Infinity – When too much is never enough, Dose 2 – Teaching Mathematics: What, When and Why

July 6, 2026

Infinity – When too much is never enough, Dose 2

Much of modern mathematics is the result of centuries of wrestling with infinity. Even in school mathematics, we find the concept pops up here and there. Each time it points to some interesting part of mathematics. This is a guided tour of some of these occurrences. As usual I focus the ideas through my personal experience, this dose especially.

Each episode begins with school mathematics, then will point into some part of the mathematics that you meet at university. These latter parts are too complex to explain in detail here. I hope the reader will be content with a taste, although the more trained might be able to fill in the gaps.

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tom

8 responses to “Infinity – When too much is never enough, Dose 2”

Terence Mills

July 6, 2026 at 10:47 am

Tom

I have a question about infinity.

Consider a square drawn in the plane with vertices (0,0), (0,1), (1,1), (1,0).

I can draw a step function from (0,0) to (1,1) with many steps (horizontal and vertical).

The length of the graph is always 2, no matter many how many steps are involved.

If I create more and more steps I record the total length of the step function as {2, 2, 2, 2, … }.

But the limiting graph has length \sqrt{2}.

Why is it so?

Best wishes Terry

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1. tom
  
  July 6, 2026 at 11:19 am
  
  Hi Terry, nice question.
  
  It reminds me of when I was working on image processing. There we are dealing with a finite grid of points “joined” by neighbours around it. The running joke was that “pi equals 4”. Because of course we only allowed movement side to side or up and down. So the American fundamentalists who claimed “pi equals 3” were even more wrong.
  
  I guess the best answer lies in the question “what is length”. For a curve break it into sections and take the length to be the limit for the sum of the bits as they become finer. So the answer then becomes how to find the length of a bit. In Euclidean geometry we use $\sqrt{x^2+y^2}$ and each point has neighbours infinitely close in any direction. So we are down to the fundamentals of the geometry. If we take the neighbours to be only directly at the sides or above and below, then your argument above is valid?
  
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2. tom
  
  July 9, 2026 at 10:07 am
  
  Hi again Terry I was expecting you to discuss the $l_p$ or $L_p$ metrics. Got to thinking … the general idea of distance would make a lovely university subject, the sort of subject that we occasionally get called upon to teach to students in the arts faculty, where there is scope for the more philosophical side of mathematics. If we start with Pythagoras, we can arrive at the distance in terms of coordinates, $d_2 = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ . But navigating city blocks, the distance becomes $d_1 = |x_2-x_1|+|y_2-y_1|$ . Can we find constraints where distance becomes $d_\infty = \max(|x_2-x_1|, |y_2-y_1|)$ ? Perhaps next the general case $d_p = \sqrt[p] {[|x_2-x_1|^p + |y_2-y_1|^p]}$ . Then defining a circle to be the path of a point moving at a constant distance from another point, getting the various shapes. Basically repurposing your first lecture in functional analysis.
  
  Many ways to go from there. Maybe distance in 3 dimensions, then onto higher dimensions. Distance and angles in $n$ dimensions? Or relativity theory where we include time coordinates in our separation, Minkowski spacetime.
  
  There is a lot of lovely mathematics around the idea of the distance between a point and a set of points. This could give a lead in to the idea of a supremum. Or maybe the path of a point moving at a constant distance from a set of points; some nice shapes and Steiner theorems.
  
  Or instead, the length of a curve. Here we are forced to the idea of a limit. Rectifiable and non-rectifiable curves, fractional dimensions.
  
  Distance between functions, lots of good engineering applications.
  
  Or, for something really fancy, distance between p-adic numbers. Surely we could find a field where $1+1+1 + \ldots = -1/12$ .
  
  This has grown from a single subject to a whole degree in mathematics.
  
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Jon

July 15, 2026 at 1:13 pm

This was a fascinating episode on how Zeno’s paradoxes inspired thinking on infinity – https://www.bbc.co.uk/programmes/b07vs3v1. I got history of maths at uni, but we didn’t learn that Cantor’s work led to the establishment of the modern foundations of maths (and set theory!) – which seems to me to have been a major missed opportunity. I use this sort of thing to inform my teaching.

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1. tom
  
  July 16, 2026 at 2:03 pm
  
  Hi Jon
  
  The site (hopefully) is a forum for people to share their experiences teaching, or learning, mathematics. So please if you would like to share some examples; I am particularly interested in incorporating history in lessons.
  
  [You can include latex, just write the word “latex” in front of an expression, and then wrap the whole thing in dollar symbols.]
  
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  1. Jon
    
    July 21, 2026 at 11:02 am
    
    Thanks Tom
    
    Firstly I should comment on your post – my initial response was based on the thoughts it prompted in me and I was a bit quick off the mark. It’s a great idea to concentrate on infinity – kids love it! Also wonderful to see a story of a teacher going the extra mile and the difference it made. I have to confess that the later parts of the post were beyond me without going back and thinking hard (which I’m not currently really wanting to do – sorry).
    
    As for teaching, I recently taught logarithms and explained their history and context. I’m old so I explained that we didn’t get calculators until Year 11. I explained about log tables and how multiplication and division are so much harder than addition and subtraction and logarithms are a way of turning one into the other. I related it to the British sailing around the world and hence Australia and colonisation (noting ongoing impacts on First Nations peoples). There had to be lots of calculations involved. And I talked about the moon landing etc etc. I explained how someone spent decades preparing the log tables.
    
    I always start a unit with history and context and often a couple of very short videos if I can find an excellent one (max 5 minutes). Takes a long time to find a worthwhile video!
    
    I don’t take very long with this – but it does seem to motivate students.
    
    Even just saying that we wouldn’t have the classroom or any technology without algebra seems to motivate students when I teach it.
    
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tom

July 21, 2026 at 1:18 pm

I love it! I too have a logarithm story. My mother was a Napier. Being vaguely descended from nobility, she and all the family, had four given names. She also had this list of the Lairds of Merchiston, the Napiers. They were mainly Generals or Admirals, called Thomas or Charles, all except one called John who wasted his life doing mathematics. One of the Charles’s even has a statue in Trafalgar square; although he suppressed a rising in India, he seems to have been quite enlightened. I especially like the story of how he banned suti.

tom = Thomas Charles

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Jon

July 26, 2026 at 3:43 pm

What a wonderful story Tom – our provenance is so important!

I don’t think we make enough use of story in maths teaching.

I would of course continue my logarithms context by explaining that the modern world is full of huge numbers and how do we handle huge numbers? – by bringing the power down of course, which logs do e.g. logarithmic graphs

And then of course I would explain that having invented the tool of logs, we discover other uses for it, such as natural logarithms and how logarithms appear in calculus. And then of course I would put up the equation E to the power i pi equals negative one, to give them a sense of wonder.

I’m surprised that no one seems to have an interesting way of relating mathematical topics when they are commenced. I tend to find the introductions in textbooks to be fairly uniformly boring.

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