Two blogs: I think the maths looks a bit better on my other blog at (using MathJax). But this WordPress blog is still my main blog!

University of Nottingham Open Days – back on campus!

I will finally be back on campus in the Mathematical Sciences Building at the end of this week for the University of Nottingham Undergraduate Open Days!

See from where I quote:

On campus at last!

Campus visits are your opportunity to soak up the atmosphere and explore some of the UK’s most green and beautiful campuses. Learn more about your course, meet academics and chat to our current students so that you can decide whether life as a Nottingham student is right for you.

Campus visit dates will be:

Friday 10 September 2021

Saturday 11 September 2021

Saturday 9 October 2021

But you don’t have to wait until then to find out more about the University of Nottingham, watch pre-recorded lectures and general talks to learn more about subjects, student life and to see some of our academics in action.

Quantifier packaging, convergence of sequences, and absorption

One of my first ever blog posts (from 2008) was on “quantifier packaging” when teaching convergence of sequences. In fact I first posted about this on Blogger, before moving over to WordPress. I have now produced an updated version of that post at

(The WordPress version is at, but the new Blogger version has a few tweaks, and uses MathJax for the maths.)

Here is a screenshot from the Blogger post (where I use MathJax for the maths) of the standard definition of what it means to say that x_n \to x as n \to \infty (working in the real numbers).

Definition of what it means for a sequence of real numbers (x_n) to converge to a real number x

That probably looks a bit scary to first-year students (and possibly beyond first year!). My idea with ‘quantifier packaging’ is to look at some easier concepts first, before building up to the full definition.

Anyway, if you are interested in this idea, see the new Blogger post at

FPM quiz question on permutations

Here is another question and (partial) answer from my FPM Piazza forum last autumn, this time related to the “challenge question” from my First quiz on permutations.

First, here is a screenshot of the relevant quiz question. (You can click on the image to view it full size.)


Challenge question from first FPM quiz on permutations

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Cartesian squares and ordered pairs

Here is another question and answer from my FPM Piazza forum last autumn.

Question: Suppose that S is a set with two elements, say S=\{a,b\}. When looking at elements of the Cartesian square S  \times S, are (a,b) and (b,a) the same element, or are they different elements? Does S \times S have four different elements, or only three?

My answer:

The key term in the definition of Cartesian squares, and generally Cartesian products, is “ordered pair”. When you use standard round brackets in this way, the order does matter. You have specified a first coordinate and a second coordinate.
For example, if you work in \mathbb{R} \times \mathbb{R} = \mathbb{R}^2, the point (1,0) (which lies on the x-axis) is different from the point (0,1) (on the y-axis).

Many of the sets S we have looked at are subsets of \mathbb{R}, and this results in S \times S being a subset of \mathbb{R}^2. When this happens, you can often think of points in our Cartesian square as being points in 2-dimensional space.

For example, if S=\{1,3\}, then S \times S has four different points, which you can think of as being points in \mathbb{R}^2 that are at the corners of a square:
         S \times S = \{(1,1), (1,3), (3,1), (3,3)\}\,.
On the other hand, if S is the closed interval [0,1]=\{x \in \mathbb{R}: 0 \leq x \leq 1\}, then S \times S really is the “unit square” \{(x,y) \in \mathbb{R}^2: 0 \leq x \leq 1, 0 \leq y \leq 1\,\}.

Best wishes,
Dr Feinstein

Taster Sessions Update September 1 2021

For other messages in this series, see

Hi everyone,

As mentioned in my previous message, we have recently added to our YouTube channel ( a video of a masterclass by Jorma Louko. Here is some information about this masterclass.

From satellite navigation to quantum black holes
Speaker: Dr Jorma Louko

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Taster Sessions Update August 29 2021

For other messages in this series, see

Hi everyone,

Taster Sessions videos (with captions) are now available on our YouTube channel ( from all six of our May/June 2021 Taster Sessions:

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What is the best way to discuss FPM quiz challenge problems?

I am wondering how best to proceed with the FPM quiz challenge problems (and maybe some other problems I post). Of course, I could simply post full solutions here on WordPress and/or on my Blogger blog (where the maths can be presented using MathJax). Or I could issue some hints first. Or readers could suggest solutions, or give hints. Or I could post some hints on social media (Twitter etc.).

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Challenge questions from FPM quizzes: First quiz on sets and subsets

My fourth FPM quiz last autumn was on sets and subsets. Again the “challenge” question isn’t too hard, as long as you understand the basic concepts and definitions, though it is easy to make mistakes.

Here is a screenshot of the question. As usual, the buttons don’t do anything, but you can enlarge the image by clicking on it.

Click on the screenshot to enlarge the image

Here \mathbb{R}, \mathbb{Q} and \mathbb{Z} have their usual meanings, and \emptyset is the empty set.

You also need to know about the operations of intersection (denoted by \cap) and set difference (denoted by a backslash, \setminus) and the “subset” relation (denoted by \subseteq). Here I use the “subset or equals” notation \subseteq to make it clear that sets which are equal do count as subsets of each other.

In particular, note that the notation Y \nsubseteq X means that Y is not a subset of X, which is equivalent to saying that there is at least one element of Y that is not an element of X.

Challenge questions from FPM quizzes: First quiz on prime factorization

My third FPM quiz last autumn was on prime factorization. As usual the final question was labelled as a “challenge question”. This one is probably relatively easy once you really understand the definition of the set S, which is closely related to the material from my classes on Bézout’s lemma. (But you don’t need to know about Bézout’s lemma to answer the question.)

Here is a screenshot of the question. As usual, the buttons don’t do anything, but you can enlarge the image by clicking on it.

Note that three of the statements are true and one of the statements is false. You are supposed to spot the false one!

Challenge questions from FPM quizzes: First quiz on rational and irrational numbers

My second FPM quiz last autumn was a First quiz on rational and irrational numbers.

Here is a screenshot of the challenge question.

Note added: Thanks for asking about whether zero is a natural number. This varies in the literature, and (as you can check) the answer here depends crucially on this. For my teaching in Nottingham, zero is not included in the natural numbers. So the natural numbers are the strictly positive integers, i.e.,


Challenge questions from FPM quizzes: GCD1

Perhaps it would be interesting to post some of my so-called “challenge” questions from my FPM quizzes. I won’t post them all at once. Here, as a png image, is the very first “challenge” question I set them, on GCDs (Greatest Common Divisors, also known as Highest Common Factors). If you click on the image of the question you can enlarge it. However, it is just a screenshot, so the buttons don’t do anything. Enjoy!

Screenshot of “challenge” question from FPM First Quiz on GCDs

Quiz question on prime factorization

See also the version of this post on Blogger at

Last year many of us at the University of Nottingham created Moodle quizzes for our students to practise on. I had some fun with the technical aspects of this, and may post on that another time. Typically my FPM quizzes would have a few relatively routine questions, followed by a “challenge” question (clearly labelled as such) which was intended to stretch them. Quite often we would discuss the challenge questions in the “live” sessions. I also made a small number of “challenge quizzes” (clearly labelled again) where all of the questions were challenge questions.

Here is a Piazza post I made to explain one of the “relatively routine” quiz questions related to prime factorization.

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Reflexive relations and the diagonal

See also the version of this post on Blogger at

In a first-year FPM workshop last year, I mentioned the diagonal in the Cartesian square of a set, and the connection with reflexive relations on the set. Relations R on a set S correspond to subsets M of S\times S in a standard way. And then the reflexive relations on S correspond to those subsets M of S\times S such that the diagonal is a subset of M.

I had a question on Piazza asking me to explain a bit more about what the diagonal is. Here is my reply.

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Maths Taster Sessions mailing list

As you know, in May and June this year we ran a series of maths Taster Sessions at the University of Nottingham (UoN). We hope to run some more this autumn.

The associated web page has details of these events, as well as links to slides and (once ready) videos from these events. There is also an opportunity to sign up for the UoN Maths Taster Sessions mailing list. If you sign up for the mailing list, I send a “welcome” email with some further information in it. Here is the current version of the welcome email (with greetings and small print removed).

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Relationships between powers of integers, Part 1

See also the version of this post on Blogger at

Here is another question and (incomplete) answer from my Autumn 2020 FPM Piazza Forum. The question under discussion comes from my Sample Exam Paper, and the student was asking how to approach it, and whether the Fundamental Theorem of Arithmetic could help.

Here is the question:

sample question
Question from FPM Sample Exam Paper, Autumn 2020
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