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Deriving e Using the Limit Definitione is a constant, usually shrouded in mystery, used mainly to terrorize students in finance and calculus classes. It's irrational (meaning it's a non-terminating, non-repeating decimal figure). It is not only similar to π and i; it's inextricably related to them (as proven by Euler). Every respectable scientific calculator has some approximation of e, and the nicer ones can represent it symbolically (e.g. 3 * e = 3e, rather than 3 * e = 8.15484549). (I have a Casio fx-300ES, and the only gripe I have about it is that it doesn't have a button for π except if you shift first. It represents e and π symbolically.) So for the person who wants to dig deeper and figure out what this e is all about, let's look at how it's derived using the limit definition:
This looks unpleasant, so let's see where this is going by trying a simpler approach at first. Consider the case of compound interest: If you invested $1 into an interest-bearing account at 100% APR, in one year your account would be worth $2. What if you were compounding twice a year? Well, then you'd be compounding twice as often at half the APR (once on June 30 at 50%, once on December 31 at 50%). In this case, you'd end up with $2.25 at the end of the year. What about different compounding schedules? A picture is worth a thousand words:
So what's really happening here? Refer back to the limit definition above, and try to think of this as just (1 + 1/n)n. We're infinitely increasing the number of times we compound, and infinitely decreasing the amount we're compounding. In a sense, this is very similar to calculating area using an integral: we're making smaller and smaller slices, but zillions of them. Before you get too bent out of shape that e is more precise in your calculator than it is calculating it using the limit as n approaches 10 billion, know that people much smarter than I have used computers to calculate it well past 500 billion, and it never terminates and never repeats. (I promise.) What might really warp the mind is that e is so named for Euler ("Euler's Constant"), though actually a guy named Napier was more directly responsible for it. Euler found an interesting property of it, though (Euler's formula):
And this elegant little identity shows the fundamental relationship of e, i, and π. |
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