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A while back, I came across this question. At the time, the question read:

What is $0 \neq 1$ in the context of Algebraic Axioms?

In this wiki link, the author included $0 \neq 1$ as the axiom that relates addition, and multiplication.

But, what I see the most is just the distributive law. What does $0 \neq 1$ mean? and how it is related to addition and multiplication?

I thought the question was very clear, so I answered it. I wrote:

Well, the statement $0 \ne 1$ means that $0$ is not the same thing as $1$. In other words, it means that the additive identity ($0$) is not the same as the multiplicative identity ($1$).

This statement is related to addition because it mentions the additive identity ($0$), and it's related to multiplication because it mentions the multiplicative identity ($1$).

I came back a while later, and I was surprised to see that my answer had received 3 downvotes and no upvotes. (I later deleted it.) So presumably there was something seriously wrong with it... but what?

I'm guessing that either I misunderstood the question, or I understood it correctly but my response didn't clarify anything.

Of the two possibilities, I think it's more likely that I misunderstood the question. In retrospect, "what does $0 \ne 1$ mean?" is a strange question, and I should have realized that someone who asks that question probably actually wants to know something else. So the only question at that point is, what is the intended question?

Matthew Daly's answer seems to have interpreted the question as, "Why is $0 \ne 1$ included as one of the ring axioms?" That answer got several upvotes and was accepted, so evidently that was the correct interpretation of the question. However, I'm very confused by that. I really don't see how that interpretation makes sense, given the question as written—to me, the questions "What does $0 \ne 1$ mean?" and "What was the motivation behind including $0 \ne 1$ in this list?" are completely different and unrelated questions.

So, in the future, when I come across questions similar to "What does $0 \ne 1$ mean," how can I determine the intended meaning, as Matthew apparently did for this question?

On the other hand, perhaps I did understand the question correctly, but my response failed to say anything that would help a student to understand anything. I admit that if someone doesn't know what $0 \ne 1$ means, then they're unlikely to have a strong understanding of the phrases "the additive identity" and "the multiplicative identity" either. Perhaps my answer would have been better if I'd given a more detailed explanation of just what the symbols $0$ and $1$ mean in the context of a ring, and of the resulting significance of the statement $0 \ne 1$.

So, I'm wondering:

  • Did I misunderstand the question? If so, how could I have known what the correct interpretation was in this case?
  • Did I understand the question correctly, but fail to provide any useful explanation?
  • Or was the problem something else entirely?
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    $\begingroup$ I didn't downvote, but I'd guess the reason is that you chose the most trivial interpretation possible for the question - which does not seem to be what was intended on careful reading. I've seen many other answers like that get downvoted even more heavily. $\endgroup$ – Bill Dubuque Nov 26 at 23:47
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    $\begingroup$ mod 1 they are the same... $\endgroup$ – Roddy MacPhee Nov 27 at 1:15
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    $\begingroup$ There's this terrible habit by people who ask questions to give priority to the title. Your interpretation is closer to the truth in my book. What the questioner intended and what they ask aren't aligned, and of course, it's not our responsibility to guess intentions. $\endgroup$ – Git Gud Dec 1 at 20:27
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The community can be fickle sometimes. Especially with questions that aren't very detailed, folks can get protective of their notions of what the poster is actually looking for. I focused on "What does $0\neq1$ mean?" while you focused on "How is it related to addition and multiplication?" Looking back on it, the real question should have been "Why would someone paste the ring axioms into an Algebra II wikibook and then not cut out the axiom that has no impact on Algebra II?"

I saw an even more profound example this morning.

How to proof that $2\sqrt{3}$ is greater than $\pi$

The first two answers got two downvotes apiece despite being completely correct. One of those answers was from User, who has 109K rep and knows a thing or two about answering questions. Why? Evidently, without any prompting from the OP, the community decided that only a geometric answer would do even though anybody who studies math even casually should know $\sqrt3$ and $\pi$ to two decimal places.

I don't get why someone would downvote a correct answer, and that goes double when they don't leave a comment explaining themselves. But everybody has 40 votes and their own standards, so here we are. I'd share my tips for always knowing what the OP wants and makes the community happy, but I've got my own list of 100 answers with 0 votes. ^_^ Do your best, don't take anything personally, and stay cheerful!

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    $\begingroup$ Your comment on the $2\sqrt{3}$ question is quite interesting since I would say that both of those answers amounted to "put them into a calculator and compare". It's an apt comparison though: a good proof provides insight, which neither of those do, and its possible that people felt that OPs answer also failed to provide insight. $\endgroup$ – postmortes Nov 27 at 9:50
  • $\begingroup$ it all comes down to ${\pi\over 3}<{2\over \sqrt{3}}$ either way. $\endgroup$ – Roddy MacPhee Nov 27 at 12:40
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    $\begingroup$ @postmortes That's a good point. If Hidden Figures taught me anything, it's that I'm a calculator. $\endgroup$ – Matthew Daly Nov 27 at 12:44
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    $\begingroup$ I have 204, 0 votes on main and a few are accepted answers. $\endgroup$ – Roddy MacPhee Nov 27 at 12:48
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    $\begingroup$ Honestly, while I know $\sqrt 2$ to three decimal places, I only know $\sqrt 3$ to one decimal place. I have a PhD in math, so certainly not merely a casual student of mathematics, but I would say my interest in anything involving explicit computation with real numbers ended at least 15 years ago. $\endgroup$ – Matt Samuel Dec 2 at 22:01
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Not quite an answer, but perhaps helpful.

When I see a question like this one where I suspect that the OP does not quite know how to phrase what they want to know I try to guess. Then I answer this way:

I think what you are asking is [rephrased/expanded question]. If so, then [possible answer].

Then the OP (or others) can agree or disagree and vote as they wish.

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Here's another not-quite-an-answer to complement the not-quite-an-answer of @EthanBolker.

Sometimes I write an answer like he does.

But there are plenty of other times that I start to write such an answer, and then get annoyed as the answer gets longer because I realize that I might well be wasting my time writing an answer that could turn out to be inappropriate.

That's when I stop, and instead just write a comment: "Are you really asking [rephrased/expanded question]?"

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