I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.
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 ey=x. Given my highly warped sense of humor, and my love of bad puns (especially bad geekpuns) , I like to call e theunnatural 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 naturallogarithm. Why is that bizarre 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: limn → ∞ (1 + 1/n)n.
It’s also what you get if you work out this very strange looking series:
It’s also the base of a very strange equation: the derivative of ex is… ex.
And of course, as I mentioned in my post on i, it’s the number that makes the most amazing equation in mathematics work: eiπ=-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 deeplynaturalnumber; 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 ofeverything.
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 surprising; it’s just another way of stating the integral definition of e, but it’s got a nice intuitiveness to it.)
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 hedidwrite 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?
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 logarithmis 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 withoutknowing the value of e.