In a paper you have to present the complete argument

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In a paper you have to present the complete argument however repulsive the details. Similarly, in your essay you must give proofs in full although you may well precede such a proof by an example or a proof of a simple case to help the reader. In talking about the background to your paper (for example if the purpose of your essay is to use a technique to do some calculations) or in describing developments of your results you may well choose to summarise and give a reference to the literature. Think of your topic as as a tree, you can assume the roots and prune the branches but you cannot remove a section of the trunk. Of course, there are exceptions to this rule. There are branches of math- ematics where full proofs of major results do not exist or are just too compli- cated and some where, to the outsider at least, it appears that practitioners leap from assertion to assertion like chamois on a mountainside. However, if you do decide to omit a proof on the grounds that it is too hard for the reader examine your conscience and ask whether you do not mean that it is too hard for you. Such omissions are precisely the kind of thing you should discuss with your assessor. 4.9 Layout is important The layout of a paper is how it looks on the page — margins, indentation, how many lines are skipped between paragraphs, which headings are underlined and so on. Here the general principle is do what is necessary to make the important bits stand out. If you write your paper by hand most problems of layout will settle them- selves. If you use L A T E X then the system will settle most problems for you. If you are a T E X fan then you do not need (or, what comes to almost the same thing, you think you do not need) any advice. If you use a typewriter or a non-T E X based word processing system then you need all the help you can get. Here are a couple of general principles. (1) All definitions and notational conventions should be clearly set out. Leaving spaces before and after definitions greatly increases their visibility. The same goes for lemmas, examples and theorems. (2) When in doubt, display. Most mathematicians include some very short formulae in their text. ‘If x = 2 then integraltext x 3 f ( t ) dt is negative.’ However, 16
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even moderately short formulae like A = uniontext i =1 ( A i B ) may become ugly and difficult to read. It is much better to use a displayed equation A = uniondisplay i =1 ( A i B ) . If you are writing for a journal the printer may be anxious to save space but, so far as your essay is concerned, paper is cheap. (Nowadays, even publishers may approve of displayed equations since they make the job of electronic typesetting much easier.) Next time you have to attend a seminar with overheads covered with minute detail (or even better when you are preparing an overhead) remember SPACE aids LEGIBILITY. LEGIBILITY aids COMPREHENSION.
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