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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.


  1. The easiest way to fix it would just to change your definition of e so that it's not so much e = (1+1/inf)^inf as e^x = (1+ x/inf)^inf.

  2. With your definition of e, you have (1 + 1/something)^something as something goes to infinity.

    Rewrite (1 + r/n) as (1 + 1/(n/r)) so you have 1 + 1/something inside the parentheses, and multiply the original exponent (nt) by r/r to get
    (1 + 1/(n/r))^[(n/r)rt].

    Rewrite as
    [(1 + 1/(n/r))^(n/r)]^(rt)

    As n goes to infinity, so does n/r, so the stuff in the brackets is e.

  3. I think this page has a reasonable proof that doesn't use calculus that e^x = lim n->inf (1 + 1/n)^n

  4. (I also apologize for making real mathematicians bleed from the eyeballs.)

  5. I usually finesse things by showing them on the calculator that (1+.05/365)^365 (the annual factor for daily compounding at 5% interest) is really, really close to e^.05 (the factor for continuous compounding at 5%). They can then verify that this is also true for any interest rate they choose, and then double check that if they compound hourly the two numbers are even closer.

  6. I concur with Whit. Use that kind of inductive reasoning. That gets us around statements containing limits.

  7. ELEGANCE ALERT! Simple.

    Define s=n/r and substitute. You get P(1+1/s)^rst. And as n -> infinity, so does s. Then rewrite once more as
    P[ (1+1/s)^s ]^rt, and let s go to infinity. The inside is clearly e, leaving Pert.

  8. This comment has been removed by the author.

  9. Sorry, same thing as doug. A little cleaner maybe.

  10. I have used doug & paul's suggested method to bridge the gap between P(1+r/n)^nrt and Pe^rt in my precalculus classes. Combined with with a numerical (table on TI calc) exploration of lim (1+1/x)^x I've found it significantly helpes to demystify where PERT "comes from" without being unduly complicated.

  11. Doug & Paul's explanation is the one I'm accustomed to using.

    As an aside, I think this is a great illustration of why reasoning with infinity makes modern mathematicians bleed from the eyeballs. We didn't ban infinity from calculus because it offends our delicate sensibilities; we did it because infinity covers up cool processes, like how the r gets into the exponent.

    If you "skip straight to infinity," as you found out, the fact that [1 + 1/(N/r)]^N goes to e^r as N grows makes no sense. It's only by watching the whole limiting process that we see what's going on. We need limits because, in calculus, the journey is more important than the destination.

  12. That's a little misleading, Aaron. Modern calculus doesn't differ from that of Newton and Leibniz by "banning infinity". It differs by banning infinitesimals. Specifically, by using rigorously-defined limiting processes instead of epistemically-sloppy "differentials", which are something like numbers, yet don't behave like real numbers in certain important ways.

  13. drmathochist---

    I don't know about Newton and Leibniz, but Euler definitely played with infinitely large numbers---check out his derivation of the power series for sine and cosine. (There's a nice walkthrough on p. 295 of "Rigor and Proof in Mathematics: A Historical Perspective.")

    The linked article says this kind of reasoning was "practiced by most 18th-century mathematicians," but I don't know any other specific examples of infinitely large numbers being used in analysis. If you do, please share! :)

  14. This is a cheesy idea, but it's worked in my classes:

    Imagine investing $1 at 100% annual interest.
    Compound it monthly, weekly, daily, every minute, every hour, every second.... what do you notice?

    Plus the music/ theater kids get to use lyrics from RENT in math class. ("525,600 minutes...")

    This motivates why lim (n-->inf) (1 + 1/n)^n converges to e.


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