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When writing a proof, what level of mathematical understanding can I assume my reader has?

For example, can I assume they know all odd integers can be represented by $2q+1$? (Right?)

Or that all even integers can be represented by $2k$?

When do I have the power to draw on a theorem?

Can I always assume that my previous problem was right and cite it to help me with my current proof?

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You can give the result before the new theorem. Then you either cite where you can find a proof or give a proof yourself. – Jonas Teuwen Sep 6 '11 at 13:55
And if the proof is not too long and not distracting, go with Jonas's second option. – J. M. Sep 6 '11 at 13:59
So I assume they know very little and try to help the reader as much as I can. and ensure that I don't get too off topic? – Dennis Hayden Sep 6 '11 at 14:01
I hope your readers don't "know all odd integers can be represented by 4q+1"... :) – t.b. Sep 6 '11 at 14:04
The question cannot be answered generally. You'll have to know which audience you're targeting, and there is no "standard audience". The assumptions can be very different according to whether you're writing an article for a professional journal, or helping a highschooler on math.SE (or something in between). – Henning Makholm Sep 6 '11 at 14:09
up vote 15 down vote accepted

In mathematical writing, as in all writing, you need to think carefully about who your readers will be (and who you want them to be). It is rarely possible to write in a way which will be equally satisfactory to every conceivable person.

In this case, although you don't say so explicitly, it sounds like you are writing up problem sets for a course, so your intended reader is the grader of the problem set and/or the course instructor (perhaps the same person). With such a small audience it is feasible to simply ask them about their preferences, and while you probably don't want to do this before you turn in every single problem set, some questions in the beginning will probably make things go smoothly. In general though, this type of reader is someone who knows the material well -- probably better than you do -- and in most cases broadly knows what you are trying to say even before you say it. However, they are also looking for gaps and mistakes in your arguments. So in this case I would start by erring on the side of including more details / supporting reasoning, while understanding that if you do not express any given idea / argument in the best possible way you are more likely to be understood anyway than with a reader who really doesn't know what you're trying to tell her.

With regard to your specific questions:

can I assume they know all odd integers can be represented by $4q+1$[?]

Gosh, I hope not, since this is not true: e.g. $3 \neq 4q + 1$.

[Added: It seems that the "$4$" was just a typo which has since been corrected to $2$. In this case there may well be something to prove. It is relatively common to define an integer to be odd if it isn't even, and then one has to justify that an odd number is of the form $2k+1$. In fact, in an honors course for future math majors that I am currently teaching this came up in the first week. I defined an integer to be even if it is of the form $2k$ for some integer $k$ and odd if it is of the form $2k+1$, but there was still something to show: every integer is either even or odd and not both. The "not both" is easy, but the first part requires something: a few days later I proved it by mathematical induction, and then later stated it as a special case of the theorem about division with remainder. So no, for my intended audience I did not want to just assume that familiar facts about even and odd numbers are true, although I probably would do so in a course pitched either at a higher or a lower level.]

Or that all even integers can be represented by $2k$?

This is a standard definition of an even number (in fact, I can't think of any other standard definition at the moment). So this may well have come up before in class or in the course text. If not, and you are not working with any other definition of even, then you can just say something like "if $x$ is even -- that is, $x$ is of the form $2k$ for some integer $k$ -- ..." and move on.

Can I always assume that my previous problem was right and cite it to help me with my current proof?

I don't see why not. But if you are unsure, ask your grader/instructor.

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I agree with all of Pete's advice. One additional point, if this is an intro number theory course that focuses on learning how to write proofs: Some people define odd numbers to be those numbers which are not of the form $2k$. In that case, it is not obvious that odd numbers are of the form $2 m+1$, and does require proof (in a more advanced course, this would count as standard knowledge). Other people define odd numbers to be numbers of the form $2m+1$, in that case it is not obvious that every number is either even or odd. (continued) – David Speyer Sep 6 '11 at 14:18
From my experience grading intro number theory at PROMYS, most students never say which definition they are using, and thus lose points. – David Speyer Sep 6 '11 at 14:19
@David: and I agree with your comment -- note that I independently added a bit about even and odd numbers to my answer. – Pete L. Clark Sep 6 '11 at 14:28
@Peter Mortensen: please do not edit my answer (especially, a personal anecdote in my answer) for style. That sort of thing may be common on some other SE sites, but if you spend more time on this site I think you'll see that it's not done here. – Pete L. Clark Sep 6 '11 at 17:30

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