[mthiim]

To me it was only when we got to radioactive decay in physics it clicked to me why the particular "e" was special and useful. Background: exp(x) was introduced as the inverse to log(x), defined as the area under 1/x etc. It was already in 1st high school year (before integrals had been introduced!). We showed how you could express other bases as e^(ax) but you could have done that with any other base, so nothing special about e. In calculus we later learned that exp'(x) = exp(x). This made it clear it was special - but not why it was useful.

The clear usefulness appeared in the case of radioactive decay. We first learned about half-life, and clearly you can write the decay function as N(t) = N0 * 0.5^(at) with 1/a being the half-life. But we then learned that the decay values in the data book are often not the half-life but this special "k" that you put in a decay equation that has e as the: N(t) = N0 * e^(-kx). My reaction was: why did they pick e here? Why not just use 0.5 as base and the book would list the half-life? But then we got to the kicker: we learned the formula: Activity = -k*N, meaning that the activity (decays/time-unit) is proportional to the amount of material with k being the proportionality constant. We hadn't learned of differential equations yet and I was confused about what k was, having first seen it in the decay equation and now in the activity equation which were seemingly very different... how on earth could the same number work in both capacities. And I read a bit ahead in the math book and it all made sense. And then I understood it was because of e as the base that the constant k got this property - which it would only get in that particular base. So this showed to me how e was special, allowing the same constant to serve in these two capacities.