Five intuitive approaches to teaching the infinitely small

The truth of infinity being central to the process is singular and univocal. I can’t imagine anyone being able to invent 5 let alone 20 ways to uniquely and intuitively approach this in their teaching. — Will Faris

Twenty might require help from my esteemed commenters, but here’s five:

1. Zeno’s first paradox

Suppose a runner is trying to get from point A to point B. To make the traversal, at some point the runner must reach halfway:

To get from the 1/2 point to B, the runner must pass through the 3/4 point. To get from the 3/4 point to B, the runner must pass through the 7/8 point. To get from the 7/8 point to B, the runner must pass through the 15/16 point …

… and so on, hence the runner must pass through an infinite number of points, which is clearly impossible, therefore motion is impossible.

This is Zeno of Elea’s first paradox, and it seems to stick very well in the minds of students; I’ve mentioned it in classes where several months later (when we started asymptotes) students immediately remembered this paradox.

The are fun extra additions, like Thomson’s lamp, if you want to keep the discussion going: suppose your runner has a light which starts at red, and switches from red to green or green to red every time he reaches a halfway point. What color is the lightbulb when the runner reaches point B?

2. .99999… = 1

The source of infinite Internet arguments can also make a source of productive classroom struggle. (I have written about a lesson that included the Internet arguments as part of the discussion.)

Not only are there a wide variety of approaches in explanation why .999… = 1, there’s a wide variety of erroneous arguments to examine, like claiming .999….9 is smaller than .999….99.

3. Archimedes and his approximation of pi

To approximate a circle, one can inscribe and circumscribe a polygon of many sides. As the number of sides approaches infinity, the perimeter of the polygon converges to that of the circle.

4. Fractals

When zooming in appropriately on a fractal like the Mandelbrot Set, the same image can be produced an infinite number of times.

The same applies to a function like as it approaches infinity:

5. The diverging harmonic series

The series

diverges, and there is a fairly elementary proof by Oresme of that, as well as a nice physical representation:

[Source.]

However, a slight modification

converges, again with an easy physical model. Many other modifications of the same series produce interesting results, so this is a good approach to take in a class with an experimental comptuational emphasis where students are playing with Matlab or Mathematica, contrasting stacks that can move to the right forever versus stacks that have some limit.