what disambiguation pages are for). This article should mention much broader interpretations of the word "arithmetic", but I think it's a mistake to focus significant attention on them. –jacobolus (t) 16:36, 21 October 2023 (UTC)

Thanks for raising this point. I agree that Wikipedia articles do not need to focus on every meaning of its title term and leave that to disambiguation pages. Delimiting the scope of this article might be a good idea before getting started with any major changes. I guess the best approach here would be not to decide ourselves on the scope of the article but to consult how reliable overview sources treat this topic and follow their lead.

• According to the Encyclopedia of Mathematics, arithmetic is The science of numbers and operations on sets of numbers. Arithmetic is understood to include problems on the origin and development of the concept of a number, methods and means of calculation, the study of operations on numbers of different kinds, as well as analysis of the axiomatic structure of number sets and the properties of numbers.
• The Arithmetic entry in the Gale Encyclopedia of Science states that Arithmetic is a branch of mathematics concerned with the numerical manipulation of numbers using the operations of addition, subtraction, multiplication, division, and the extraction of roots". Its entry includes discussions of different types of numbers (natural, rational,...), numbering systems, and axioms.
• From the Facts On File Encyclopedia of Mathematics: The branch of mathematics concerned with computations using numbers is called arithmetic. This can involve a number of specific topics—the study of operations on numbers, such as ADDITION, MULTIPLICATION, SUBTRACTION, DIVISION, and SQUARE ROOTs, needed to solve numerical problems; the methods needed to change numbers from one form to another (such as the conversion of fractions to decimals and vice versa); or the abstract study of the NUMBER SYSTEMS, NUMBER THEORY, and general operations on sets as defined by GROUP THEORY and MODULAR ARITHMETIC, for instance.

I'm not sure if the scope discussed in these definitions is roughly what you had in mind. If you have some reliable overview sources that provide a very different outlook then I would be interested to have a look at them. I agree with you that there are many advantages to focusing on the simpler aspects of this topic. For example, even if number theory should be included, it would probably be a bad idea to dive into all its intricacies. Phlsph7 (talk) 18:00, 21 October 2023 (UTC)

I don't think these encyclopedias pick a very reader-relevant scope, to be honest. They are overly focused on recent theoretical developments in pure mathematics (past 2 centuries or so) at the expense of punting on covering the primary topic beyond a bare sketch.

In theory this article could cover anything with either "arithmetic" in the name or broadly relevant to calculation. Here are some:

Arithmetical hierarchy, arithmetical set, true arithmetic, Peano axioms, second-order arithmetic, Robinson arithmetic, Büchi arithmetic, Skolem arithmetic, Heyting arithmetic, Presburger arithmetic, primitive recursive arithmetic, elementary function arithmetic, bounded arithmetic, ordinal arithmetic, cardinal arithmetic, non-standard model of arithmetic, hyperarithmetical theory;

Higher arithmetic (number theory), fundamental theorem of arithmetic, arithmetic function, arithmetic dynamics, arithmetic topology, arithmetic geometry, arithmetic combinatorics, arithmetic group, arithmetic Fuchsian group, arithmetic variety, arithmetic surface, arithmetic hyperbolic 3-manifold, arithmetic of abelian varieties, arithmetic number, field arithmetic, arithmetic derivative, arithmetical ring, arithmetic progression topologies;

Arithmetic mean, arithmetic progression, Dirichlet's theorem on arithmetic progressions, Roth's theorem on arithmetic progressions, arithmetic–geometric mean, arithmetico-geometric sequence;

Modular arithmetic, residue arithmetic, lunar arithmetic, saturation arithmetic, finite field arithmetic, surreal number (combinatorial game theory), arithmetic billiards;

Hilbert's arithmetic of ends;

Significance arithmetic, interval arithmetic, affine arithmetic, Logarithmic arithmetics;

Binary arithmetic, Location arithmetic, counting board, abacus, counting rods (rod calculus);

slide rule, nomogram;

Floating point arithmetic, arbitrary-precision arithmetic, fixed-point arithmetic, mixed-precision arithmetic, symmetric level-index arithmetic, arithmetic coding, serial number arithmetic, arithmetic logic unit;

Arithmologia, The Foundations of Arithmetic, Philosophy of Arithmetic;

Etc.

But I just don't think most of these topics are that helpful to try to cram together with an article about ordinary "arithmetic" as used on common language. –jacobolus (t) 19:05, 21 October 2023 (UTC)

The Concise Oxford Dictionary of Mathematics has a reasonable definition: Arithmetic: The area of mathematics relating to numerical calculations involving only the basic operations of addition, subtraction, multiplication, division, and simple powers. The term ‘higher arithmetic’ refers to elementary number theory. –jacobolus (t) 19:27, 21 October 2023 (UTC)

I agree that this article should not strive to be a complete compendium of everything associated with arithmetic. One way to decide what to include in an article is WP:PROPORTION: An article should...strive to treat each aspect with a weight proportional to its treatment in the body of reliable, published material on the subject. For example, Arithmetic Fuchsian group and many of the other topics mentioned by you should probably not be discussed in detail (or at all) because they do not receive much treatment in reliable sources on the topic. A good way to determine this is to consult reliable overview sources, like the ones I mentioned in my last reply (by the way, they include discussions of the history of arithmetic and do not restrict themselves to pure mathematics in the past 2 centuries). Speculating on what the readers of this article might be interested in is a tricky business and different contributors might have very different opinions on this issue. It could be that many among them are university students who have to take a course in this area and who already know well how to add and multiply numbers. I think the content policy WP:PROPORTION is a better guide in deciding what to include.

I'm not sure whether our disagreement is actual or merely verbal. For example, you "recommend predominantly focusing on ... base-ten arithmetic". I'm in agreement if this means that base-ten arithmetic deserves more weight than other number systems. But I'm in disagreement if this means that other number systems should not be discussed in the article (especially since alternative number systems, like the basics of the binary system, are commonly discussed in school). A similar potential disagreement might be in relation to the foundations of arithmetic. As I see it, they are not the prime focus of the article but they are still an important topic (given their treatment in reliable overview sources) and the article would be incomplete without them. My concern is that it may not be possible to get this article to GA status if these topics are left out. I hope we can resolve these potential points of tension and arrive at some form of compromise that works for both of us. Phlsph7 (talk) 07:41, 22 October 2023 (UTC)

My point is just that "arithmetic" means very different things to different people in different historical time periods, and you shouldn't decide that you need to describe all of them just because people used the word "arithmetic" for them.

Conceivably the word "arithmetic" could be considered to mean "anything related to concrete calculations with numbers or similar formal systems".

But I don't think that really gives a good idea of what people typically mean by "arithmetic" in modern times. The number theorists have mostly by now settled on the name "number theory" rather than "higher arithmetic". The logicians and philosophers are often happy with high-level names like "foundations of mathematics" rather than always putting "arithmetic" in the name.

Many of the broader topics can be covered in articles called e.g. computation, calculation, axiomatic system, number, algebraic structure, mathematical operation, etc.

I think it's fine to mention some of these topics but I wouldn't make them nearly so much the primary focus of the article as some of the other encyclopedia articles you have linked.

The most basic subject should in my opinion be the basic structure of rational numbers and written algorithms for calculations with integers, rational fractions, and decimal fractions, including square roots and possibly a bit about other kinds of calculations and the use of pre-printed tables. Then a summary of the history of arithmetic should focus predominately on the history of these, with some side mention of other kinds of calculating methods/tools.

But there are a bunch of other topics that I think are important to mention/discuss at an article about arithmetic that are often not adequately covered. For example, the cognitive basis for arithmetic and development of number sense (not sure our article about has a definition quite matching broad use), mental arithmetic, finger counting and finger reckoning (we don't have a good article about this), the use of arithmetic in society and its changing role(s) over time, the displacement of arithmetic practice by handheld calculators, school pedagogy and curriculum and the role of conceptual understanding vs. memorization in learning arithmetic, arithmetical word problems, the relationship between arithmetic and algebra and difference in problem solving methods and mental models involved, estimation and approximate calculation methods (rounding, significant figures, etc.), the differences between "arithmetical" vs. "instrumental" (originally based on measurements with dividers and various scales or tools like sundials or globes, later the use of slide rules and more sophisticated nomograms) vs. "geometrical" solutions of problems. Ultimately we might consider analytic geometry to be a kind of "arithmetization" of geometry, or more generally modern science to be a kind of arithmetization of the world; some kind of discussion of these seems just as (if not more) important than a detailed description of axiomatization by Peano & al.

–jacobolus (t) 21:12, 22 October 2023 (UTC)

Thanks, that is a great overview of possible topics to include! I'm trying to conceive how the topics mentioned in the last paragraph of your reply could be organized into subsections. Several fall under psychology or numeracy. Some could be grouped together as techniques for counting and calculating, either with or without external tool. Maybe this could be combined with simple algorithems. Some would fit into the current section "Arithmetic in education", which could be expanded. Some concern how arithmetic impacts other areas in mathematics and the sciences. The part about the (changing) role of arithmetic in society might be best included in the history section. Phlsph7 (talk) 08:25, 23 October 2023 (UTC)

IMO, in its modern meaning, arithmetic is essentially the art of representing numbers and computing with them. So, the article must focus on this, and have sections (without details that belong to specific articles) on

• Integer arithmetic (Hindu-Arabic numerals, bases 10 and 2, algorithms for the operations, from addition with carries to multiplication in $O(n\log n)$
• Rational arithmetic, that is, the arithmetic of fractions
• (Approximate) real arithmetic:
  • Decimals and binary numbers as approximations of real numbers
  • Rounding, truncation, relative and absolute error (this may be called "arithmetic of approximations")
  • Scientific notation, floating point arithmetic and their specificities, such as non-associativity of the addition. The problem of exact floating point operations (IEEE754)
• Interval arithmetic
• Modular arithmetic and its application to integer arithmetic through Chinese remainder theorem and Hensel lifting
• Mental arithmetic
• History: This section must explain clearly how the meaning of "arithmetic' has evolved. In particular, despite its name, Arithmetica is not about arithmetic. IMO, counting rods, abaci and slide rules belong to the history section.

This is evidently not a complete list, but all these items belong clearly to arithmetic, and the above items seem a good way to structure the article. D.Lazard (talk) 10:54, 23 October 2023 (UTC)

These are all good ideas, I have the impression that we are getting somewhere. My rough idea on how to organize the material into different sections is the following

• Definition
• Basic concepts
  • Numbers
  • Numbering system
  • Arithmetic operations
    • Addition and subtraction
    • Multiplication and division
    • Exponentiation and logarithm
    • Modular arithmetic
    • Compound unit arithmetic
• Laws and fundamental theorems
• Techniques, tools, and algorithms
• Foundations
• History
• In various fields
  • Education
  • Psychology
  • Philosophy
  • Computer
  • Other areas of mathematics and the sciences
  • Everyday life

With this number of sections and subsections, each one would be relatively short and only provide an overview with a link to the main articles that treat the topic in more detail. The section "Definition" covers the basic definition and mentions some of the problems already discussed here, like the difficulty in delimiting its scope and its relation to number theory. It also mentioned the etymology. The subsection "Numbers" explains the different types of numbers (natural, integers, rational,...). It could also cover floating point numbers in relation to rational numbers and maybe rounding and truncation. The subsection "Numbering system" explains the differences between positional and non-positional systems and shows how the same number can be expressed in different systems, for example, as a Roman numeral in contrast to the decimal and binary systems. Maybe we can also mention the Scientific notation here.

The section "Laws and fundamental theorems" discusses things like commutativity, associativity, and the Fundamental theorem of arithmetic. The subsection "Psychology" deals with numeracy, Mental arithmetic, and similar issues. The subsection "Philosophy" mentions some philosophical problems, like whether numbers are real entities or mere fictions. The subsection "Computer" includes information on how arithmetic operations are implemented, including the technical level (Arithmetic logic unit) and things like floating-point arithmetic. Maybe we could also mention cryptography like RSA there.

I'm not sure if this way of dividing the topic can properly deal with your suggested sections of "Integer arithmetic", "Rational arithmetic", and "Real arithmetic". Part of it would be covered in the sections "Numbers" and "Arithmetic operations". If this is not sufficient then we could include them as separate subsections. Interval arithmetic could be discussed in the section "Other areas of mathematics and the sciences". Sorry for the rather lengthy explanation. Phlsph7 (talk) 11:54, 23 October 2023 (UTC)

The "fundamental theorem of arithmetic" is about number theory, and definitely does not deserve a separate section. It can be briefly mentioned in the history section if you like. –jacobolus (t) 16:17, 23 October 2023 (UTC)

I would say the whole article is about "Techniques, tools, and algorithms", so it's also weird to make that a dedicated section. I'm not sure a "Definition" section is particularly necessary. I'd get rid of "Basic concepts" as a top-level section, and aim for a flatter structure, and avoid splitting numbers / number systems / operations which seems like a division which unnecessarily slices single topics into multiple pieces and then scatters them around in a way that will be unnecessarily confusing to readers. I think D.Lazard has a better top-level structure, and you should keep at least arithmetic with integers, rational fractions, and decimal fractions among the first few top-level sections, though his imagined article has a broader scope than what I'd cover if I were writing the article myself. –jacobolus (t) 16:50, 23 October 2023 (UTC)

(Hopefully the above doesn't seem too negative. I'm not trying to be a jerk or rain on parades here.) –jacobolus (t) 18:08, 23 October 2023 (UTC)

I have no problem with removing the heading "Basic concepts" to have a flatter structure. The idea behind the distinction between numbers, numbering systems, and arthimetic operations is the following: numbers are the objects, numbering systems are ways of representing those objects, and arthimetic operations are ways of combining and manipulating those objects. This seems to be a natural rather than an artificial distinction: it's possible to discuss different types of numbers (natural vs rational) without discussing different ways of representing them (decimal or binary) or what operations can be used on them (addition or multiplication). This type of division is also used in some overview works, such as the entry "Arithmetic" in the UXL Encyclopedia of Science. In the process of writing those topics, I'll see if this structure makes sense or if there is a better way to arrange them. I'll try to follow the suggestion of having distinct sections for integer, rational, and real arithmetic. Phlsph7 (talk) 11:18, 24 October 2023 (UTC)

The problem is not that the distinction is unnatural. The problem is that your structure then looks like: [Numbers: [integers, common fractions, decimal fractions, (binary fractions, complex numbers, ...?)], number systems: [various representations of integers: [...], representations of fractions; [...], ...], Operations: [operations on integers, operations on fractions, operations on decimal fractions, (operations on binary fractions, operations on complex numbers, ...)], which unnecessarily chops material into little bits and then rearranges it in a way that readers will be continually hopping back and forth between different sections to make sense of it. –jacobolus (t) 11:39, 24 October 2023 (UTC)

It was not my plan to subdivide the sections on numbering systems and arithmetic operations by different types of numbers. Phlsph7 (talk) 16:04, 24 October 2023 (UTC)

The first few sections of this article should in my opinion consist of concrete descriptions and explanations of various arithmetical concepts and methods. Stuff like: the number line and counting in a base-ten positional number system, multi-digit addition/subtraction with carrying, addition and subtraction with negative numbers (integers) possibly mentioning double-entry bookkeeping, "Egyptian" multiplication, long multiplication (and the lattice method) and some mention of the multiplication table, long division with remainder, addition and multiplication of common fractions, etc. –jacobolus (t) 18:10, 24 October 2023 (UTC)

Also remember, we don't have to reproduce the content of number, numeral system, algebraic structure, etc.; this article can focus on arithmetic per se. Aside: I looked at the UXL Encyclopedia's article "Arithmetic" and it's a mediocre mess: poorly organized, poorly written, weirdly conversational, full of speculation and vague nonsense (and significant factual inaccuracies; please don't cite that as a reliable source), and never actually gets around to discussing the topic. –jacobolus (t) 12:02, 24 October 2023 (UTC)

The UXL Encyclopedia belongs to Gale (publisher), which is considered a reliable publisher. It is part of the Wikipedia library, see Wikipedia:Gale. See my response to D.Lazard for the parts on number and our earlier discussion on the scope of arithmetic. Phlsph7 (talk) 16:06, 24 October 2023 (UTC)

I don't really care who the publisher is. The article itself is bad. It reads like a sloppy and informal paraphrase of some other source (which itself had a sort of weird scope/organization) which was then never read over by anyone with expertise in the subject or willingness to double-check factual claims. –jacobolus (t) 18:12, 24 October 2023 (UTC)

I agree that the distinction between numbers and numerals is fundamental This the reason for which it must be distinguished between the algebraic properties of arithmetic operations (commutativity, associativity, etc) that are properties of numbers and belong to number theory and .algebra, and the application of these operations to specific numbers that is the true object of arithmetic. Also, if you categorize operations by number systems, you will be faced to many problems such as the following: although decimals and fractions of integers are all rational numbers, addition of decimals is very different from addition of fractions ($3+{\frac {1}{2}}={\frac {7}{2}}$ vs. $3+0.5=3.5$). Also, multiplication is not associative with long division ($1=(3\times 1)/3\neq 3\times (1/3)=0.999\ldots$). The fact that this inequality of numerals is an equality of numbers is a result of number theory, not really a result of arithmetic, even if it is important in arithmetic. You wrote it's possible to discuss different types of numbers (natural vs rational) without discussing different ways of representing them (decimal or binary) or what operations can be used on them (addition or multiplication). This is exactly the reason for which the sectioning of this article must not follow the sectioning of Number. D.Lazard (talk) 14:07, 24 October 2023 (UTC)

Let me see if I understand you correctly. According to you,

• arithmetic only studies the application of arithmetic operations to specific numbers
• arithmetic does not study the properties or laws of those operations, like commutativity and associativity
• arithmetic does not study any other properties of numbers

Could you provide some reliable overview sources that present the topic in a way that reflects your view on the scope of arithmetic? Because I'm having trouble reconciling your views with the reliable sources I'm aware of. For example, the ones I cited above explicitly discuss things like commutativity and associativity as part of arithmetic but you are saying that they do belong to number theory instead. These sources also paint a different picture of the relation between arithmetic and number theory, for example, the Encyclopedia of Mathematics. From the [1]: Arithmetic is The science of numbers and operations on sets of numbers and includes the ... analysis of the axiomatic structure of number sets and the properties of numbers. From [2]: Number theory is The science of integers. This would mean that number theory is much more narrow than arithmetic.

I'm not in principle against using a more narrow scope for this article but I can only write the article this way if the sources support it. If you know of a few high-quality overview sources that present the topic this way then I would be happy to take a look at them to see if it makes sense to follow the more narrow scope. Phlsph7 (talk) 16:12, 24 October 2023 (UTC)

Number theory is a theoretical pure math subject, in which the topic about which proofs are made is integers/rational numbers (and e.g. Diophantine equations), but in which any method whatsoever can be used to write proofs of the theorems of interest, meaning that modern number theory draws on more or less every branch of pure mathematics. The scope is incredibly broad.

Arithmetic by contrast (at least, as I would use the term) refers to the explicit calculation of concrete numerical operations, and the employment of those calculations to solve concrete problems. Once someone is making general proofs, they start to stray away from arithmetic per se. –jacobolus (t) 18:19, 24 October 2023 (UTC)

Specifically, I would not consider the logical formalization of arithmetic to be part of arithmetic itself. –jacobolus (t) 18:29, 24 October 2023 (UTC)

I fully agree. It is a very good summary of what I was trying to explain with examples. D.Lazard (talk) 19:23, 24 October 2023 (UTC)

@D.Lazard 192.145.175.198 (talk) 00:57, 1 December 2023 (UTC)

I started a draft of the section "Integer arithmetic" at User:Phlsph7/Integer_arithmetic to implement some of the talk-page suggestions here. I haven't done any copyediting and I haven't added any references. The section requires at least one more image to visualize how long multiplication works. I was hoping to get some feedback on the selected topics and their explanation before I get the other aspects of this draft in order. There are many more algorithms that could be described step by step but my impression was that this is better left to the corresponding child articles. Also, feel free to edit the draft directly if you have improvement ideas. Phlsph7 (talk) 09:58, 29 October 2023 (UTC)

The diagram you have there is an explanation but doesn't reflect any actual practice. I wonder if anyone has the time/ability to make some animated examples. It would be neat to compare e.g. multi-digit addition or subtraction using (a) the kind of counting board common in medieval Europe, (b) a soroban, (c) Hindu-Arabic numerals on a dust board using erasure as a fundamental technique, (d) some variant of the pen-and-paper algorithm usually taught in schools today. –jacobolus (t) 17:18, 29 October 2023 (UTC)

I think even a summary section on (positional decimal) integer arithmetic can be extended quite a bit. I'd maybe make sub-sections for: (1) counting, (2) addition and subtraction within 20, (3) a number line concept, (4) multi-digit addition/subtraction with a general concept of borrowing/carrying, (5) addition/subtraction with negative numbers, (6) skip counting, (7) single-digit multiplication and a multiplication table, (8) multi-digit multiplication methods including "peasant multiplication", lattice multiplication, long multiplication, (9) division by repeated subtraction, (10) long division with remainder (11) the Euclidean algorithm, greatest common divisors, and continued fractions. There are probably other worthwhile subsections I'm leaving out here. I'd defer discussion of asymptotically faster multiplication algorithms implemented in computer systems to a later part of the article. –jacobolus (t) 18:17, 29 October 2023 (UTC)

Thanks for the feedback and the many suggestions. I made new diagrams to more accurately present addition with carry and long multiplication. You presented many interesting expansion ideas for this section. I mentioned some of them. My goal now is to first get the essentials of the new sections down, like rational arithmetic and real arithmetic. I will also have to adjust the pre-existing sections accordingly. I hope to revisit the expansion ideas once the main ideas are implemented. Phlsph7 (talk) 13:36, 3 November 2023 (UTC)

Many thanks for your great work. I did not paricipate further to the above discussion because of other occupations, and also because I had the feeling that my few a priori concerns were well understood. I just read the new version (without comparing with the older one), and I find it excellent; this is a rare case where everything is better written than what I could do myself. Congratulation again. D.Lazard (talk) 11:41, 12 December 2023 (UTC)

Thanks a lot for taking the time to review the new version and for all your initial help in ensuring that this project set off in the right direction! Phlsph7 (talk) 12:53, 12 December 2023 (UTC)

Sources

Let's start a collection of relevant sources here. Feel free to modify the below list. –jacobolus (t) 20:10, 24 October 2023 (UTC)

Thanks for listing the sources. Several of the ones listed so far should be useful for the parts that deal with education and psychology. It will take me a while to familiarize myself with them. My idea was to get started with the sections on integer arithmetic, rational arithmetic, and real arithmetic. Do you know of any sources that provide a good overview of one or several of these topics? Phlsph7 (talk) 07:58, 25 October 2023 (UTC)

These "education" sources deal extensively with these topics. –jacobolus (t) 08:01, 25 October 2023 (UTC)

History:

• Berggren, J.L. (2016), "Arithmetic in the Islamic World" in Episodes in the Mathematics of Medieval Islam, Springer, doi:10.1007/978-1-4939-3780-6_2
• Saidan, Ahmad S. (1996) "Numeration and arithmetic" in Roshdi Rashed (ed.) Encyclopedia of the History of Arabic Science, vol. 2, Routledge.
• Saidan, Ahmad S. (1978) The Arithmetic of Al-Uqlīdisī: The Story of Hindu-Arabic Arithmetic as told in Kitāb al-Fuṣūl fī al-Ḥisāb al-Hindī, Reidel.
• Herreman, Alain (2001), "La mise en texte mathématique: Une analyse de l’«Algorisme de Frankenthal»", Methodos 1, doi:10.4000/methodos.45

Education:

• "Number and Arithmetic" in the International Handbook of Mathematics Education, doi:10.1007/978-94-009-1465-0_4.
• Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, Eds. (2001), Adding it Up: Helping Children Learn Mathematics, National Academies Press, doi:10.17226/9822.
• Hart, K. M., ed. (1981) Children's Understanding of Mathematics: 11–16, John Murray. https://archive.org/details/childrensunderst0000unse_p8x0
• Williams, J. D. (1965). "Understanding and Arithmetic – II: Some Remarks on the Nature of Understanding". Educational Research, 7(1), 15–36. doi:10.1080/0013188640070102
• Ma, Liping (2020) [1999], Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States (3rd edition), Routledge.
• M. G. Bartolini Bussi, & X. Sun (Eds.) (2018), Building the foundation: Whole numbers in the primary grades, Springer
  • Ma, L., & Kessel, C. (2018), "The theory of school arithmetic: Whole numbers", in Bartolini Bussi & Sun (2018), pp. 437–462, doi:10.1007/978-3-319-63555-2_18
• Ma, Liping, and Cathy Kessel (2022), "The theory of school arithmetic: Fractions", Asian Journal for Mathematics Education 1(3): 265–284 doi:10.1177/27527263221107162
• David Eugene Smith (1909), The Teaching of Arithmetic, Ginn, https://archive.org/details/teachingofarith00smit/
• Howe, Roger, and Susanna Epp (2008), "Taking place value seriously: Arithmetic, estimation and algebra", Resources for PMET (Preparing Mathematicians to Educate Teachers), MAA online, https://maa.org/sites/default/files/pdf/pmet/resources/PVHoweEpp-Nov2008.pdf

Cognitive science:

• Brian Butterworth (1999) The Mathematical Brain, Macmillan. https://archive.org/details/mathematicalbrai0000butt/
• Lakoff & Núñez (2000), Part I: "The Embodiment of Basic Arithmetic" in Where Mathematics Comes From, Basic Books. https://archive.org/details/wheremathematics00lako/
• Steffe, Leslie P., and Paul Cobb (2012), Construction of arithmetical meanings and strategies, Springer.
• Roi Cohen Kadosh Ann Dowker (2015), The Oxford Handbook of Numerical Cognition, Oxford.

Word problems:

• Wim Van Dooren, Lieven Verschaffel and Patrick Onghena (2002) "The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems", Journal for Research in Mathematics Education, Vol. 33, No. 5, pp. 319-351
• Andrei Toom (2005) "Word Problems in Russia and America", https://web.archive.org/web/20190428212118/http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf

Algebra vs. arithmetic:

• Herscovics, Nicolas, and Liora Linchevski. "A cognitive gap between arithmetic and algebra." Educational studies in mathematics 27, no. 1 (1994): 59-78. doi:10.1007/BF01284528
• Carraher, David W., Analúcia D. Schliemann, Bárbara M. Brizuela, and Darrell Earnest. "Arithmetic and algebra in early mathematics education." Journal for Research in Mathematics education 37, no. 2 (2006): 87-115. doi:10.2307/30034843
• Filloy, Eugenio, and Teresa Rojano. "Solving equations: The transition from arithmetic to algebra." For the learning of mathematics 9, no. 2 (1989): 19-25. https://flm-journal.org/?showMenu=9,2

Computer algorithms:

• Donald Knuth (1997) [1969], Ch. 4 "Arithmetic" in The Art of Computer Programming, Vol 2. (3rd edition)

"Arithmetization"

• Felix Klein (1896) "The Arithmetizing of Mathematics", Bulletin of the AMS, https://www.ams.org/journals/bull/1896-02-08/S0002-9904-1896-00340-2/

multiref2 template

I'd recommend against using this {{multiref2}} template for lists of short "harvnb" references. It puts an unreasonably large amount of space between lines and ends up taking up like 2.5x more space overall than just putting the short references on the same line separated by semicolons, or similar. –jacobolus (t) 21:59, 25 November 2023 (UTC)

Thanks for pointing this out. I replaced it to avoid the extra space between citations. I hope the reference section is tidier this way. Phlsph7 (talk) 09:04, 26 November 2023 (UTC)

I think it's better. The one with more space would work better between longer notes or full citations. –jacobolus (t) 19:45, 26 November 2023 (UTC)

GA Review

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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This review is transcluded from Talk:Arithmetic/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: History6042 (talk · contribs) 15:25, 30 December 2023 (UTC)

Hi I will be reviewing this article.

Rate	Attribute	Review Comment
1. Well-written:
	1a. the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct.
	1b. it complies with the Manual of Style guidelines for lead sections, layout, words to watch, fiction, and list incorporation.
2. Verifiable with no original research:
	2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.	There are hundreds of organized references.
	2b. reliable sources are cited inline. All content that could reasonably be challenged, except for plot summaries and that which summarizes cited content elsewhere in the article, must be cited no later than the end of the paragraph (or line if the content is not in prose).	There's an inline citation for nearly every sentence, and when that is not true, there is one for every paragraph.
	2c. it contains no original research.
	2d. it contains no copyright violations or plagiarism.
3. Broad in its coverage:
	3a. it addresses the main aspects of the topic.	It addresses addition, subtraction, multiplication, division, and exponentiation.
	3b. it stays focused on the topic without going into unnecessary detail (see summary style).
	4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each.
	5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute.	There is no edit war.
6. Illustrated, if possible, by media such as images, video, or audio:
	6a. media are tagged with their copyright statuses, and valid non-free use rationales are provided for non-free content.
	6b. media are relevant to the topic, and have suitable captions.	Every image has a caption.
	7. Overall assessment.

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Should methods / tools be a top-level section?

I still am not entirely sold on all of the high-level organization here. The subsections about 'Numeral systems' and 'Kinds' of numbers seem fundamentally unalike and don't really fit in the same top-level section in my opinion. The numeral systems section is also kind of a mess in my opinion; tally marks are not really a "numeral system" except as a kind of anachronistic modern imposition, and non-positional vs. positional numeral systems are fundamentally different (in particular, non-positional systems such as Roman numerals were not ever used with written "arithmetic" methods, but were more like a written input/output for calculations done with fingers, tokens, or some kind of counting board; the computer-programming analogy I sometimes use is that these are more like a serialization format than a calculation tool). Likewise, the binary system is also not like Hindu–Arabic numbers, in the sense that neither humans nor computers commonly write "1101" or whatever and do arithmetic with those written symbols; instead humans write numbers in decimal or hexadecimal, and computers store/transmit bits of data and operate using logic gates rather than writing. So lumping these all together the way is currently done is in my opinion sort of misleading.

I wonder if it would work better to leave 'Numbers' as a top-level section and eliminate 'Kinds' as a heading, and then split information about numerals into multiple subsections of a new top-level section. My proposal would be to make some kind of top-level section about "Methods" or "Tools" or "Approaches" or similar (not sure the best name), which could include 2nd-level subsections on e.g. finger counting, tally marks / counting by tokens, positional counting boards / counting rods / bead abacuses, positional written arithmetic, mechanical calculators, electronic calculators (mentioning binary), mental arithmetic. @Phlsph7 what do you think? I don't want to stomp too much on your hard work. –jacobolus (t) 18:47, 3 February 2024 (UTC)

@Jacobolus: Thanks for sharing your improvement ideas. I think your suggestion of presenting different forms of arithmetic tools/methods in a common place could work. I'm not sure that this topic is important enough to have a top-level section. Some of these tools/methods are mentioned in the sources but they are usually not discussed in great detail. I'll have a look to see if I can come up with something.

Numeral systems are discussed in various overview sources on arithmetic, like Romanowski 2008, EoM staff 2020a, Nagel 2002, and Lockhart 2017. I think it makes sense to discuss them somewhere. Since they represent numbers, I thought having them as a subsection of the section "Numbers" is the most obvious choice. Tally marks are used as an example of a unary numeral system following Mazumder & Ebong 2023. As an example, it is not essential. It was intended to help the reader by making the discussion more concrete. We could replace it with another example of a unary numeral system. Various overview sources of arithmetic discuss the binary system, like Lockhart 2017 and Nagel 2002.

There are different ways to organize the material into a section structure. They all have their advantages and disadvantages and there is probably not one single "right" way. It's usually easier to make several smaller changes within the current structure than to make a more radical reorganization, which often requires rewriting various parts, ensuring that they properly represent the sources, and taking care not to introduce new errors in the process. Unless there is a weighty reason otherwise, I would suggest that we first try to implement several smaller changes to address specific problems one at a time. Phlsph7 (talk) 09:44, 4 February 2024 (UTC)

I saw that you separated mental arithmetic to form a distinct subsection. I followed your lead and made a first attempt to implement your idea by expanding this subsection to cover tool use in general. It's probably not exactly what you had in mind but it goes in the same direction. Phlsph7 (talk) 14:21, 4 February 2024 (UTC)

Maybe a title like "systems and tools" would work better. As I said, I'd put this immediately after the section about "numbers" and merge the "numeral systems" section into there, split into multiple subsections; the part about "unary" can go into a subsection about counting including tally marks, piles of tokens, etc., the part about "binary" is best contextualized in a later section about electronic calculators, and the parts about positional vs. non-positional number systems should be separated as they are fundamentally different in their purpose and uses.

As another example, the slide rule was arguably the most important method of computation throughout the 18th–20th centuries, and is currently not mentioned on this page; more generally there were other analog "instruments" used for calculation such as the sector and various scales used together with a pair of dividers, which were essential calculation methods of the 16th–19th century. If you go earlier than that, most serious calculation throughout history was done with some kind of counting board, which is currently unmentioned on this page; counting rods are also unmentioned, and the discussion about bead-frame abacuses is sort of misleading and limited.

A tool used for more precise calculations, also essential throughout math, science, and engineering for centuries, was printed tables of trigonometric functions and logarithms, also not mentioned here. While on the subject of tables, tables for things like reciprocals, squares, etc. were an important calculation aid in ancient Mesopotamia.

I had in earlier conversations a few months ago conceived of the scope of this article as potentially being mostly about positional decimal pen-and-paper arithmetic, with other topics sent to other pages like calculation, computation, history of computing, mechanical calculator, abacus, etc., but the scope you settled on here is very broad; in that case we should actually try to cover that full scope (at least in a compressed summary; we don't need to be excessively detailed about any particular part). –jacobolus (t) 19:32, 4 February 2024 (UTC)

Those are all good expansion ideas, I tried to fit the main ones into the current setup. I'm still hesitant to go for a full-scale implementation of your ideas since the sources that I'm aware of give more importance to numeral systems than to calculation instruments. I'm not in principle against the suggestions but I fear that they could conflict with WP:PROPORTION. Maybe the root of the disagreement is that our outlooks are based on different sources that present the relative importance of those topics differently. I'll respond to your comments below later. Phlsph7 (talk) 10:49, 6 February 2024 (UTC)

I guess what I mean is, I think "numeral system" as an overarching concept is actually somewhat off topic, in the sense that you can't really do "arithmetic" in a meaningful sense with a "unary numeral system" per se, and the concept of "unary numeral system" is an anachronistic modern imposition on a range of past practices which were much more flexible and creative, and frankly not really a "number system" at all. What's really important about it is the direct representation of natural numbers by an equivalent count of marks or tokens (e.g. pebbles or shells). No historical culture that we know about ever limited itself to only representing numbers as tally marks, which are a record-keeping tool more than a calculation tool.

I don't think your summary here accurately reflects the sources you mentioned.

• Lockhart's book doesn't talk about number systems in at all the way this article does: he tells a kind of (partly imagined) story about the different ways of representing numbers and their relationships, but doesn't try to strictly categorize or label them.
• The Encyclopedia of Mathematics article (aside: you should credit this to the authors A. A. Bukhshtab and V. I. Pechaev not to "EoM staff"; their article is unchanged from their original) does not mention "unary numerals" or "numeral systems" at all.
• Neither Nagel nor Romanowski discusses these topics either (and in my opinion both are poorly written and poorly organized mishmashes aimed I assume at an audience of children or non-English-speakers which should be avoided as sources for Wikipedia).
• Mazumder & Ebong is a weird source. It's a book ostensibly about circuit design, and the section about miscellaneous number representations doesn't really connect to the rest. It reads to me like they had a page count they were aiming for and were just padding it out with fluff. YMMV. I'd recommend avoiding this as a source about "arithmetic" per se (it might be a good source about the concrete circuits needed to implement hardware for binary-coded decimal arithmetic; I didn't read those parts carefully).

–jacobolus (t) 17:31, 4 February 2024 (UTC)