My new course “Mathematical Structures” begins next week. (This is my slightly perverse way of celebrating my retirement.)

Every week, after the lectures, I will post the lecture notes and supplementary material on the course web page. This will be done in the by-now old-fashioned way, on a public web page rather than on a closed ILE page or whatever the fashionable acronym is.

I hope that this experiment will be of some interest to anyone who is involved with teaching or learning mathematics at first-year undergraduate level. We have found that there is an increasing disconnection between what our students learn in the exam-centred environment at school and what we would like them to know to tackle the study of university mathematics. So this course is at least as much about raising the students’ game as it is about actual mathematical content.

There are some other new features about the course (at least as far as Queen Mary is concerned): the tutorials will be in small groups, with the academic adviser as tutor; and the students will be giving presentations to their tutorial groups.

In an informal document to the course tutors, I described the course as having three main aims:

- to introduce some of the basic objects of mathematics (sets, functions and numbers);
- to get across to the students some good mathematical habits, such as using definitions and how to construct proofs;
- to make the students enthusiastic about our beautiful subject.

As the course progresses, we will see to what extent these aims have been achieved!

The notes will be in ten chapters, and will be posted week by week on the course page, except for week 7 when we have no lectures. I will also try to post a brief discussion here about what that section of the notes is trying to achieve and why I made some of the decisions I did. In my Gresham College lecture next year on 14 May, I want to look back over the process and provide some kind of summing-up. The week-by-week plan goes as follows:

- Introduction
- Sets
- Infinity
- Functions and relations
- Natural numbers
- Integers and rational numbers
- Real numbers
- Complex numbers
- Proofs
- Constructing and debugging proofs

Each chapter of the notes will also include a section on study skills, and

will be accompanied by supplementary material.

I would be very interested in your comments on the course material, though I don’t promise to act on them! Please bear in mind, as you read what I have written, that the slogan of the course (taken from T. S. Eliot) is “precise but not pedantic”. So, for example, real numbers will not be defined by Dedekind cuts or by Cauchy sequences. This doesn’t mean that I think there is anything wrong with these constructions.

Naturally, if you are a student taking the course, and find your way here, your opinions are especially welcome and valuable!

It seems a little odd talking about infinity before you’ve set up functions (and thus the means of talking about countability) or even the infinite sets that first year undergraduates will spend most of their time playing with – the N, Z, Q, R and C.

In the first year undergraduate course I teach countability is introduced at the end of the sections of the course discussing N, Q Z and R as one of the properties that distinguishes R from Q but otherwise not a huge amount is said about infinity. It will be interesting to see what you have to say about it!

If I were Bourbaki, I would do things differently.

When I was an undergraduate, the mathematics department was on the side of a hill; you came in at the ground floor and could go either up or down. This is my model. The students come in with at least some idea of what natural and real numbers, finite and infinite are; with the aim of being precise but not pedantic, I build on this to some extent.

I have no idea whether it will work, but it will be interesting to find out!

Some of my thinking on functions and relations:

PlanetMath • Relation Theory

I have just glanced at the first coursework assignment, looks good. I especially like the inclusion of the project question.

Something you wrote above caught my eye as well sir:

“Please bear in mind, as you read what I have written, that the slogan of the course (taken from T. S. Eliot) is “precise but not pedantic”. So, for example, real numbers will not be defined by Dedekind cuts or by Cauchy sequences. This doesn’t mean that I think there is anything wrong with these constructions.”

I would agree that there is nothing wrong with these constructions if one is interested in more advanced stuff ,say topology, but for beginners is there anything wrong with just defining them as decimals?

Just speaking from personal experience, I don’t recall (when I was an undergrad at least) feeling enlightened by learning the Dedekind cut approach and certainly not the Cauchy sequence approach (spit). On the other hand I feel that going through the construction of R via decimals gave me a (slightly) better appreciation of elementary real analysis. Perhaps its just my applied bias talking.

Kind Regards

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Out of curiosity……

Analysis books seem to have an irrational (no pun intended) pedagogical bias against the decimal approach to R despite the alleged point of the first course in analysis being to understand Calculus better. Can anyone out there explain why?

The definition of real numbers as infinite decimals has a serious problem which you may have overlooked. How do you add or multiply them, when you can’t “start at the right”? Given two infinite decimals, you want to know a particular digit of their sum, say just after the decimal point; there is no a priori bound on how long you have to wait until you know what it is.

Here is an example which I will put in the notes. Suppose you have to calculate the digit after the decimal point in 0.414213562…+0.585786437…. Adding the digits so far we get 0.999999999…. But we don’t know whether this pattern continues. If each pair of digits sums to 9, then the sum of the numbers is 1, and the digit is zero. Otherwise, consider the first position where the digit sum is not 9. If it is less than 9, then the digit we are looking for is 9; if it is greater than 9, the required digit is 0. We have no idea how long we will have to wait to resolve this. Yet, of course, the longer we have to wait, the closer the sum is to 1.

Thank you for your response sir.

Every decimal based approach to constructing the real number system that I have seen relies on defining addition/multiplication of two infinite decimals as limits of addition/multiplication of truncations of those decimals. So for example (0.111111…) x (0.222222…) will be defined as the limit of the sequence of multiplications:

0.1 × 0.2

0.11 × 0.22

0.111 × 0.222

etc

An approach that is quite natural when one considers that infinite decimals are themselves limits of sequences of finite decimals (since an infinite series is just the limit of a sequence of partial sums).

Kind Regards.

The point is that addition and multiplication do indeed give you closer and closer approximations, which fits in perfectly with the Cauchy sequence approach that you don’t like – what is a Cauchy sequence, after all, but a sequence of better and better approximations? But they fit very badly with the decimal approach, where you can’t guarantee to tell after any given finite number of steps whether the answer is 0.9… or 1.0…

I am using the decimal approach because it is familiar to the students; the entry price for Cauchy sequences or Dedekind cuts is much higher. But ultimately I do not think it is satisfactory. Perhaps we just have to agree to differ.