Writing a Short Description - Writing Well

Mathematical Writing - Vivaldi Franco 2014

Writing a Short Description
Writing Well

Writing a synopsis of a mathematical topic is a common task. It could be the abstract of a presentation, the summary of a chapter of a book, an informal explanation of a theorem. It could be the closing paragraph of a large document, which distils the essence of an entire subject.

Writing a short essay is difficult—the shorter the essay, the greater the difficulty. We shall adopt the format of a MICRO-ESSAY: 100—150 words (one or two paragraphs) and no mathematical symbols. Within such a confined space one is forced to make difficult decisions on what to say and what to leave out; the lack of symbols gives further prominence to the concepts. Our command of the syntax will be put to the test.

Our first MICRO-ESSAY is a summary of Sect. 2.1 on sets. We have access to all relevant material, and the main difficulty is to decide what are the highlights of that section. We select two ideas: how to define a set, and how to construct new sets from old ones.

A set is a collection—finite or infinite—of distinct mathematical objects, whose ordering is unimportant. A small set may be defined by listing all its elements explicitly; a large set is instead defined by specifying the characteristic properties of the elements.

Combining numbers with arithmetical operators gives new numbers. Likewise, combining sets via set operators (union, intersection, difference) gives new sets. But this is a bit like recycling what we already have. A class of brand-new sets are the so-called cartesian products, which are constructed by pairing together elements of existing sets. A well-known example is the cartesian plane, which is made of pairs of elements of the real line, each representing a coordinate.

The description of set operators exploits an analogy with arithmetical operators, and for this reason the first two sentences of the second paragraph have the same structure. We avoid a precise definition of the term ’cartesian product’ (not enough space!), opting instead for an informal description and an illustrative example.

Next we write a MICRO-ESSAY on prime numbers, a synthesis of our knowledge of this topic.

A prime is a positive integer divisible only by itself and unity. (However, 1 is not considered prime). The importance of primes in arithmetic stems from the fact that every integer admits a unique decomposition into primes. The infinitude of primes—known from antiquity—and their unpredictability make them object of great mathematical interest.

Many properties of an integer follow at once from its prime factorization. For instance, looking at the exponents alone, one can determine the number of divisors, or decide if an integer is a power (i.e., 4 a square or a cube).

Primality testing and prime decomposition are computationally challenging problems with applications in digital data processing.

This essay begins with a definition. The technical point concerning the primality of 1 has been confined within parentheses, to avoid cluttering the opening sentence. The following two sentences deliver core information in a casual—yet precise—way. We state two important theorems (the Fundamental Theorem of Arithmetic, and Euclid’s theorem on the infinitude of the primes) without mentioning the word ’theorem’. We also give a hint of why mathematicians are so fascinated by primes. The second paragraph elaborates on the importance of unique prime factorisation, by mentioning two applications without unnecessary details. The short closing paragraph, like the last sentence of the first paragraph, is an advertisement of the subject, meant to encourage the reader to learn more.

Now a real challenge: write a MICRO-ESSAY on Theorem 4.2, which we reproduce here for convenience.

Theorem. Let $$X$$ be a set, let $$A,B\subseteq X$$, and let $${\fancyscript{P}}_A$$, $${\fancyscript{P}}_B$$ be the corresponding characteristic functions. The following holds (the prime denotes taking complement):

We must extract a theme from a daunting list of inscrutable symbols. We inspect the material of Sect. 4.3 leading to the statement of this theorem: it deals with characteristic functions, and the main idea is to link characteristic functions to sets. We can see such a link in every formula: on the left there are logical operators, on the right set operators. Given the specialised nature of this theorem, some jargon is unavoidable, so we write for a mathematically mature audience.

Every characteristic function corresponds to a set, and vice-versa. If we let a logical operator act on characteristic functions we obtain a new characteristic function, and with it a new set. How is this set related to the original sets? This theorem tells us that for each logical operator there is a set operator which mirrors its action on the corresponding sets. In other words, we have a bi-unique correspondence between these two classes of operators. For instance, the logical operator of negation is represented by the set operation of taking the complement. More precisely, the negation of the characteristic function of a set is the characteristic function of the complement of this set. Analogous results are established with respect to the other logical operators.

Here words are better than symbols to describe the structure, but they can’t compete with symbols for the details. So, to avoid tedious repetitions, we have chosen to explain just one formula carefully (the easiest one!), mentioning the other formulae under the generic heading ’analogous results’.

Exercise 6.1

Answer concisely. [ $$/\!\!\varepsilon ,30$$ ]

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Exercise 6.2

Consider the following question:

Why is it that when the price of petrol goes up by 10 $$\%$$ and then comes down 10 $$\%$$, it doesn’t finish up where it started?

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Exercise 6.3

Consider the following question:

I drive ten miles at 30 miles an hour, and then another ten miles at 50 miles an hour. It seems to me my average speed over the journey should be 40 miles an hour, but it doesn’t work out that way. Why not?

Write an explanation for the general public, clarifying why such a confusion may arise. You may perform some basic arithmetic, but do not use symbols as most people find them difficult to understand. [ $$/\!\!\varepsilon $$ ]

Exercise 6.4

Consider the following question:

I tossed a coin four times, and got heads four times. It seems to me that if I toss it again I am much more likely to get tails than heads, but it doesn’t work out that way. Why not?

Write an explanation for the general public, clarifying why such a confusion may arise. You may use symbols such as $$H$$ and $$T$$ for heads-tails outcomes, but avoid using other symbols. [ $$/\!\!\varepsilon $$ ]

Exercise 6.5

Consider the following question:

In a game of chance there are three boxes: two are empty, one contains money. I am asked to choose a box by placing my hand over it; if the money is in that box, I win it. Once I have made my choice, the presenter—who knows where the money is—opens an empty box and then gives me the option to reconsider. I can change box if I wish. It seems to me that changing box would make no difference to my chances of winning, but it does not work out that way. Why not?

Write for the general public, explaining what is the best winning strategy. [ $$/\!\!\varepsilon $$ ]

Exercise 6.6

Write a MICRO-ESSAY on each topic. [ $$/\!\!\varepsilon ,150$$ ]

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Exercise 6.7

Write a two-page essay on each topic.

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Footnotes

1

This terminology is due to Blaise Pascal (French: 1623—1662).

2

Physicists use Greek letters for subscripts and superscripts.

3

Bernhard Riemann (German: 1826—1866).

4

Abbreviation for the Latin id est, meaning ’that is’.