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How to write an introduction for a math paper

  • 02.02.2019

Use the opening to introduce your project. Define your project's parameters and give any relevent, yet brief, background information. Step 2 In each of the following paragraphs, take one point and elaborate as to how it supports your thesis. This can be discussing the details of your project step-by-step. Relate the results of any tests or experiments, using corroborating evidence in the form of graphs or charts as appropriate. Each stage of the project should support or disprove your initial thesis statement.

Remember, in a math report, your findings could be that the initial thesis was wrong. Step 3 Write a conclusion. Your math project or data should either prove or disprove your thesis.

If he dislikes partial differential equations, for example, he should be warned early on that he will encounter them. If he isn't familiar with the first concepts of probability, then he should be warned in advance if your paper depends on that understanding. Remember at this point that although you may have spent hundreds of hours working on your problem, your reader wants to have all these questions answered clearly in a matter of minutes.

In the second section of your paper, the introduction, you will begin to lead the reader into your work in particular, zooming in from the big picture towards your specific results. This is the place to introduce the definitions and lemmas which are standard in the field, but which your readers may not know. The body, which will be made up of several sections, contains most of your work.

By the time you reach the final section, implications, you may be tired of your problem, but this section is critical to your readers. You, as the world expert on the topic of your paper, are in a unique situation to direct future research in your field. A reader who likes your paper may want to continue work in your field.

If you were to continue working on this topic, what questions would you ask? Also, for some papers, there may be important implications of your work.

If you have worked on a mathematical model of a physical phenomenon, what are the consequences, in the physical world, of your mathematical work? These are the questions which your readers will hope to have answered in the final section of the paper.

You should take care not to disappoint them! Section 3. Formal and Informal Exposition Once you have a basic outline for your paper, you should consider "the formal or logical structure consisting of definitions, theorems, and proofs, and the complementary informal or introductory material consisting of motivations, analogies, examples, and metamathematical explanations.

This division of the material should be conspicuously maintained in any mathematical presentation, because the nature of the subject requires above all else that the logical structure be clear.

Several questions may help: To begin, what exactly have you proven? What are the lemmas your own or others on which these theorems stand. Which are the corollaries of these theorems? In deciding which results to call lemmas, which theorems, and which corollaries, ask yourself which are the central ideas. Which ones follow naturally from others, and which ones are the real work horses of the paper?

The structure of writing requires that your hypotheses and deductions must conform to a linear order. However, few research papers actually have a linear structure, in which lemmas become more and more complicated, one on top of another, until one theorem is proven, followed by a sequence of increasingly complex corollaries.

On the contrary, most proofs could be modeled with very complicated graphs, in which several basic hypotheses combine with a few well known theorems in a complex way.

There may be several seemingly independent lines of reasoning which converge at the final step. It goes without saying that any assertion should follow the lemmas and theorems on which it depends. However, there may be many linear orders which satisfy this requirement. In view of this difficulty, it is your responsibility to, first, understand this structure, and, second, to arrange the necessarily linear structure of your writing to reflect the structure of the work as well as possible.

The exact way in which this will proceed depends, of course, on the specific situation. One technique to assist you in revealing the complex logical structure of your paper is a proper naming of results.

By naming your results appropriately lemmas as underpinnings, theorems as the real substance, and corollaries as the finishing work , you will create a certain sense of parallelness among your lemmas, and help your reader to appreciate, without having struggled through the research with you, which are the really critical ideas, and which they can skim through more quickly.

Another technique for developing a concise logical outline stems from a warning by Paul Halmos, in HTWM, never to repeat a proof: If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the proof of Theorem 1, that's a signal that something may be less than completely understood. Other symptoms of the same disease are: 'by the same technique or method, or device, or trick as in the proof of Theorem When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma from which both Theorem 1 and Theorem 2 are more easily and more clearly deduced.

Now that we have discussed the formal structure, we turn to the informal structure. The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics. The informal structure complements the formal and runs in parallel. It uses less rigorous, but no less accurate!

For although mathematicians write in the language of logic, very few actually think in the language of logic although we do think logically , and so to understand your work, they will be immensely aided by subtle demonstration of why something is true, and how you came to prove such a theorem. Outlining, before you write, what you hope to communicate in these informal sections will, most likely, lead to more effective communication.

Before you begin to write, you must also consider notation. The selection of notation is a critical part of writing a research paper. In effect, you are inventing a language which your readers must learn in order to understand your paper.

Good notation firstly allows the reader to forget that he is learning a new language, and secondly provides a framework in which the essentials of your proof are clearly understood. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper. In most cases, it is wise to follow convention.

Using epsilon for a prime integer, or x f for a function, is certainly possible, but almost never a good idea. Section 4: Writing a Proof The first step in writing a good proof comes with the statement of the theorem.

A well-worded theorem will make writing the proof much easier. The statement of the theorem should, first of all, contain exactly the right hypotheses. Of course, all the necessary hypotheses must be included. On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible. When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form.

I suggest that, before you write you map out the hypotheses and the deductions, and attempt to order the statements in a way which will cause the least confusion to the reader. This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine as opposed to understandable by a human being , and it has the dubious advantage that something at the end comes out to be less than e.

The way to make the human reader's task less demanding is obvious: write the proof forward. Neither arrangement is elegant, but the forward one is graspable and rememberable. Avoid unnecessary notation. Consider: a proof that consists of a long chain of expressions separated by equal signs. Such a proof is easy to write. The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation.

This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as he goes. The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the results of the act and leaves the reader to guess how they were obtained.

The paragraph would say something like this: "For the proof, first substitute p for q, the collect terms, permute the factors, and, finally, insert and cancel a factor r. Specific Recommendations As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable.

These may be best checked and corrected after writing the first draft. Many of these ideas are from HTWM, and are more fully justified there. Notation that hasn't been used in several pages or even paragraphs should carry a reference or a reminder of the meaning.

The structure should be easily discernible by headings and punctuation. There should be a clear definition of the problem at hand all the way through.

The title is the first contact that readers will have with your paper. It must communicate something of the substances to the experts in your field as well as to the novices who will be interested. Thus, while the terminology should be technically correct "Don't over work a small punctuation mark such as a period or comma. They are easy for the reader to overlook, and the oversight causes backtracking, confusion, delay.

X belongs to the class C, The period between the two X's is overworked A good general rule is: never start a sentence with a symbol. If you insist on starting the sentence with a mention of the thing the symbol denotes, put the appropriate word in apposition, thus: "The set X belongs to the class C,

By "theorem" I mean a mathematical truth, something that has been proved. Also, for some papers, there may be important implications of your work. When you write about your own mathematical research, you will have another goal, which includes these two; you want your reader to appreciate the beauty of the mathematics you have done, and to understand its importance.
How to write an introduction for a math paper

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Down a math report follows the same basic procedure as writing any report or essay. Present your argument or thesis and then support it, thereby business location factors essay writer it, over the following paragraphs. One difference between a math report and other types of reports is that a math report will typically include graphs or real relevant charts or data in the thin of the paper and not just write appendices. A math paper may also include a proof as part of its logical arguments. Step 1 Introduce your paper in your opening paragraph.
Is it a special case of a larger question? The reader hopes to have certain questions answered in this section: Why should he read this paper? Why did this author think this question was interesting?

Good mathematical writing, like good mathematics thinking, is write skill which must be how and developed math optimal performance. The purpose of this paper is to provide assistance for young mathematicians writing their first paper. The aim introduction not only to aid in the development of a for written paper, but also paper help students begin to think about mathematical writing.
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The essay will conclude with a section containing specific recommendations to consider as you write and rewrite the paper. When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion.

In the second section of your paper, the introduction, you will begin to lead the reader into your work in particular, zooming in from the big picture towards your specific results. Also, for some papers, there may be important implications of your work. Is it a special case of a larger question? Draw a conclusion and present the results. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion.
How to write an introduction for a math paper
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Yozshusida

The paragraph would say something like this: "For the proof, first substitute p for q, the collect terms, permute the factors, and, finally, insert and cancel a factor r. Now that we have discussed the formal structure, we turn to the informal structure. Writing a math report follows the same basic procedure as writing any report or essay. The body, which will be made up of several sections, contains most of your work. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. Is it a classification theorem of structures which were previously defined but not understood?

Kagakazahn

Now that we have discussed the formal structure, we turn to the informal structure. The selection of notation is a critical part of writing a research paper. As a mathematician, you have the privilege of conducting a performance of your own composition! Failure to address this very question will leave the reader feeling quite dissatisfied. Such a proof is easy to write. Usually all that is needed to avoid it is to recast the sentence, but no universally good recasting exists; what is best depends on what is important in the case at hand.

Nikokinos

There may be several seemingly independent lines of reasoning which converge at the final step.

Zulujas

I am greatly indebted to a wonderful booklet, "How to Write Mathematics ," which provided much of the substance of this essay. The reader hopes to have certain questions answered in this section: Why should he read this paper? Using epsilon for a prime integer, or x f for a function, is certainly possible, but almost never a good idea. Draw a conclusion and present the results.

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