Gee, I appear to have touched on a nerve here. Unfortunately, my copy of Infinity and the Mind is at the office, so I’m going to have to wing this one.
Okay, many of you have probably heard of John Conway, the 20th century’s most eccentric mathematician. He invented the “game” of Life, discovered the Conway group, and almost in his spare time built up the framework of modern game theory. All these are amazing things, but I expect that the thing he will be most remembered for among mathematicians is the surreal numbers.
A very quick course on the surreal numbers:
[/list][li]A surreal number is a pair of sets of surreal numbers, a left set and a right set, with every element of the left set less than every element of the right set; conversely, every such pair of sets is a surreal number. If the left set is A and the right set is B, we denote the number by (A|B).[/li]li <= (C|D) if every element of A is less than (C|D) and (A|B) is less than every element of D. (A|B) = (C|D) if (A|B) <= (C|D) and (C|D) <= (A|B). (A|B) < (C|D) if (A|B) <= (C|D) and ((C|D) <= (A|B) is false).[/li][li]Hi Opal![/li]li + (C|D) = (a + (C|D), c + (A|B) | (A|B) + d, (C|D) + b), where the lower case letters are numbers that are elements of the respective capital letters.[/li][/list]
There is a similar rule for multiplication but it gets a little tricky. Since the empty set is (vacuously) a set of surreal numbers, the “simplest” surreal number is ( | ) (when writing out sets in surreal numbers, we leave the braces off), which we call 0. The next-simplest numbers are (0| ) and ( |0). You cannot have (0| ) <= 0, since by our second rule that would mean 0 < 0; but 0 <= 0, so 0 < 0 is false. (I know, the definitions are a little circular, but you can make it work.) However, 0 <= (0| ), so we have 0 < (0| ); likewise ( |0) < 0. We call (0| ) 1 and ( |0) -1. It turns out by the addition rule we have above, 0 is the additive identity, and by the multiplication rule that I didn’t write out, 1 is the multiplicative identity. By simple repetition of the rules, we can construct (among many other things) all the real numbers in a pretty straightforward way, so that the addition and multiplication on surreal numbers is the same as conventional addition and multiplication. I’m going to ask you to buy that for a moment.
It turns out that there is a very natural “binary expansion” for every surreal number. I’m not going to go into enormous detail, except to say that it matches up pretty closely with our usual concept of binary expansion; i.e., 1 = 1.0000…, 1/2 = 0.100000…, 1/3 = 0.0101010101… and so on. The tricky bit is, every binary expansion turns out to correspond to a unique surreal number. Where does that leave us? Well, it leaves us with 0.11111… being a different number than 1.00000…, for starters, but that isn’t a lot of help. What number is it?
Well, let’s go back to our other notation. (Sometimes, having two notations can help explain things that one notation alone can’t.) 0.10000… = (0|1) = 1/2, 0.110000… = (0,1/2|1) = 3/4, 0.1110000… = (0,1/2,3/4|1) = 7/8, … so intuitively we should have 0.111111… = (0,1/2,3/4,7/8,15/16,…|1), where the left set is infinite (but each of its elements is strictly less than 1). This surreal number is less than 1, but greater than every real number between 0 and 1. The normal way to express it is 1 - 1/omega, where you can think of omega as being infinity.*
How does this work with the usual proof that .999999… = 1?
The proof as mathematicians usually write it out goes as follows. Call the value of 0.9999999… S. Then
10S = 9.9999999…
S = 0.9999999…
Subtracting on both sides, we get
9S = 9
Therefore, S = 1. But clearly we’re missing something if we try this out with .999999… = 1 - 1/omega. The problem is, we can get at 9.99999… two different ways - we can multiply .999999… by 10, or we can add 9 to it instead. (This is the trick that makes the proof work.) In the world of surreal numbers, the two methods are not compatible and give you different answers: 9 + (1 - 1/omega) = 10 - 1/omega, but 10(1 - 1/omega) = 10 - 10/omega, which is a distinct surreal number, although the two are only infinitesimally different.
Bored yet?
- In fact, omega is the surreal number (0,1,2,3,…| ); it’s the first of the “transfinite” surreal numbers, but it’s neither the greatest nor the smallest. For instance, (|omega) < omega < (omega|), and so on. Reciprocals of infinite surreal numbers exist; in fact, every surreal except 0 has a unique inverse. If this sort of stuff is endlessly fascinating to you, see if you can track down a copy of Donald Knuth’s little book Surreal Numbers. If you’re still not satisfied, then try Conway’s book On Numbers and Games, preferrably the second edition. Or send me an e-mail; I find this stuff endlessly fascinating too.