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Wednesday, January 5, 2011

e is a Slippery Little Devil

Can someone please explain this to me in a few sentences that make sense, without calculus?

If we start with P, increase it at a rate of r, compounded n times per time period for t time periods, we end up with


If we compound it more and more frequently to the point where we're compounding it all the time, we're basically doing... (I know this is not rigorous. I'm sorry. I'm trying to make this graspable to any old 16 year old. Not just the future engineers etc.)

$P(1+\frac{r}{\infty})^{\infty t}$

And since


(Again, I realize this would make a real mathematician bleed from the eyeballs. Sorry sorry sorry.)

That means we can replace part of our percent change equation with e, and calculate continuous growth with


To recap:

My question is, why is r in the exponent now? I don't get that.