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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 \frac{1}{2}^x as it approaches infinity:

5. The diverging harmonic series

The series

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

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


However, a slight modification

\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots

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.


9 Responses

  1. I’m a high school math teacher and I sometimes run into issues explaining why an infinite geometric series can have a finite sum.

    Unlike the surprising 0.9999… = 1 example, I like to consider a repeating decimal they *have* seen lots before. Students are FINE understanding that 0.333333… = 1/3, a finite result. Yet 0.3333… is the sum of an infinite geometric series: 0.3 + 0.03 + 0.003 + …

    Students can’t deny either fact. It’s certainly an infinite series. And it certainly has a finite sum–they’ve known that since grade school. So they come to this understanding in their own mind, and their lives are forever changed :-).

  2. this is a very nice blog. keep it up. i have a related article on this one. you may want to check it out.


  3. Mr. Dyer,

    Is it okay to borrow the fractals part to add to my blog intro to limits blog?


  4. By the way, I have endorsed this link in my blog. You can check it out. 🙂

  5. […] Mr. Jayson Dyer, author of The Number Warrior has another excellent explanation on the concept of limits in his blog Five intuitive approaches to teaching the infinitely small. […]

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