Explaining mathematics

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.,

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!

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/08/quiz-question-on-prime-factorization.html

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.

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/08/reflexive-relations-and-diagonal.html

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 on a set correspond to subsets of in a standard way. And then the reflexive relations on correspond to those subsets of such that the diagonal is a subset of .

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

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/08/relationships-between-powers-of.html

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:

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/07/cartesian-squares-and-relations.html where the maths (using MathJax) looks a bit clearer.

I’m continuing to transfer some selected posts from the Piazza forum for my Autumn 2020 first-year pure module Foundations of Pure Mathematics (FPM) to my two blogs (on WordPress and on Blogger).

Don’t forget that videos from this module are available (complete sets of recordings from the 2014-15, 2018-19 and 2019-20 editions of FPM).

Here is a question and answer concerning Cartesian squares and relations.

Question

Hi, can someone please explain how are Cartesian squares and relations connected and why the number of relations on a finite set is the same as the number of subsets of ? 

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/07/squares-and-fourth-powers-modulo-k.html where the maths (using MathJax) looks a bit clearer.

Here is another post from my Autumn 2020 introductory pure maths module’s Piazza Forum, continuing the discussion of squares and other powers in modular arithmetic. This has quite a lot in common with my post on squares modulo .

See also the version of this post on Blogger at https://explaining-maths.blogspot.com/2021/07/squares-modulo-k.html where the maths (using MathJax) looks a bit clearer.

Here is another post from my introductory pure maths Piazza forum from autumn 2020. This time I was asked how we determine, in general, which integers are squares modulo (where is a positive integer which, in this module, is usually relatively small).