Should I include obvious steps in mathematical paper?

Question

I am writing a formal school paper (by formal I mean, with an abstract, a bibliography and $20$ pages in length) on mathematics (it is called "Extended Essay", for those having knowledge of the International Baccalaureate) and I was wondering whether I should include algebra steps as in $$\frac{(2k+1)^{-2}}{1-(2k+1)^{-2}} = \frac {1}{4k(k+1)}.$$ In fact I can do this in my head and I am sure that the external examiner will not have the slightest problem understanding steps. However, I only fear that when you don't show any steps it is as if you copied it from a book or so, which is not case.

In general for academic papers (like important undergraduate mathematical projects), should I include such steps or other non-algebraic steps that are sort of obvious and that the reader can pretty much understand even by grabbing pen and paper in the worst case?

Thanks in advance.

Answer 1

Always best to error on the side of providing "too much" information, than not enough. If the examiner is trained in mathematics, then you could probably get by without providing excessive detail.

But take care not to assume "too much" in the way of what you think others will know. What is "obvious" is in "the eye of the beholder." In the example you provide, I'd suggest including at least an intermediate step or explanation as to why the left-hand side follows from right hand side of your equation.

The important thing, of course, is that you understand what you are doing.
But you also want to take care to ensure others understand what you are saying.
And certainly not least, you want to be sure that others (as in examiners) understand that you understand what you are saying!

Answer 2

First, if this is an examination in which you will be awarded marks, you need to find out from the examiner what can be omitted.

Second, what is obvious to one may not be to another, so you need to write in such a way that it caters to your intended audience.

In your example, assuming it is the IB, it would be good to fill in the missing steps as it is not immediately obvious why equality holds.

Answer 3

If it's a paper to be read by people trying to learn what it contains that's new to them, I might say something like "By completing the square, we see that . . . . .", and skip the details of completing the square if it's for an audience for whom that is routine.

But for a "school" paper, the point may be to demonstrate what you know about the topic rather than to make the topic known to your audience. Sometimes in that situation one should include more than in the other situation. Whether that includes details of how to complete the square (or whatever) might depend on whether that's part of the prerequisites to a course or something included in the course.

At any rate, the difference in purpose is there, and can influence what ought to be included.

Answer 4

I am not a mathematician, but I teach engineering and computer science, and I've read quite a number of papers and reports from students. I don't know if this is just me being particularly lazy, but I'd rather avoid having to decode things that are only "sort of" obvious, or that I need pen and paper to get through.

When a student has worked with a project, maybe for months, many things seem obvious to that student, so they omit them from the report or paper. Perhaps they even feel ashamed to put such very simple things in their text. But the things that are obvious to someone who has just spent weeks or months on them, sometimes are much less obvious to another reader.