Talk:Arithmetic

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)
 * 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 История_арифметики &#91;machine translation&#93; ), with just a summary (no more than maybe 2000 words) at . 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)
 * 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 Arithmetic entry in the Gale Encyclopedia of Science states that . Its entry includes discussions of different types of numbers (natural, rational,...), numbering systems, and axioms.
 * From the Facts On File Encyclopedia of Mathematics:
 * 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: 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 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 ($$3+\frac12= \frac 72$$ 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 . 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 : Arithmetic is and includes the . From : Number theory is . 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)

multiref2 template
I'd recommend against using this 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)

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)


 * 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)
 * Please correct me if I'm wrong, but doesn't Lockhart have several chapters dedicated to the problem of the representation of numbers? For example, the chapter "Language" talks using rocks, sticks, or fingers, to scratches on bones, to verbal utterances (i.e., words), to abstract symbols to represent numbers while stating that the story of arithmetic ... in a large part ... is a history of representation. The chapter "Repetition" contains a detailed discussion of tally marks and how they lead to more complex representational systems, which is continued in the following chapters. The chapter "Egypt" discusses the Hieroglyphic numerals and the chapter "Rome" discusses the Roman numeral system, similar to our section. The binary system is also explicitly discussed in a later chapter. When I have the time, I'll take a more detailed look at the other sources. Phlsph7 (talk) 08:51, 7 February 2024 (UTC)
 * Yes, Lockhart's whole book is more or less about the representation of numbers (what Lockhart means by arithmetic is "the art of counting and arranging things").
 * But my point is that his book is not really any clear precedent for the way this article was presenting the subject, as I took your implication to be in your previous comment. It's a fine book, but in my opinion its scope or organization shouldn't be adopted as the basis for an encyclopedia article: it's a meandering personal essay written in literary style, rather than an organized encyclopedic overview trying to summarize scholarly consensus. –jacobolus (t) 16:00, 7 February 2024 (UTC)
 * Sorry for the misunderstanding, my intention was not to argue that we should reorganize our article to match Lockhart's approach but that the emphasis he gives to numeral systems throughout his book underlines that this topic is important enough to merit its own subsection.
 * In regard to EoM staff 2020a: they discuss the hieroglyphic, Babylonian, Greek, and Indian-Arabic numeral systems and consider the advantages and disadvantages of the different systems. I was also thinking about mentioning the original authors (A. A. Bukhshtab and V. I. Pechaev) but I'm not sure to what extent the article was modified and whether other authors would need to be mentioned as well (see, for example, their revision history). At the bottom of the page, they suggest citing the entry without an author and state that it is an adapted version of the original one. I'm not sure if there is a good way to cite sources using the sfn format without any author at all, so I used "EoM staff" as a compromise. Phlsph7 (talk) 09:04, 8 February 2024 (UTC)
 * None of the EoM articles I've ever looked at was nontrivially modified away from the original (you can see the page history by clicking somewhere down at the bottom IIRC); the changes are stuff like fixing OCR typos and changing the rendering method for mathematical formulas. I would cite the original authors, and not bother figuring out who made minor typo fixes.
 * If you want another good "encyclopedia" source about fairly basic topics (though it's not organized into named articles in quite the same way as a traditional general encyclopedia), the VNR concise encyclopedia of mathematics (2nd ed 1989; originally published in 1965 in German) seems excellent. –jacobolus (t) 17:39, 14 February 2024 (UTC)
 * At least for the articles we use, there does not seem to be any significant difference so I followed your suggestions to use the original authors instead of "EoM staff". I had a look at the VNR concise encyclopedia of mathematics. Its first few entries would have been quite useful for sourcing this article. Phlsph7 (talk) 09:39, 15 February 2024 (UTC)
 * Another good (albeit relatively old) source: Halsted, On the Foundation and Technic of Arithmetic. –jacobolus (t) 21:23, 15 February 2024 (UTC)

Approximate arithmetic
Hello and thanks for adding the detailed and accessible discussion of how to deal with measurement uncertainty. Is the term "Approximate arithmetic" a technical term found in the reliable sources for this form of arithmetic? I didn't find it in your first source (Drosg 2007, pp. 1–5) and a short google search did not turn up much either except for electronic components known as approximate arithmetic circuits. Phlsph7 (talk) 09:55, 7 March 2024 (UTC)


 * "Approximate arithmetic" is just a descriptive phrase, not an existing jargon term. The section heading could certainly change. –jacobolus (t) 15:21, 7 March 2024 (UTC)
 * I do not know any better title for the present content of this section. However, this section deals with two very different problems, the approximations and errors in measurement, and the approximations resulting from the representation of numbers in a computer, mainly through floating-point arithmetic. IMO, this deserves to be split into two different sections, which could be called respectively "Approximations and errors" and "Computer arithmetic".
 * Computer arithmetic is a well established area that belongs to both computer science and mathematics, has an annual international conference (ARITH) and two standards (IEEE 754 and GNU GMP). So, it deserves to have a specific article. Unfortunately, Computer arithmetic is a redirect that was targetted to Arithmetic logic unit; I changed the target to Floating-point arithmetic, but it would much better to have an article that covers all aspects of this subject. Having here a section "Computer arithmetic" could be a good starting point for this lacking article. D.Lazard (talk) 15:36, 7 March 2024 (UTC)
 * I don't think you want to lump "computer arithmetic" as a section here, as computers do arithmetic of a wide variety of types, including symbolic arithmetic, integer arithmetic, modular arithmetic, floating point arithmetic, complex floating point arithmetic, matrix arithmetic, etc.
 * The section I called "approximate arithmetic" could plausibly be split or shortened a bit. I added it because I didn't feel that "real arithmetic" was really the appropriate heading for the subject, and thought concepts like "significant digits" etc. should be at least slightly described, since they are essential context for understanding scientific notation, which was included before.
 * Aside: In looking around, the existing articles Error analysis (mathematics) and Propagation of uncertainty do quite a poor job at providing explanations which are complete and accessible to a lay / student audience (or even an audience of scientists or engineers, frankly), and the article Significant figures is a weakly sourced and somewhat confusing mess. Observational error and Measurement uncertainty don't particularly clearly distinguish these terms and Instrument error is essentially a stub. Accuracy and precision talks about measurements per se but doesn't get into error propagation at all. I should maybe go ask at science/statistics/... wikiprojects if someone can figure out a way to insert a high school accessible explanation somewhere on Wikipedia in a place where folks who need can find it. –jacobolus (t) 15:54, 7 March 2024 (UTC)
 * I agree that Computer arithmetic should be a dedicated article though. –jacobolus (t) 16:08, 7 March 2024 (UTC)
 * We have to be careful about how we name the sections. If we call a section "Approximate arithmetic" then readers assume that this is an established technical term for a type of arithmetic. If have a paragraph on floating-point arithmetic and put it into a section called "computer arithmetic" then readers get the impression that this is all or most there is to computer arithmetic.
 * Currently, we have a paragraph in the section "Others" that discusses how arithmetic can deal with uncertain measurements and errors using interval arithmetic and affine arithmetic. This would probably be the best place to include the newly added information. We might have to condense it down a little but topic-wise, this seems to be the best match. What do you think? Phlsph7 (talk) 17:08, 7 March 2024 (UTC)
 * I don't think readers are going to assume this is an "established technical term". "Real number arithmetic" is also not really an established technical term, but just a descriptive phrase which people most often (very misleadingly) use to mean binary floating point arithmetic which is decidedly not about real numbers per se, so by that standard shouldn't be a heading either.
 * If there's going to be a section about "real numbers" it should more clearly describe the way arbitrary real numbers cannot be represented as completed strings of decimal digits and how non-definable numbers cannot really be represented at all, so that the real number system is not really an arithmetic system at all, in any proper sense. Some real numbers can be represented as e.g. programs for generating decimal digits (which are very problematic for arithmetic because it takes potentially unbounded amounts of work to compute even a single digit of the result of a basic arithmetic operation applied to a pair of such numbers, related to the "table maker's dilemma") or programs for generating terms of a continued fraction (Bill Gosper had a proposal for doing arithmetic with such programs as part of HAKMEM and expanded into a longer unpublished manuscript which has been somewhat influential), or programs for generating rational approximations within any specified tolerance, called computable numbers. Most of what might be called "real number arithmetic" consists of symbolic computations (currently covered on Wikipedia at Computer algebra).
 * The (idealized) assumption of real number arithmetic is the basis for many computational geometry algorithms, but since real number arithmetic doesn't actually exist in practice, this causes severe problems because these algorithms end up pathologically breaking when implemented naïvely using IEEE floats. This has led to various kinds of workarounds, e.g. "exact geometric computation", which means something like "as precise as necessary to exactly describe the geometry" (see also Shewchuk's "robust predicates").
 * It would be good to have some clearer discussion of the distinction between floating point arithmetic of a specific precision, arithmetic of expanded precision (sometimes implemented using integer hardware or sometimes implemented using floating point hardware), arithmetic of arbitrary precision, etc.
 * D.Lazard has a decent point that splitting separate sections about uncertainty (not sure the right title) vs. computer binary floating point could be clearer. I would merge material about interval arithmetic into the section about uncertainty.
 * While we're here though, the high level organization of "Types of arithmetic" seems substantially problematic to me. Maybe we can think about possible alternative organization schemes. –jacobolus (t) 17:41, 7 March 2024 (UTC)
 * The part about non-definable real numbers sounds interesting. Do you know of a source that could be used to support a sentence on this topic?
 * Regarding the name of the subsection, I'm not opposed to using D.Lazard's suggestion "Approximations and errors" instead of "Approximate arithmetic". In principle, we could leave the paragraph on floating-point arithmetic there since it is used as an approximation for computers. The paragraph on interval arithmetic and affine arithmetic could be included there as well. If you feel that this topic does not fit well under the main heading "Types of arithmetic", we could change that heading to "Areas", which is sufficiently vague to include all the subsections. Phlsph7 (talk) 09:17, 8 March 2024 (UTC)
 * The "types of arithmetic" name of the section is probably okay, and I probably shouldn't recommend a more significant reorganization unless I can think it through and be more specific and concrete. But even within this section, I'm not sure about what the best flow is to make the article read smoothly.
 * I think it's helpful somewhere to give readers a sense that numerical or mathematical expressions are manipulated or evaluated (1) to find mathematically exact concrete results by following specific set of rules on various structured data types (stuff like integers, fractions, quadratic surds, finite-symmetry-group elements, graphs, matrices with integer entries, etc.), which often boil down to something like fancy counting (this is the subject of discrete mathematics) (2) to make symbolic combinations of abstract quantities which might remain as undetermined symbolic variables or might represent a specific quantity but which might not be exactly representable in the common ways we ordinarily represent numbers (this is the subject of big parts of mathematics, but in particular mathematical analysis), (3) to make finite (approximate) calculations of mathematically exact quantities that we might not have any nice closed-form symbolic expression for, but can in principle be approximated to any desired precision (this is the subject of numerical analysis), (4) to combine, transform, or model inherently uncertain physical quantities, usually done in terms of some finite-precision approximate calculations using the tools of #3 (this is the subject of science, engineering, and statistics).
 * If we have sections about integer arithmetic and rational arithmetic, I don't quite know how to make a parallel section about "real arithmetic" because it's fundamentally different in character (#2 vs. #1 in my list above). I'm not sure what the right section title for that should be but I think the focus should be about how real number "arithmetic" is inherently abstract and theoretical, rather than concrete and practical. To make it practical we need some kind of method of approximation, and there are a variety of approaches that might be taken.
 * Then I think it's important to keep some section about uncertainty/error. But maybe this should be separate from sections about finite representations of infinite things, since it's a somewhat different idea again (#4 vs. #3 in my list above).
 * @D.Lazard does my discussion here make any sense, or am I just confusing myself? How do you think we should try to order / organize these sections? –jacobolus (t) 18:15, 8 March 2024 (UTC)
 * I renamed the new subsection as suggested and moved the main part of the discussion of interval and affine arithmetic there. Do you know if the four-fold distinction you mentioned is discussed like this in reliable sources? I know that concepts like arithmetically definable number and computable number are discussed. Their definitions are not that straightforward so I'm divided on whether we should explain them in the article. Phlsph7 (talk) 18:09, 9 March 2024 (UTC)
 * I'm not sure what kind of sources would talk about this in a very deliberate and organized comparative fashion. I'm also not at all an expert in definable or computable numbers, etc. (you can see I edited my comment a few times to try to get it sort of right, but I wouldn't trust the above to not have some inaccuracies). –jacobolus (t) 20:54, 9 March 2024 (UTC)
 * I have moved Computer arithmetic into a stub. For the moment, it does not contain much more than the last paragraph of, and the prose of the latter is much better. I agree that the stub must be extended to include the various arithmetics that you cite. However, as the core of computer arithmetic is primarily floating point arithmetic and multiple-precision arithmetic, I do not see any objection for having here a section on computer arithmetic, with a template . D.Lazard (talk) 17:12, 7 March 2024 (UTC)
 * I agree that there should be an article "Computer arithmetic". There is already a wikidata item (and corresponding articles in 2 languages, but not much in them) and a category (which could be useful to write the article). — Vincent Lefèvre (talk) 14:08, 8 March 2024 (UTC)