π at least has an obvious job, it is the ratio of a circle. e just turns up in compound interest, in calculus, and in probability, and I have never understood what it is the ratio of.
What is the actual definition, and why does that particular value fall out of it?
Accepted Answer
e is the base for which the exponential function is its own derivative, and that is the whole of it.
For any base a, the derivative of a to the power x is a to the power x, multiplied by some constant that depends on a. For a = 2 that constant is about 0.693, for a = 3 about 1.099. Somewhere between them there is a base where the constant is exactly 1.
That base is e. It is not chosen for being 2.718; 2.718 is what you measure once you insist on the constant being 1.
Every other appearance follows from that. Compound interest, the normal distribution, radioactive decay: all of them involve something whose rate of change is proportional to its size, and e is the number built for exactly that.
One thing worth unlearning here: e is not the ratio of anything, and looking for a geometric picture like π has will waste your time.
π is defined by a shape. e is defined by a behaviour, namely growth proportional to current size. They feel like the same kind of constant because they are both irrational and both turn up everywhere, but they arrive for different reasons.
The nearest thing to a picture is the area under 1/x from 1 to e, which is exactly 1. Even that is a consequence rather than the definition.
The compound interest limit is the same statement in disguise: e is the limit of (1 + 1/n) to the power n as n grows. That is what continuous growth at 100 percent means, and you get 2.718 because the compounding compounds.
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