Introducing a Concept - Writing Well

Mathematical Writing - Vivaldi Franco 2014

Introducing a Concept
Writing Well

Opening the exposition with a formal definition is rarely a good way to introduce a new concept, particularly if we write for non-experts. The following examples show how to set the scene for a gentle introduction to a new idea. In each case we ask a question—a simple rhetorical device to engage and prepare an audience.

EXAMPLE. Introducing recursive sequences.

Let $$n$$ be a natural number. How is the power $$2^n$$ defined? We could use repeated multiplication

$$ 2^n:=\underbrace{2\times 2\times \cdots \times 2}_n, $$

but we could also write

$$ 2^1:=2\qquad \text{ and }\qquad 2^{n}:=2\times 2^{n-1}\quad n > 1. $$

The second formula is an example of a recursive definition of a sequence $$(a_n)$$. When $$n=1$$, the first term $$a_1=2$$ is defined explicitly; then, assuming that $$a_{n-1}$$ has been defined, we define $$a_{n}$$ in terms of it.

[The recursive definition of a general sequence follows (see Sect. 9.4 )].

The opening question leads to the familiar definition of exponentiation in terms of multiplication. The less familiar recursive definition comes after, illustrated by a simple example and supported by an appropriate notation. After this preamble, the reader should be ready to confront an abstract definition.

EXAMPLE. Introducing the exponential function.

The process of differentiation turns a real function into another real function. For example, differentiation turns the sine into the cosine. Are there functions that are not changed at all by differentiation?

[The definition of the exponential function follows.]

We recall a structural property of differentiation and present a familiar example. The question that follows suggests how the argument will develop.

EXAMPLE. Introducing the rational numbers.

A rational number is represented by a pair of integers, the numerator and the denominator. As these integers need not be co-prime, we may choose them in infinitely many ways. How are we to construct a single rational number from an infinite set of pairs of integers? How do we define the set $$\mathbb {Q}$$ from the cartesian product $$\mathbb {Z}\times (\mathbb {Z}{\backslash }\{0\})$$ ?

[The definition of the set of rational numbers follows.]

The first two sentences recall elementary facts. A question then invites the reader to think about this problem more carefully. A second question, which echoes the first, uses proper terminology and notation, in preparation for a formal construction.

EXAMPLE. We design an exercise structured as a list of questions. The topic is the number of relations on a finite set.

What is a relation on a set?

Can a relation be defined on the empty set?

How many relations can one define on a two-element set?

Let n be the number of relations on a set. What values can $$n$$ assume?

The first question checks background knowledge; the other questions gently explore the problem, guiding the reader from the specific to the general. Textbook exercises are sometimes structured in this way, to encourage independence in learning.

Formulating questions is not just a method to grab attention or structure exercises. Asking the right questions—those that chart the boundaries of our knowledge—is the essence of research.