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Franco Vivaldi - Mathematical Writing

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Franco Vivaldi Mathematical Writing
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    Mathematical Writing
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Mathematical Writing: summary, description and annotation

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This book teaches the art of writing mathematics, an essential -and difficult- skill for any mathematics student.
The book begins with an informal introduction on basic writing principles and a review of the essential dictionary for mathematics. Writing techniques are developed gradually, from the small to the large: words, phrases, sentences, paragraphs, to end with short compositions. These may represent the introduction of a concept, the abstract of a presentation or the proof of a theorem. Along the way the student will learn how to establish a coherent notation, mix words and symbols effectively, write neat formulae, and structure a definition.
Some elements of logic and all common methods of proofs are featured, including various versions of induction and existence proofs. The book concludes with advice on specific aspects of thesis writing (choosing of a title, composing an abstract, compiling a bibliography) illustrated by large number of real-life examples. Many exercises are included; over 150 of them have complete solutions, to facilitate self-study.
Mathematical Writing will be of interest to all mathematics students who want to raise the quality of their coursework, reports, exams, and dissertations.

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Springer-Verlag London 2014
Franco Vivaldi Mathematical Writing Springer Undergraduate Mathematics Series 10.1007/978-1-4471-6527-9_1
1. Some Writing Tips
Franco Vivaldi 1
(1)
School of Mathematical Sciences, Queen Mary, University of London, London, UK
Franco Vivaldi
Email:
The following short mathematical sentences are poorly formed in one way or another. Can you identify all the errors and would you know how to fix them? Compare your answers with those given at the end of the book.
Exercise 1.1
Improve the writing.
Picture 1 is positive.
Two is the only even prime.
If Picture 2Picture 3 .
We minus the equation.
Picture 4 has no real solution.
When you times it by negative Picture 5 , Mathematical Writing - image 6 becomes Mathematical Writing - image 7 .
The set of solutions are all odd.
Mathematical Writing - image 8Picture 9Picture 10 is integer.
An invertible matrics is when the determinant is non-zero.
This infinete sequence has less negative terms.
In this chapter you will learn how to recognise and correct common mistakes, the first step towards writing mathematics well. By the time you reach the exercises at the end of the chapter you should already feel a sense of progress. You should return to this chapter repeatedly, to monitor the assimilation of good practice.
1.1 Grammar
You are advised to use an English dictionary, e.g.,)see, for instance, [37, pp. 8995].
  • Write in complete sentences. Every sentence should begin with a capital letter, end with a full stop, and contain a subject and a verb. The expression A cubic polynomial is not a sentence because it doesnt have a verb. It would be appropriate as a caption, or a title, but you cant simply insert it in the middle of a paragraph.
  • Make sure that the nouns match the verbs grammatically.
    Bad:
    The set of primes are infinite.
    Good:
    The set of primes is infinite.
    (The verb refers to the set, which is singular.)
  • Make a pronoun agree with its antecedent.
    Bad:
    Each function is greater than their minimum.
    Good:
    Each function is greater than its minimum.
    (The pronoun its refers to function, which is singular.)
  • If possible, do not split infinitives.
    Bad:
    We have to thoroughly examine this proof.
    Good:
    We have to examine this proof thoroughly.
    Bad:
    I was taught to always simplify fractions.
    Good:
    I was taught always to simplify fractions.
    (The infinitives are to examine and to simplify.) In some cases a split infinitive may be acceptable, even desirable.
    Good:
    This is a sure way to more than double the length of the manuscript.
  • Check the spelling. No point in crafting a document carefully, if you then spoil it with spelling mistakes. If you use a word processor, take advantage of a spell checker. These are some frequent spelling mistakes:
    Bad:
    auxillary, catagory, consistant, correspondance, impliment, indispensible, ocurrence, preceeding, refering, seperate.
    These are misspelled mathematical words that I have found in mathematics examination papers:
    Bad:
    arithmatic, arithmatric, derivitive, divisable, falls (false), infinaty, matrics, orthoganal, orthoginal, othogonal, reciprical, scalor, theorom.
  • Be careful about distinctions in meaning.
    Do not confuse its (abbreviation for it is ) with its (possessive pronoun).
    Bad:
    Its an equilateral triangle: its sides all have the same length.
    Do not confuse the noun principle (general law, primary element) with the adjective principal (main, first in rank of importance).
    Bad:
    the principal of induction
    Bad:
    the principle branch of the logarithm
    Do not use less (of smaller amount, quantity) when you should be using fewer (not as many as).
    Bad:
    There are less primes between 100 and 200 than between 1 and 100.
  • Do not use where inappropriately. As a relative adverb, where stands for in which or to which ; it does not stand for of which .
    Bad:
    We consider the logarithmic function, where the derivative is positive.
    Good:
    We consider the logarithmic function, whose derivative is positive.
    The adverb when is subject to similar misuse.
    Bad:
    A prime number is when there are no proper divisors.
    Good:
    A prime number is an integer with no proper divisors.
  • Do not use which when you should be using that . Even when both words are correct, they have different meanings. The pronoun that is defining, it is used to identify an object uniquely, while which is non-defining, it adds information to an object already identified. The argument that was used above is based on induction.
    [ Specifies which argument. ]
    The following argument, which will be used in subsequent proofs, is based on induction.
    [ Adds a fact about the argument in question. ]
    A simple rule is to use which only when it is preceded by a comma or by a preposition, or when it is used interrogatively.
  • In presence of parentheses, the punctuation follows strict rules. The punctuation outside parentheses should be correct if the statement in parentheses is removed; the punctuation within parentheses should be correct independently of the outside.
    Bad:
    This is bad. (Superficially, it looks good).
    Good:
    This is good. (Superficially, it looks like the bad one.)
    Bad:
    This is bad, (on two accounts.)
    Good:
    This is good (as you would expect).
1.2 Numbers and Symbols
Effectively combining numbers, symbols, and words is a main theme in this course. We begin to look at some basic conventions.
  • A sentence containing numbers and symbols must still be a correct English sentence, including punctuation.
    Bad:
    Picture 11Picture 12
    Good:
    We have Picture 13 and Picture 14 .
    Good:
    We find that Picture 15
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