Sunday, December 27, 2009

One Plus One

I've been thinking about this for a long time.

And I do think there is something (I do not know precisely what) that is very essential about moving forward from counting to adding and then from adding to multiplication and then from multiplication to exponentiation. This does invite going forward still further (beyond exponentiation).

However; to move from one to the other (counting to adding to multiplication to exponentiation) you can do so by just saying "repeat". Another way to move from one to the other is to think of it geometrically. Both are sort of covered in the post "One Plus One"

I actually wrote the post a couple of days before Christmas and for a Christmas present this year my brother-in-law Frank gave me "ONAG". A couple of years ago I purchase "Surreal Numbers" and quickly thought that ONAG would be the perfect follow on. Of course the post "One Plus One" is light years from what Conway is doing but it really did strike me when I got the Christmas present, I thought "Wow I was just thinking about what numbers are?"


  1. In the post the equation a+b=c is used so one way to generalize this would have been to say, are there three numbers so that a^2 * b^2 = c^2 which would have been the more natural extension of a+b=c (and equally as easy to answer) than the equation a^2 + b^2 = c^2. So this last equation, the one that was used in the post, can be thought of as sort of a 'mixed-extension' that skips over multiplication and mixes addition and exponentiation. That 'skipping-over' bit introduces an extra layer of complexity. Which when extended to a^3 + b^3 = c^3 does not make it too surprising that the complexity takes over and prevents an integer solution. I really still do think that there is some natural geometric way of thinking about all this that sheds light on it. And that feeling is really universal among all the people who have ever thought about this problem.

  2. My brother sent me this link and I think it fits right in with what I posted above: