Different definitions of e (Part 2): Discrete 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 line

In yesterday’s post, I used a numerical example to justify the compound interest formula A = \displaystyle P \left(1 + \frac{r}{n} \right)^{nt} when interest is compounded n times a year. In a couple of days, I’ll discuss how the above formula naturally leads to the formula A = P e^{rt} when interest is continuously compounded. Today, I’d like to give some pedagogical thoughts about both formulas.

The mathematics in yesterday’s post was pretty straightforward: apply the simple interest formula I = P r t a few times and see if a pattern can be developed. However, my observation is that college students have no memory of being taught how the compound interest formula A = P \displaystyle \left( 1 + \frac{r}{n} \right)^{nt} can be seen as a natural consequence of the simple interest formula. In other words, they’d just use the compound interest formula without having any conceptual understanding of where the formula came from.

For my math majors who aspire to become secondary teachers in the future, I’ll make my observation that there’s absolutely no reason why students couldn’t discover this formula on their own similar to the outline above. Doubtlessly, it would take more time that I use in my college class… I can usually cover the points in yesterday’s post in about 10 minutes or less, even allowing for students to pause and interject the next step of the calculation. So while the pace would be slower for a class of high school students, the mathematical ideas are simple enough to be understood by high school students.


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