# Boxing Pythagoras

## More on 0.999…=1

In my last post, I discussed a particular video which I found to be more than a bit misleading. The discussion centered around a simple, but extremely counterintuitive notion of mathematics: the fact that the number 0.999…, or zero-point-nine-repeating, is equal to 1.

Well, as I mentioned, the very counterintuitive nature of the result led at least one of my readers to question its validity. As such, I thought I would lay out one proof of this concept, in order to make it easier for those who do not accept the result to pinpoint exactly where they disagree. I’ll break my proof down into numbered steps, to ease in that venture.

## Yet another failed attempt at showing 0.999…≠1

I’ve discussed before how mathematics can sometimes lead to very counterintuitive results. One of the most common, and famous, of these counterintuitive properties of math is that the number 0.999… (that is, zero point nine, nine, nine, repeating) is equal to 1. This one is so well known that it is fairly often taught even to Elementary and High School students. If you are unfamiliar with this discussion, I highly recommend that you watch this video from Vi Hart, in which she discusses 10 different reasons to accept this concept. Additionally, you may have fun watching this video, in which she lampoons the common objections to the concept.

Despite the fact that it is fairly simple to prove that 0.999…=1, the concept is so counterintuitive that I find people try to struggle against it– even when they know and accept the reasoning behind the equality. One such attempt comes from Presh Talwalkar. In the following video, Mr. Talwalkar attempts to demonstrate that on the Surreal number system, 0.999…≠1.

Unfortunately for Mr. Talwalkar, he is wrong. Even on the Surreals, it is still true that 0.999…=1.