A few months ago I saw a really cool blog post by Jim Propp about the surreal numbers. It inspired me to try explaining the some of the ideas to my kids and it went really well:

Walking down the path to the surreal numbers Part 1

Walking down the path to the surreal numbers Part 2

Then, a few weeks ago, I saw Donald Knuth’s book about the Surreal Numbers:

which inspired me to revisit these prior projects:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

The second time around we explored slightly more complicated ideas -> 1/4 and , for example.

These two visits to the surreal numbers made me think that kids would really have fun seeing them, so I’ve decided to do a dive into the surreal numbers at next week’s 4th and 5th grade Family Math night.

My goal will be to make the hour long lesson / discussion look a lot like the two prior excursions with my kids:

(1) Introduce the game of “checker stacks” from Propp’s blog.

(2) Explain that the game positions can be represented by numbers and that the game has special properties when the position is positive, negative, and 0. The kids (and parents) will have to take this part for granted.

(3) Begin to explore the positions by looking at game positions with integer values. I’ll have the kids play 1 on 1 games versus their parents here.

(4) Explore the game positions that have value 1/2 and -1/2. This is where things get fun ðŸ™‚

Now – based on how things go up to this point we can either try to explore the positions that have value 1/4 and -1/4 or just skip ahead to infinity. With only an hour for the activity, I’ll probably skip 1/4 to be sure that we don’t have to skip infinity!

(5) To get to infinity I have to introduce the “deep blue” and “deep red” checkers from Propp’s blog. Hopefully a couple of examples will help, but this part should also be a neat way for the parents and kids to explore the properties of these checkers.

(6) Once we discover that these checkers have values of plus and minus infinity, we can talk about infinity +1 and things like that.

(7) I’ll wrap up with with the sort of inverse of infinity -> the stack whose value is positive, but smaller than any positive number you can think of!

(8) At the end I’ll mention Siobhan Roberts’s biography of John Conway:

and the quote from the book that the surreal numbers sort of break your brain when you learn about them for the first time.

I hope this will be a fun lesson for the kids in addition to breaking their brains – ha ha ðŸ™‚

## Comments

Until your posts, I had barely heard of surreal numbers. In fact, just a passing comment of Jacob Lurie’s that stuck with me, but I’d always thought was a joke. I wonder if they now appear anywhere in a “formal” math education?

From our own playing with checker stacks today, I would emphasize the two most exciting moments were:

(a) finding the value of 1/2 for BR stack. Not clear why, but they quickly build a confident assumption that the value of positions should be an integer, so the fractional value was a delightful surprise. I think this is heightened by playing first with 4 or 5 games that do have integer values. FWIW, mine preferred to call it 0.5 as yours did, while I naturally thought of it as one-half.

(b) the infinitesimal value for BRRRR….. I think this really got them because it is such an interesting concept and a surprising reversal of expectations based on the value of the deep red when standing as a stack alone.

If at all possible, I’d encourage you to try to get this into your family math night exploration.

In comparison, the deep blue was slightly anti-climactic. Basically, asking them to conceive of a checker with that behavior wasn’t that different from their idea of infinity, so the value it got ended up seeming completely natural.

Also, skipping the deep purple makes a lot of sense to me. I don’t think the kids really cared much about finding a position whose value isn’t a dyadic rational. That aside, if you do include it again, I think it was much easier to play deep purple + RB to show the value was bigger than 1/2 than to work through 2 deep purples + R.