Exploring 0.99999…. = 1 using fractions and binary

Last week my younger son read chapter 2 of Jordan Ellenberg’s How not to be Wrong. In that chapter Ellenberg discusses the the nunber 0.999999…. and whether or not it equals 1.

We discussed his thoughts on that chapter here:

https://mikesmathpage.wordpress.com/2021/01/09/talking-through-chapter-2-of-jordan-ellenbergs-how-not-to-be-wrong-with-my-younger-son/

Today I thought it would be fun to approach the idea from the (slightly) different perspective of using fractions and binary.

We started with a review / refresher of how to write integers in binary since we haven’t talked about that in a while:

Then we talked about how you write fractions in binary including fun problem of writing 1/3 in binary:

Now I posed the question of how could we write 1 in binary – this part turned out to be the rare discussion that was as fun as I’d hoped it would be 🙂



Finally, having found an interesting way to write 1 in binary, we moved on to the question of how to write 1 in base 10:

This was a enjoyable project. The discussion of infinite series in How not to be Wrong is fascinating and accessible to a wide audience. Talking through the ideas in that chapter with my younger son has been really fun!