Exploring infinity and other Surreal Numbers

[post publication update on March 14th, 2016. Lazily doing a google search to get a link for this blog post I learned of this article in Discover Magazine from 1995:


I’d not seen this article before – though I should have as it is linked in Jim Propp’s “The Life of Games” blog post which was the seed of my interest in the Surreal numbers. The original title of this post was completely by accident the same as the title of the Discover Magazine article. After learning about the prior article I have revised the title of this post.]

Yesterday we revisited the surreal numbers by looking at the game “checker stacks”

Revisiting the Surreal Numbers

We explored the values of some of the positions in the game and found some simple stacks that had values of 1/2 and 1/4.  Today we studied some of the more unusual ideas in the game, looking at positions that seem to have infinite and infinitesimal values.

Just to be super clear from the start -I’m not trying to be even remotely formal about the surreal numbers in this project.  Rather, I’m stating a few rules and ideas and we are exploring some simple consequences for fun.  The ideas here are something that I think that many many kids will find fascinating.

So, on to the game following the ideas and terminology in Jim Propp’s Life of Games.

The first thing we looked at was the “deep blue” checker.

The boys seemed to catch on to the idea that the deep blue checker had a value of infinity fairly quickly, though the idea that it was strange that a piece could have an infinite value didn’t become clear until the next part of the talk.  I was pretty happy to see that they wanted to explore what happened when a deep blue played against a deep red – that game seems shows that infinity minus infinity equals 0!


After the discussion about infinity in the last part of the talk, we explored some of the strange properties of these new numbers. First we looked at infinity + 1, which the boys assumed would be the same as infinity. Surprise 🙂

For the last part of the project this morning we looked at a new stack – a blue with a deep red on top of it. I had to do a little bit of review of the game for my younger son at the beginning of this part of the talk, but once we got the rules straight he understood the strange property of the blue + deep red stack – it has a positive value, but that value seems to be less than any positive number we can think of.

So, a fun project showing the boys some neat, though odd, ideas from math. I love how easy it is to lay out some basic properties of the game “checker stacks” and have kids explore the implications of these properties. To me, this is what learning math looks like.


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