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?

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*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.)

History

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*e* has less history to it than the other strange numbers we’ve talked about. It’s a comparatively recent discovery.

The first reference to it indirectly by William Oughtred in the 17th century. Oughtred is the guy who invented the slide rule, which works on logarithmic principles; the moment you start looking an logarithms, you’ll start seeing *e*. He didn’t actually name it, or even really work out its value; but he *did* write the first table of the values of the natural logarithm.

Not too much later, it showed up in the work of Leibniz – not too surprising, given that Liebniz was in the process of working out the basics of differential and integral calculus, and *e* shows up all the time in calculus. But Leibniz didn’t call it *e*, he called it b.

The first person to really try to calculate a value for *e* was Bernoulli, who was for some reason obsessed with the limit equation above, and actually calculated it out.

By the time Leibniz’s calculus was published, *e* was well and truly entrenched, and we haven’t been able to avoid it since.

Why the letter *e*? We don’t really know. It was first used by Euler, but he didn’t say why he chose that. Probably as an abbreviation for “exponential”.

Does *e* have a meaning?

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This is a tricky question. Does *e* mean anything? Or is it just an artifact – a number that’s just a result of the way that numbers work?

That’s more a question for philosophers than mathematicians. But I’m inclined to say that the *number* *e* is an artifact; but the *natural logarithm* is deeply meaningful. The natural logarithm is full of amazing properties – it’s the only logarithm that can be written with a closed form series; it’s got that wonderful interval property with the 1/x curve; it really is a deeply natural thing that expresses very important properties of the basic concepts of numbers. As a logarithm, some number had to be the base; it just happens that it works out to the value *e*. But it’s the logarithm that’s really meaningful; and you can calculate the logarithm *without* knowing the value of *e*.