Eric the Blue

I’ve found that little kids are interested in numbers, especially big numbers. A big number could refer to the times I’ve had conversations more or less like:

Kid: What do you do at Valley View?

Me: I teach math to some of the older kids, 4th- and 5th-graders.

Kid: Math?

Me: Yeah.

Kid: Ok, what’s 20 kerjillion plus 20 kerjillion?

Me: [after a pause, as if deep in thought] That would be 40 kerjillion.

Kid: [after a pause] Well, what’s 40 kerjillion plus 40 kerjillion?

And it can go on like this for awhile. Fortunately it doesn’t occur to them to ask what’s 137 kerjillion cubed.

Though old I still can experience a little gee-whiz-that’s-wild when it comes to contemplating large numbers. People talk about “exponential growth,” but what’s really boggling is how big the so-called factorials get, almost immediately. Five factorial, which means

5 x 4 x 3 x 2 x 1,

is only, as you can easily compute, 120. Ten factorial–10 x 9 x 8 etc.–is more than 3.6 million. What’s cool is that the concept of the factorial was developed to solve problems that come up here on earth. How many different ways can the 52 cards in a standard deck be arranged? That turns out to be 52 factorial. How big is 52 factorial? According to Warren Weaver, author of Lady Luck, a book on probability theory:

If every human being on earth counted a million of these arrangements per second for twenty-four hours a day for lifetimes of 80 years each, they would have made only a negligible start in the job of counting all these arrangement–not a billionth of a billionth of one percent of them! You can see that to handle even the very mild case of a deck of 52 cards it is necessary to have some mathematical tools with which to work.

A “tool” mentioned by Weaver concerns the following curious expression:

In words, this says that if n is “good-sized,” then n factorial is approximately the product of the square root of 2 times pi times n and n divided by e to the nth power. There are two irrational numbers here, pi and e, the former being the ratio of the circumference of a circle to its diameter. If you think about it, that’s pretty crazy, too: what do circles have to do with factorials? Beats me. If you’re as big a nerd as I am, you can read the Wikipedia article on Stirling’s approximation, which is the name for the above expression. Warning: lots of calculus. You can also read AI’s answer to “What is pi doing in Stirling’s approximation” here. I did and still have no clue.