Uncovering the Discovery of e=2.7

[arildno]

Back in my day, if a math exam paper asked you to prove three results about X, it was perfectly acceptable to define X three different ways for the three proofs, without bothering to show that the definitions were equivalent!

As it should be, since you weren't asked to prove the equivalence of those definitions.

 

[bahamagreen]

Wish I could remember where I read about the early origin of e, long before it was known.

What I read stated that farmers that grow seed crops need to put a certain amount of the seed crop aside to plant the next season... but a portion of that put aside will actually be accounting for the seed crop to be put aside for the season after that, and so on ad infinitum, converging to what would later be found as e.

So staying above e was crucial to surviving... anyone ever hear about this?

 

[jbunniii]

Personally I find the connection with $\lim_{n \rightarrow \infty}(1 + \frac 1 n) ^n$ the least motivational approach.

This was how I recall first learning about $e$ in high school. It was motivated by the subject of compound interest. We start with some initial sum $C$ and put it in the bank at an interest rate of $r$.

How much will this grow after one year? It depends on how often the interest is paid. If paid annually, we will simply have $C(1+r)$.

More generally, if it is paid $n$ times per year, then we will have $C(1+r/n)^n$, which is larger than $C(1+r)$ because we were able to earn interest on each interest payment in addition to our original sum.

The best case would be if interest was paid continuously, in which case we get $Ce^r$, where we have defined $e = \lim_{n\rightarrow \infty}(1+1/n)^n$.

Of course, we have to prove that this limit exists and perform the simple change of variables to evaluate $\lim_{n\rightarrow \infty}(1+r/n)^n$.

I certainly found this more motivating than any of the competing definitions. But I like money.