| I'll give it a shot. For me, I think about e in terms of compound interest. I.e. you have a bank account and put in a dollar. Its a great bank, they give you 100% interest. So, after a year, you have (1 + 1) = 2 dollars. But wait, instead, they compound twice. That is, every 6 months, you get 50% interest. So now you have (1 + 1/2)^2 = 2.25 dollars. Instead of compounding twice, they compound 4 times. So now you have (1 + 1/4)^4 = 2.44 dollars. Each time you get interest, its given in proportion to what you have in the account at the time. So, if they are compounding 4 times: 1 --> (1.25 * 1) = 1.25 --> (1.25 * 1.25) = 1.56 --> (1.25 * 1.56) = 1.95 --> (1.25 * 1.95) = 2.44. (Please note: Rounding error. I just typed in to two places after the decimal point). Why 2 or 4? Lets just call it compounding n times. As n gets larger, you get more money, but it doesn't go on forever... For n times, the formula for how much money you have is (1 + 1/n)^n. What if you want to continuously compound? I.e. at every instant, growth is proportional to the current value? Well, then you would want something like (1 + 1/infinity)^infinity. But, we can't really do that, so instead we say: lim (n --> inf) of (1 + 1/n)^n And thats what we call e. Start putting in larger and larger values of n and you'll see that it converges to 2.718... The point is that e is pretty much defined as the number you get when you take a number starting at 1 and have it grow in proportion to itself continuously. Instead, if you use 5 instead of e, your growing too fast. In terms of johnaspden's explanation, you'd move too far around the circle, past (-1, 0). Also, johnaspden, your explanations are truly great, really
helped me understand Euler's formula. Edit: Small change. |