While others have already answered your question, there's another thing to bear in mind: the most elegant way to think about some concept in mathematics is not

While others have already answered your question, there's another thing to bear in mind: the most elegant way to think about some concept in mathematics is not necessarily the best way to introduce it to someone who is new to the topic. There are multiple ways of defining either the number e or the function that takes x to e^x (you can define one of them in terms of the other; if you know e, you can define e^x in the same way you usually define exponentiation: first you define it for integers, then rationals, and then you "fill in the gaps" for the irrational arguments (let's leave out complex numbers for now); while, if you know how to compute e^x for any x, then e is just that function evaluated at 1). Each of them highlights a different aspect of that all-important number. What the GP was trying to explain was that the exponential function to the base of e, up to a multiplicative constant, is the only function whose derivative (roughly: slope) is equal to itself. That is: any such function is of the form a*e^x for some real number a. This is certainly one big reason for why e plays such a central role in mathematics. But I think for someone not very familiar with all these topics, it's much easier to think of e as some number somewhere between 2,5 and 3. There are also some explanations for where exactly this number comes from, e.g. as the basis for continuously compounded interest (which is useful in economics), but they do require the student to be comfortable with some sort of, at least informal, limiting process.