What’s the Biggest Number in the World?
They go on forever, of course, but one of the entertaining things about mathematics is its habit of defining and naming fabulously large numbers. This can also be useful. The Romans could barely write down the number of citizens in their empire. These days Roman numerals couldn’t write down the number of people on the planet. Not all number systems are created equal.
Millions
The Roman’s could count up to a few million, which turned out to be enough for them. (It is not a coincidence that people never “need” math that hasn’t been invented yet. That’s for essentially the same reason that programmers never need features the programming languages they know don’t have. It’s hard to think things you don’t have language for.)
Googols
Indian place notation made “orders of magnitude” meaningful - every new column increases the size of the largest number by a factor of 10, and that turns out to be plenty for the physical world. There are about 10^80 elementary particles in the universe. And even if we turn out to be missing 99.999999% of the universe, that still doesn’t takes us to 10^90.
Stupid Big
But while the number of “things” is relatively small, the number of relationships between things grows shockingly fast. If there are 20 people at your party there are 171 introductions to make. If you have four tables of five there is a truly ridiculous 11 billion ways of assigning guests to tables.
Does that matter? Sometimes it does. There are 53,644,737,765,488,792,839,237,440,000 different ways to deal a deck of cards to four players. This is a giant number - at one deal per millisecond that’s still about a hundred million times the age of the universe. And yet, if you play cards you care about this number. This is the basis of card odds in Bridge. Sometimes money is at stake.
Stupid Big, But Still Useful
If you think about it, it is extremely odd that such a practically uncountable number could ever have a practical use. This, I think, is one of the distinctive features of mathematics, and one that can either make it seem pointless - why should I care about numbers so big we can never count them? - or almost supernatural.
Here we have tools that let us calculate numbers so large we can’t possibly count to them, and yet we can reason about them, and come to practically important conclusions. We start in reality - 52 cards - we fly off into a fairy land of numbers so big we sometimes need new notations to write them down, and we come back with a discovery that lets us beat the other guy at poker.
Let’s hear it for fairy land.