When I was in grade school, I often had late evening talks with my friend, Peter. Topics of discussion varied wildly from day to day, sometimes about video games like Doom, sometimes about girls, and sometimes about math. On one specific evening, we talked about infinitely small numbers.
I think the topic held our attention because the books we had access to said, in no uncertain terms, that the decimal expansion .9̅ is equivalent to 1. This left no room for a smaller value, in between .9̅ and 1, but which nevertheless was distinct from 1. We found this perplexing, as there seemed to be nothing logically incoherent with the idea of having an infinitely small positive value which we could subtract from 1 — yet the textbooks made it clear that that resulting difference could not be .9̅, since .9̅=1.
This idea is not unique. Many people make the same error, thinking that .9̅ should be strictly less than 1. In the sci.math newsgroup, the main FAQ's top entry for decades showed that .9̅=1. When I took a look at sci.math just now in November 2021, one of the most recent entries is literally someone making the argument that they are distinct. This is a topic that gets brought up again and again, and there's always someone more knowledgeable around that works tirelessly to correct these misunderstandings. (I used to be that guy on the old skeptic forums, though thankfully not on math ones.)
But in order to tread new frontiers in mathematics, you sometimes have to take a "yes, and..." approach. Sometimes when you do so, you're able to reach new ground that later ends up bearing significant fruit. This is how it was with negative numbers, this is how it was with imaginary numbers, and maybe something similar could be said with the idea of a positive number so small that even adding an infinite number of them together will not sum up to 1.
I first discovered the concepts in Berlekamp, Conway, and Guy's Winning Ways for Your Mathematical Plays back when I was working my way through Feynman's Lectures on Physics. I had been gifted a very nice three volume set as a teenager, and while the first book wasn't terribly difficult to get through, I was having trouble understanding books 2 and 3. At the time, I had dropped out of school, and so my only way to read these Feynman lectures required me to first teach myself more complex mathematics. I went to the local library, taking out texts that would help me to get through what Feynman had written, and, occasionally, I'd use the internet to supplement my understanding. Back then we did not have 3Blue1Brown; the best online math explainers were merely paragraphs of html text alongside slowly loading jpgs. So it was hard going. Nevertheless, I kept at it and eventually learned what I needed in order to properly enjoy my boxed set of Feynman lectures.
|Hackenbush girl from WWfYMP.|
It was during one of these online math excursions that I came across Andy Walker's excellent late 1990s-era html maths-explainer: Hackenstrings, and the .999?=1 FAQ. Walker walks us through a simplified version of Conway's Hackenbush idea, showing the beginnings of what we now call surreal numbers. Here, Conway and Knuth take seriously the idea that there could be a positive number so small that adding it to itself infinitely many times would never add up to any traditional real number. This is the first time that an infinitesimal is taken seriously enough to warrant the creation of a new system of numbers. (At least it's the first time since limits replaced infinitesimals in our teaching of calculus.)
At the time, I was too immature to think that I should purchase for myself Winning Ways for Your Mathematical Plays, but I ever so much wish that I had. It's an amazing book and well worth the read.
If you're interested in learning more, I highly recommend this excellent video by Owen Maitzen that does an absolutely amazing job of explaining Hackenbush. While it's an hour long, this is nevertheless one of the most entertaining introductions to a new type of math that I've seen anyone on the internet create. (He's even composed a soundtrack that suits his video perfectly!) Well done, Owen.