Calculus: where did e come from?

I assembled the following in response to a question from a calculus student. What he asked was literally, “Who first found e?” What he was really asking, I suspect, was more along the lines of: “Where the heck did e come from, and how on earth did anyone find it?”

I think the article here nicely conveys the correct impression: it took mathematicians about 100 years to really appreciate the number e.

My own casual summary — few precise dates and no citations to justify what I say — is as follows.

Natural logarithms appear in 1618: they were not used for tables of logarithms, at that time, but they were mentioned in an appendix. It was a couple of generations before the connection was made between logarithms and exponentials.

The first appearance of the number e is in 1690, in a letter by Leibniz. So it would be easy to say that “he found it”, but it’s more accurate to say that he first played with it.

The first full treatment of the properties of e was in 1748 by Euler, but I know that some of its properties were already known. For example, this URL tells me that Jacob Bernoulli found the limit

e = \lim_{n\to \infty } \, \left(\frac{1}{n}+1\right)^n\ ,

which says that we should use e for computing interest compounded continuously; and since he died in 1705, he did it sometime before then — and in fact might have computed this even before Leibnitz’ letter.

If that were the only important property of e, it might have been called Bernoulli’s constant. But it is more important as a function, exp, in calculus, where e = exp(1).

I also know that Newton knew (about 1665) the infinite series for ln(1+x)

x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}+O\left(x^6\right)

and that he knew the infinite series for e^x

1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+O\left(x^6\right)

and, more importantly, he got one from the other as an inverse function (specifically, the exponential as the inverse of the logarithm).

A very casual summary, then, might be to say that any number of people had been stepping around this thing in the road… Bernoulli took a good look at it as he stepped around it… Leibnitz picked it up, played with it, and put it back down… Euler picked it up, took it home, and mounted it on his wall.

In duller words, e was far more important than people realized for a long time. (OK, I’m on really shaky ground when I talk about what Newton in particular, and all the rest of them in general, “realized”.)

Why is it so damned important?

Its crucial property (OK, one of two) is that “the” exponential function is its own derivative:

\frac{d}{dx} e^x = e^x\ .

In a sense, e was really found — really recognized for what it was — when people understood that property.

I’m not sure I can explain why that property is more than just “cute”. It means that the exponential function is, effectively, the real bulding block in calculus. Powers of x are obvious — but in a very real sense they are barely the skin on a body — the exponential function is the bones and musculature of the body.

(The technical description — which might ring a bell for some of you — is that the exponential function is an eigenfunction of the derivative operator. In less precise terms, it is an eigenvector of the fundamental linear operator of calculus.)

So let’s look quickly at related derivatives, i.e. let’s look at slightly more general exponential functions. Here’s another way to view the question, “How the heck did someone find this number?”

The notion of more general exponential functions isn’t that strange. From logarithms base 10 or logarithms base 2, people were eventually led to look at the inverse functions 2^x and 10^x\ , and more generally a^x\ .

The second URL wrote

\frac{d}{dx}a^x=a^x\ \lim_{h\to 0} \,\frac{a^h-1}{h}

but we can compute that limit if we use calculus. Since “we know” that the exponential and logarithm are inverse functions, we write (like multiplying a fraction by a special form of 1)

a^x = e^{ln (a^x)} = e^{x ln (a)}\ ,

and therefore we have the general formula

\frac{d}{dx}a^x=a^x\ (ln\ a)\ .

(All I’ve done is give the correct name to that limit in the second URL: the limit is simply ln a.)

Now, a calculation which I believe Huygens did was to take two special cases, a = 2 and a = 3:

\frac{d}{dx}2^x= (ln\ 2)\ 2^x= .6931\ 2^x\ .

and

\frac{d}{dx} 3^x= (ln\ 3)\ 3^x = 1.0986\ 3^x\ .

Well, since we see ln(2) <1\ and ln(3) >1\  , “surely” there is a number b between 2 and 3 such that ln\ (b) = 1\ .

Leibnitz called it “b” — Euler called it “e” and so do we. That is, if we take a = e, we get

\frac{d}{dx}e^x=e^x\ (ln\ e) = e^x\ .

This is what makes e so important (in the calculus of functions of real or complex variables).

(In case you’re wondering – no, I have not shown that this e is same as the e for continuously compounded interest. But it is.)

And, just to be complete if confusing, the other crucial property of e pertains to the calculus of functions of a complex variable:

e^{i \theta} = \cos(\theta)+ i\ \sin(\theta)\ .

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