Teaching Mathematics: What, When and Why

An in-depth examination of mathematics education, topic by topic

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

  1. Terence Mills Avatar
    Terence Mills

    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 Avatar
      tom

      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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