# Is arithmetic mathematics and can we answer questions about it?

Yesterday, I answered this question (later deleted, see below for the text) and was a little surprised when the question was closed. A user in the comments stated, "This is not mathematics. It is arithmetic. Voting to close." The question was eventually closed. However, Wikipedia defines arithmetic as

a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them.

Furthermore, the FAQ states, "Mathematics - Stack Exchange is for people studying mathematics at any level and professionals in related fields."

This meta post also seems to come to a consensus that arithmetic questions are allowed on the site. And finally, the original question was not just arithmetic—it was dimensional analysis, which can get pretty tricky in certain circumstances. So can questions about arithmetic be answered on the site?

The question that is discussed

Title: Say, hypothetically, there were 4 seconds in 1 minute instead of 60 seconds. How many seconds would 24 hours (1 day) be?

Body:

Let 1 minute = 4 seconds.

Let 1 hour = 60 minutes.

Let 1 day = 24 Hours.

How many seconds would be in 1 day?

• Yes, arithmetic is clearly mathematics, and it is very puzzling as to why the question was closed with that particular close reason. It's not surprising that it was closed due to the level, because that happens all the time on this site. It's true that in principle the site is for those studying mathematics at any level, but in practice lower level questions get hammered with downvotes and are closed. – Matt Samuel May 30 '20 at 18:14
• I think one can make a good argument to close this post independent of the issue of "arithmetic" - the post lacks context. It's a PSQ with no explanation on why the question matters, what's interesting about it, or why the asker is having trouble with it. – KReiser May 30 '20 at 18:15
• Yes, studying mathematics though, granted at any level, so first year undergraduate is fine, too. Note that if "any level" would mean anything at all, then what is the point of "professionals in related fields"? Why would a professional in an unrelated field not be included while the mathematics an average 10 year old is supposed to master is supposedly completely fine? Contrary to what is claimed in the the thread linked there is ambiguity there and imo the description as written only makes any sense if it means something else then what some want it to mean, but that's a tangent. – quid May 30 '20 at 18:40
• "Is arithmetic math and can we answer questions about it?" (title of this question): To address the second conjuct of your title: You certainly can answer well written questions with context and other details, that concern arithmetic. Was the question you answered well written with context and other details? – amWhy May 30 '20 at 19:33
• @amWhy No, it was not, and it should stay closed for not providing context. However, the reason for closing the question in the comments was, "This is not mathematics. It is arithmetic. Voting to close." I'm not asking for it to be reopened, I'm just concerned about that specific reason. – N. Bar May 30 '20 at 21:44
• As I clearly stated, N.Bar, I was addressing the second conjunct of your title question: "Can we answer questions about it." Yes you can, but only when you answer well written questions with context and other details. The question you answered doesn't meet that criteria. (You asked a conjunctive question: "Is x okay, AND is y okay?" I commented only on the "Is y okay" portion of your title.) – amWhy May 30 '20 at 22:33
• As of now there are 3 reopen votes on linked question. That's probably more strange than the issue being discussed here. – Paramanand Singh Jun 1 '20 at 1:38
• @MattSamuel The phrase at any level must be interpreted in context of the site, where people 13 years of age or younger are prohibited from participating by the terms of service. Admittedly I would have no qualms about a 12-year-old version of Terry Tao posting here, so there's that. – Jyrki Lahtonen Jun 1 '20 at 4:38
• Anyway, in my opinion questions that Wolfram Alpha can answer could/should be declared off-topic. – Jyrki Lahtonen Jun 1 '20 at 4:43
• "Mathematics is the queen of the sciences, and arithmetic the queen of mathematics." – bof Jun 1 '20 at 9:31
• "Question askers want to understand a concept." I think most of them just want the answer. – Joel Reyes Noche Jun 1 '20 at 15:14
• @N.Bar Quoting Franklin here feels a bit misguided given that closed questions can be immediately edited and, if the edit cured the problem, also reopened. This is not a (criminal) court of law. Rather a game of football, where we have refs calling "offside" and whatnot. Furthermore, here the crowd can talk it over with a ref, and also overrule, when judged prudent. Much gentler. The goal of these rules is not to punish, but rather to keep the game interesting for all the parties. – Jyrki Lahtonen Jun 2 '20 at 3:10
• @kludg I'm afraid that statement marks you as ignorant about number theory.. – Jyrki Lahtonen Jun 2 '20 at 15:33
• @kludg Having had a reasonable Google, I found almost no versions of that quote with "arithmetic", instead they used number theory. Of course, the original quite was in German, and apparently "die Zahlentheorie" is the relevant topic, and Google translate translates this into "number theory". – user1729 Jun 6 '20 at 9:01
• "from what I recall, Gauss' work didn't go far beyond modulo arithmetic and a 'first course in number theory'" @user1729 I suggest you refresh your memory, perhaps by having a look at what Gauss actually did. The theory of cyclotomy goes well beyond a first coourse in Number Theory, as does biquadratic reciprocity, and class numbers of quadratic fields (although presented in the context of quadratic forms, rather than number fields), for starters. – Gerry Myerson Jun 7 '20 at 0:46

The question you mentioned should be closed. The argument given for closing it is arguably prone to cause confusion and dissent though.

In our guidelines How to ask a good question. it is mentioned that some context should be provided.

Some consider lack of context as a reason to close. There is a community specific close reason for just this:

Missing context or other details

Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.

It links again to How to ask a good question. (in this case a particular answer).

However, would it follow the guidelines, the mere fact that a main part of the work to do is a relatively basic computation would not, or at least should not, result in its direct closure.

Note that even the top-answer in the thread you invoke as justification says (my emphasis):

If someone asks a well-crafted question about elementary school level mathematics that is not obviously answered by a cursory internet search and has not been asked before on this site, to me it seems clear that this should be okay.

To put it differently, yes, arithmetic question are not off-topic in general, but the specific one should still be closed as it does not follow the guidelines regarding context.

• The question has been deleted, so only users with more than 10000 reputation points can see it. This means rank-and-file users can't see directly what was wrong with it any more. – Robert Furber Jun 13 '20 at 3:01
• That is indeed not ideal; I included the text in the meta-question, which seemed better than to put it in my answer. Thanks for the heads-up. – quid Jun 13 '20 at 10:34
• Why is multiplication of integers commutative?
• Why does long division work the way it does?
• Does rounding too early make the bottom-line answer wildly wrong?
• Is there a more efficient way to find greatest common divisors than Euclid's algorithm?
• How can one find the smallest number representable in two different ways as the sum of two fifth powers?
• Can the sum of two odd numbers be odd?
• Why should fractions be in lowest terms?
• After subtracting page numbers, why does one add $$1$$ to get the number of pages?
• If $$a_1$$ and $$a_2$$ numbers leave the same remainder when divided by $$8$$, and similarly for $$b_1$$ and $$b_2$$, then will $$a_1 b_1$$ and $$a_2 b_2$$ leave the same remainder when divided by $$8$$?
• In the fraction $$\dfrac{18\times 24}{6\times5}$$, shouldn't the $$6$$ be canceled against both the $$18$$ and the $$24$$, rather than just one of them?
• How can one convert a repeating decimal to a fraction in lowest terms?
• If a base-$$10$$ expansion of a number repeats, will the base-$$12$$ expansion repeat?

All of these are questions on arithmetic. And all are questions on mathematics. And they are appropriate questions for m.s.e.

• +1 especially for the mantra of counting pages via subtraction of page numbers. I often do it to calculate the number of days passed between two dates (including both) in a month. – Paramanand Singh Jun 6 '20 at 12:22
• Great questions to ask on an interview. Even to non-mathematicians, most can be answered by deduction and will show reasoning skills. But the best answer should always be: check MathExchange :). – Abel Jun 6 '20 at 19:12
• The one about fifth powers, "Can the sum of two odd numbers be odd?" and "If a base-10 expansion of a number repeats, will the base-12 expansion repeat?" are number theory, not arithmetic. – Rosie F Jun 7 '20 at 10:38
• Many of these are ok. Some were taught to me and class mates before our 13th birthday, and I would vote to keep them out. I would also keep modular arithmetic out of this list (we have a separate tag for that), but that is just my personal point of view according to which modular arithmetic is better kept under the tag elementary-number-theory. Like Paramanand Singh, I think the example on counting pages, even though school level, is ok for general usefulness. Mind you, all these are relatively weak questions in the sense that they are highly likely to be duplicates. – Jyrki Lahtonen Jun 7 '20 at 20:09
• My list of good fit arithmetic questions would include the likes of 1 and 2. – Jyrki Lahtonen Jun 7 '20 at 20:13
• @RosieF wikipedia > The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory. – Džuris Jun 13 '20 at 13:15
• The issue is not the history of the usage of the word "arithmetic" and what meanings people used to use it for, but whether questions on arithmetic (with its present meaning) should be on math.SE. – Rosie F Jun 13 '20 at 13:29

I was the one who posted the comment, and I stand by my close vote. As I see it, the question was simply "What is $$4\times 60\times 24$$?" And that is not mathematics, as I understand the term. I would not like to see the site cluttered up with trivia like this.

• I agree that the post needed to be closed, and your close vote is not the problem. Rather, the problem is the reason you cited for closure, which was was over-generalized, and inappropriate w.r.t. any/all arithmetic questions. There was no reason for you to slam any/all arithmetic questions (invalidly claiming arithmetic is off-topic on math.se), some of which are not trivial. – amWhy May 31 '20 at 17:02
• @amWhy, just so we're on the same page: can you point me to a good question that is purely arithmetical? I promise not to vote to close it :-) (I did have a brief scan through the arithmetic tag, but I didn't find any purely arithmetical questions there.) – TonyK May 31 '20 at 17:05
• @TonyK I also agree that the post should be closed (I never said it should be reopened). I'm just concerned with the reason cited. As for a good question that's purely arithmetical, hows this – N. Bar May 31 '20 at 17:10
• I think realizing that the question is essentially "What is $4\times60\times 24$" is quite different from being able to answer "What is $4\times60\times 24$" per se. – user9464 May 31 '20 at 17:46
• Can you elaborate, @T.S, because I'm having trouble making sense of your comment. – amWhy May 31 '20 at 17:55
• @TonyK: Does this count? – user21820 May 31 '20 at 18:03
• @user21820: Yes, that counts, as does the link in N.Bar's comment. Both these questions are about the best way to evaluate arithmetical expressions; this goes beyond simply asking us to multiply three small integers together. – TonyK May 31 '20 at 18:12
• – amWhy May 31 '20 at 19:22
• "This is not mathematics, this is arithmetic." "This is not mathematics, this is algebra." "This is not mathematics, this is geometry." " This is not mathematics, this is analysis." One statement is as nonsensical as another. – bof Jun 1 '20 at 7:05
• "This is not mathematics – this is theology!" Famous quote, attributed to Gordan (quite possibly incorrectly) in response to David Hilbert's proof of Hilbert's basis theorem. webusers.imj-prg.fr/~michael.harris/theology.pdf – Gerry Myerson Jun 6 '20 at 1:34
• @TonyK I have written many answers to purely arithmetic questions; some of them are among my favorites. This search produces many that might satisfy your request, although not all will. But in particular: math.stackexchange.com/questions/703771 math.stackexchange.com/questions/757807 math.stackexchange.com/questions/683774 math.stackexchange.com/questions/376365 – MJD Jun 6 '20 at 18:32
• @MJD: Yes indeed, those are all good questions. They ask how to do arithmetic. They don't ask, effectively, What is $4\times 60\times 24$? – TonyK Jun 6 '20 at 18:47
• I agree with your point. I was only trying to address your request for “a good question that is purely arithmetical”. – MJD Jun 6 '20 at 18:49
• @MJD: See my comment ("Yes, that counts..."). – TonyK Jun 6 '20 at 19:56
• I strongly disagree with the (currently 2) votes to delete here! This answer explains the actions the question is about by the user who did them, and so should remain. You may not agree with their opinions, but then the standard action is to downvote it, not delete it. – user1729 Jun 6 '20 at 21:24

"This is not mathematics. It is arithmetic. Voting to close."

I am afraid such a definition of "mathematics" is too narrow to be valid and cannot be used to justify a close vote.

As for the context business mentioned in quid's answer and this user's comment, I would say that the "sin" in your mentioned question is the level of difficulties.

Some users consider relatively low-level questions are less useful for the site and thus have a much more strict standard to so such questions. Look at the sharp contrast between your mentioned question and this two-line linear algebra question: Do eigenvalues depend on the choice of basis? and the review history of the latter question, which even made a conceptual mistake.

Yes, the second question may be more likely to be "useful" for future readers. But sneering a question like the first one is a clear violation of the intended scope of the site. Moreover, I don't think the first question could have been written in a different way, within the ability of the asker, that could have made the post survive:

• adding the "context" of where the question is from like "I have been staring at the clock for the whole afternoon and think that ..."?
• adding the definitions of seconds/minutes/hours from a dictionary?
• any attempt? "I don't even know how to start" since this is not even a damn homework problem.
• etc.

Context alone can hardly explain why it is hard for such questions to survive.

• Yes, you are right that the question at issue is unlikely to be helpful for future visitors since it is highly localised, whereas the linked question in your answer can at least serve as a duplicate target, though I think both posts are equally badly written. – YuiTo Cheng Jun 1 '20 at 14:47
• I disagree regarding the impossibility to add context within the users ability. Arguably there is no guarantee that the post would have survived even with that context, but the chances would have been improved. To ridicule the idea is unhelpful. Things that could be done, which ought to be within the users ability: (a) make clear if the problem is given to them (assigned exercise, seen in a textbook, etc) or if they came up with it by themselves (out of idle curiosity or for a more specific reason). (b) make clear if they know how to solve the classical case. – quid Jun 2 '20 at 12:01
• It is not clear to me what the exact point of "Context alone can hardly explain why it is hard for such questions to survive." is. It is well-known, reasonable (in my opinion), and should even be documented somewhere (I think some, including me, wrote about it at times explciitly) that users when voting of course evaluate the question according to multiple criteria, including context, formatting, genuine interest, general applicability and potential to contribute to the site as a resource for knowledge, etc. Which criteria exactly and how they balance them will depend on the user. – quid Jun 2 '20 at 12:24
• Regarding scope and "is a clear violation of the intended scope of the site." As commented on the main post this is far less clear than many want to make it look. I often don't get a response to my question, but I'll try again: What is the relevance of mentioning "professionals in related fields" in the target audience when the site is supposedly clearly intended for everyone and anything at all that might fall under the purview of mathematics? As far as I can tell there would be none. Therefore "studying mathematics at any level" ought to something else than "everyone and everything." – quid Jun 2 '20 at 12:33
• To understand such things historical context is sometimes useful. First there was a site, MathOverflow, for people studying mathematics at the graduate level and professionals in the field. This was complemented by a more open site for people studying mathematics at any level and professionals in related fields. The target audience of the site is not everyone at all. This is no contradiction to there not being any strict cut-off on the level and that we might also accept all kinds of questions from all kinds of persons. But the target audience is more narrow. – quid Jun 2 '20 at 12:41