The number e, sometimes called Euler's number, may be contemplated by recreational mathematicians in the way that others gaze fondly at a painting without knowing how much time has passed. It is, like pi, irrational, and its importance in calculus is roughly analogous to that of pi in geometry. But you don't meet up with pi anywhere except in geometry, whereas e keeps popping up in the strangest places.

The most common definition of e is the limit, as x increases without bound, of (1 + 1/x)^{x}. The first time I had to think about this, I thought the answer must be either 1 or infinity. In favor of the former, the expression in the parenthesis goes to 1 when x is very large, and 1 to any power, no matter how huge, is 1. On the other hand, the expression in the parenthesis is always greater than 1, and, as exponents are powerful, it seems possible the expression "blows up" when x is very large. But it is as if my two arguments to myself fight to a draw, neither able to win in a gazillion rounds, with the result that the limit turns out to be "e," or 2.7182818284. . . . If you are skeptical, get out a calculator and perform, successively, (1.1)^{10}, (1.01)^{100}, (1.001)^{1000}, and so on. The display on the calculator will get closer and closer to 2.7182818284. . . .

The physicist Richard Feynman created a stir with his pastime of cracking safes while working on the Manhattan Project. His methods, however, were not magical. For example, he got access to some top secret papers when, trying combinations that he thought would occur to a physicist, a safe swung open after he dialed 27-18-28.

Suppose you were asked the sum of the following infinite series:

1/0! + 1/1! + 1/2! + 1/3! + 1/4! + . . . .

Would you believe that this, too, is e? That two expressions, one exponential in nature and the other a sum involving factorials, should both equal the same transcendental number--pretty weird, I think. Possibly the diverse means of deriving e is connected to its all-around mathematical versatility. Do you have a problem involving interest compounding continuously? You need e. In calculus, the derivative of the function

f(x) = e^{x}

is e^{x}, a vital fact that turns out to be exceedingly useful. It follows that the slope of a line drawn tangent to the graph of y =e^{x} will be equal to e^{x} at every point. Calculus teachers around the world have made use of these qualities to invent clever, marginally tricky problems. Q: What is the slope of the line tangent to f(x) = e^{x} at (0,1)? A: 1. Q: What are the x- and y-intercepts of a line tangent to f(x) = e^{x} at (1,e)? A: 0, 0.

My favorite uxexpected appearance of e is in certain probability problems. If there is a 1/n chance of winning a casino game, and you play n times, what's the chance you never win? Notice that if n is 2, the answer comes in at (1/2)^{2} = 1/4, or 0.25; if it's 3, it rises to (2/3)^{3} = 8/27 = 0.296; and it it's 4 it rises again to (3/4)^{4} = 0.316. What if it's 100? That would be given by (0.99)^{100}, which is very close to 0.366, which is getting very close to. . . (1/e). There is a famous problem involving a butler checking hats at a party. Each guest has a space reserved for his hat, but the butler is mute and doesn't know anyone's name, so can only guess. If there are n guests and thus n spots for hats, what is the chance that the butler does not put a single hat in its proper place? For a real big party, that is 1/e, too!