Good afternoon,

I am a high school junior in the process of preparing a real analysis paper for submission to a mathematics journal. I have written my proof and conducted an exhaustive literature review and now I would like to take the next step towards finishing my paper but I am unclear as to the convention with regards to including in my paper both an abstract and introduction or just an introduction.

I have read published papers such as this one by Erdos which include both an abstract and introduction but I have read many others such as this short proof of the prime number theorem which includes just an introduction. Then there are some like this one that have what appears to be an abstract without an explicit heading.

I have also read guides regarding how to write mathematical papers. There is this one which includes an introduction but no abstract. Then there are other guides like this one which make mention of both an abstract and an introduction.

My question is, is there a convention regarding the inclusion of an abstract in a math paper? If so, what is it? I understand that scientific papers traditionally employ both but I do not see a trend when it comes to math papers.

Thank you in advance.

  • It depends on the journal. Some journals encourage informal writing and abstracts are not required. In most journals, abstracts are required. Nov 14, 2023 at 11:57

3 Answers 3


Yes, mathematics papers generally have an abstract.

The question of whether it has the word "Abstract" as a heading is a stylistic question, usually determined by the journal where the paper is published. They might feel that the page layout makes it clear that this is an abstract, without needing a heading.

The Zagier paper you cite was published (in 1997) in the American Mathematical Monthly, which is a special case. They used to not print abstracts for very short articles. However, looking at some recent issues, even the Monthly does routinely print abstracts now.

So you should certainly write an abstract for your paper. Write it after the paper is completely finished, so that it accurately reflects what the final version of the paper contains. Format it in any reasonable way you like (e.g. the abstract environment in your preferred LaTeX document class.)

Include the abstract when you distribute the paper or submit it for publication. It will ultimately be up to the journal whether the abstract is included when the paper is published, and how to format it, but most likely it will be.


The purpose of an abstract and an introduction are very different, so it may be useful to include either or both. An abstract is a summary of the whole paper, focusing on the main result. It might exist separately from the paper itself as, say, an online "teaser" to get people to read the paper. But an introduction is intended to give background on the problem to be addressed in the paper and to situate it within the literature. You would almost always include an introduction. You might also write the abstract, but not actually include it within the paper itself.


The purpose of an abstract is to summarize the entire paper. It should have all the major findings, actual numbers (if applicable), etc. An introduction gives the history of the problem, including earlier attempts or related problems as well as the motivation for the research. They are different animals.

Perhaps you are confused because introductory paragraphs on high school essays often have an outline summary of what the essay will be. But that's really not the main objective of a journal article introduction section. (It could have that sort of outline explanation but probably more commonly doesn't.)

Also, you should just look at what journal you want to go to (pick something decent but not too high ranking). Then just ape their format.

All that said, I am skeptical of a high school junior contributing an analysis paper. Anything is possible of course. But I suspect you don't know what you don't know about the topic, previous literature, etc. For one thing, your question here shows some naivete about the structure of journal articles, so perhaps you haven't done a good lit search in your area. But...don't let me hold you back. Nothing like learning by experience (as I told my friend before he opened a restaurant),

  • 3
    Sorry, but your skepticism is misplaced here. Or the expression of it in any case. It reads as very discouraging. I wrote a decent paper when still in HS, and much more is taught there nowadays. If you have seen the paper, of course, you might want to comment on the ranking of the journal. But I doubt that you are doing anything but expressing a prejudice of the young.
    – Buffy
    Jan 23, 2019 at 19:15
  • @Buffy is High School now on topic here?
    – Solar Mike
    Jan 23, 2019 at 19:46
  • 2
    @SolarMike, not really, but the question seems a bit more sophisticated than what might normally happen. It is also a question that might occur at university for a new researcher/writer. I decided that the age was immaterial and wasn't essential. It stands without that information.
    – Buffy
    Jan 23, 2019 at 19:53

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