Mathematical Writing (Springer Undergraduate Mathematics Series)
Franco Vivaldi
Language: English
Pages: 213
ISBN: 1447165268
Format: PDF / Kindle (mobi) / ePub
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.
equation are the points at which the two functions assume the same value. (The expression ‘equating two functions’ may be appropriate for functional equations, see (3.21).) 3.4 Expressions The generic term expression indicates the symbolic encoding of a mathematical object. For instance, the string of symbols ‘’ is a valid expression, and so is ‘’, while ‘’ is incorrect and does not represent an object. It would seem that any correct expression should have—in principle, at least—a unique
give a concise version of the same proof. Proof. Let be a real number with eventually repeating decimal digits . Without loss of generality, we may assume that the integer part of is zero, and that the fractional part is purely periodic: (1) (Any real number may be reduced to this form by first multiplying by a power of 10, and then subtracting an integer, and neither operation affects the tail of the digit sequence.) Defining the decimal integer , we find, from (1) (7.11) We see that is
high-level organisation—number of chapters, outline of their content—is normally considered at a relatively early stage of the project. Planning will also suggest appropriate headings and will aid the sequencing of the arguments. Periodic reviews of work in progress may alter priorities, or even re-direct the research. The writing process culminates with the exercise of proof-reading, in itself a valuable, if painstaking, experience. In this chapter I give an overview of the structure of a
functions Computer-assisted proof Conclusion Conductor Congruence class operator Congruent Conjecture Conjunction conjuncts Consequent Continuous Contradiction, see proof Contrapositive Convergence Converse Coset Countable Counterexample D De Morgan laws Decimal point Dedekind Definition recursive Degree total Descending Difference Dimension Dirichlet's box principle Disc Discontinuous Disjoint Divergence Divide and conquer method Divisibility operator
of standard definition can be very effective. The set of rational numbers—ratios of integers with non-zero denominator—is defined as follows: (2.7) The property is phrased in such a way as to avoid repetition of elements. This is the so-called reduced form of rational numbers. The rational numbers may also be defined abstractly, as infinite sets of equivalent fractions—see Sect. 4.6. One might think that in the expression for a set we could choose any property . Unfortunately this doesn’t