Looks like I’ve accidentally created a series of articles here about fundamental numbers. I didn’t intend to; originally, I was just trying to write the zero article I’d promised back during the donorschoose drive.
Anyway. Todays number is *e*, aka Euler’s constant, aka the natural log base. *e* is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn’t expect it.
What is e?
*e* is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the natural logarithm. That means that by definition, if ln(x)=y, then *e*y=x. Given my highly warped sense of humor, and my love of bad puns (especially bad *geek* puns) , I like to call *e* the *unnatural natural number*. (It’s natural in the sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)
But that’s not a sufficient answer. We call it the *natural* logarithm. Why is that bizzare irrational number just a bit smaller than 2 3/4 *natural*?
Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. *e* is the point on the x axis where the area under the curve from 1 is equal to one:
It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …)
It’s also what you get if you take the limit: *lim*n → ∞ (1 + 1/n)n.
It’s also what you get if you work out this very strange looking series:
2 + 1/(1+1/(2+2/(3+3/(4+..))))
It’s also the base of a very strange equation: the derivative of *e*x is… *e*x.
And of course, as I mentioned yesterday, it’s the number that makes the most amazing equation in mathematics work: *e*iπ=-1.
Why does it come up so often? It’s really deeply fundamental. It’s tied to the fundamental structure of numbers. It really is a deeply *natural* number; it’s tied into the shape of a circle, to the basic 1/x curve. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure of *everything*.
Wikipedia even points out that if you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly *e* dollars. (That’s not too suprising; it’s just another way of stating the integral definition of *e*, but it’s got a nice intuitiveness to it.)