Different definitions of e (Part 4): Continuous compound interest

In this series of posts, I consider how two different definitions of the number e are related to each other. The number e is usually introduced at two different places in the mathematics curriculum:

  1. Algebra II/Precalculus: If P dollars are invested at interest rate r for t years with continuous compound interest, then the amount of money after t years is A = Pe^{rt}.
  2. Calculus: The number e is defined to be the number so that the area under the curve y = 1/x from x = 1 to x = e is equal to 1, so that

\displaystyle \int_1^e \frac{dx}{x} = 1.

These two definitions appear to be very, very different. One deals with making money. The other deals with the area under a hyperbola. Amazingly, these two definitions are related to each other. In this series of posts, I’ll discuss the connection between the two.

I should say at the outset that the second definition is usually considered the true definition of e. However, compound interest usually appears earlier in the mathematics curriculum than definite integrals, and so an informal definition of e is given at that stage of the curriculum.

logareagreen lineAt this point in the exposition, I have justified the formula A = \displaystyle P \left(1 + \frac{r}{n} \right)^{nt} for computing the value of an investment when interest is compounded n times a year. We have also seen that A increases as n increases, but that A appears to level off as n gets very large. This observation forms the basis for the continuous compound interest formula A = P e^{rt}.

To begin, let’s consider plug in variables to make the compound interest formula as simple as possible. Let’s start with 1 dollar (so that P = 1) that earns 100% interest (so that r = 1) for one year (so that t = 1). This isn’t financially realistic, of course, but let’s run with it. Then the compound interest formula becomes

A = \displaystyle \left( 1 + \frac{1}{n} \right)^n

As before, let’s see what happens as n increases. As before, I’ll plug numbers into a calculator in real time, asking my students to use their calculators along with me.

  1. If n = 1, then A = (1+1)^1 = 2. I’ll usually double-check with my class to make sure that they believe this answer… that $1 compounded once at 100% interest results in $2.
  2. If n = 2, then A = (1.5)^2 = 2.25.
  3. If n = 4, then A = (1.25)^4 \approx 2.441.
  4. If n = 10, then A = (1.1)^{10} \approx 2.593.
  5. If n = 1000, then A = (1.001)^{1000} \approx 2.71692
  6. If n = 1,000,000, then A = (1.000001)^{1000000} \approx 2.71828

As before, the final amount A appears to be increasing toward something. That something is defined to be the number e. So, as n tends toward infinity, we’ll define the limiting value to be the number e.

ecalculatorIn my experience, college students have no memory of learning how they first saw the number e when they were in high school. They remember is it as coming out of nowhere, as a number in a formula or as a button on a calculator. It really shouldn’t be this way. The above calculation is a natural consequence of the discrete compound interest formula, which makes the appearance of the number e to be a bit more natural.

Of course, this “definition” of the number e is highly informal. What we’re really claiming is

\displaystyle \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n = e.

At this point in the mathematical curriculum, students only have the haziest notion of what a limit actually means, let alone the more formal treatment that’s presented in calculus… not to mention a proper \delta-\epsilon treatment of limit in an honors calculus class or in real analysis. So, mathematically speaking, the above argument should not be considered a proper definition of the number e, but a working definition so that high school students can get comfortable with the number e before seeing it again in their future mathematical courses.

 

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