Over the past eight years I have worked as a tutor for hundreds of high school students, mostly for mathematics, and sometimes for physics. I have gradually come to dislike how math is taught in the United States- to me it is too much about memorizing formulas and procedures, and not enough about understanding where the formulas come from. Whenever I have time to explain how a particular formula came to be, the student almost always seems to pleased to learn it. Also, it’s usually forgotten that the creators of this subject, the mathematicians, were people (often very weird people), whose story is usually worth mentioning.

For this reason, many of my posts are actually attempts to explain some of this background that textbooks and teachers could cover, but usually don’t.

This post is in that vein – to give some background about the constant e. Math students normally encounter e (2.7182818284590452…) in their Precalculus chapter on exponentials and logarithms. They learn that exponential growth and decay can be modeled by the functions ae^kx and ae^-kx respectively, and that e forms the base of what are called Natural Logarithms.

Students often wonder where this rather odd number comes from, and what is special about it. The textbooks that I have seen don’t try to answer this question, perhaps assuming that if the student later takes calculus, they will learn something about e then.

Now, it is true that a knowledge of some calculus is required to fully understand some of the situations where this magic number shows up, and I covered one of these in a recent post. In this article however, I will describe how  e relates to two elementary subjects: compound interest, and exponential functions.

Jacob Bernoulli and Compound Interest

Most readers will have seen the formula for compound interest:

\[A\: =\: P(1\, +\, \frac{r}{n})^{nt}\]

Here, A is the final value of the initial deposit P at the end of t years. r is the annual rate, and n is the number of compounding periods per year. Other things being equal, A increases when the number of compounding periods n increases – monthly compounding (n = 12) produces a bigger return than annual compounding (n = 1).

In 1687, the  Swiss mathematician Jacob Bernoulli (1654 – 1705) wondered what would happen if the compounding periods became ever shorter; say, second by second compounding, and even continuous compounding (n -> infinity).

Clearly, we have to understand what happens to

\[(1\, +\, \frac{r}{n})^{nt}\]

as n approaches infinity. The expression is easier to interpret if we introduce a new variable u, defined such that:

\[\frac{r}{n}\: =\frac{1}{u}\;\; ,\: n\,= \,ru\]

The above expression then becomes:

\[(1\, +\, \frac{1}{u})^{rut}\: =\: [(1\, +\,\frac{1}{u})^{u}]^{rt}\]

Since u goes toward infinity when n does, we need to evaluate

\[\lim_{u\to\infty }(1\, +\, \frac{1}{u})^{u}\]

Let’s try a few numbers:

\[\left ( 1\, + \, \frac{1}{1000} \right )^{1000}\: =\: 2.71692..\]

\[\left ( 1\, + \, \frac{1}{1000000} \right )^{1000000}\: =\: 2.71828.. \]

It’s easy to accept that this expression is converging on a finite value, and that value is, of course, e. The compound interest formula can thus be written as:

\[A\; = \; Pe^{rt}\]

a formula the Precalculus books normally include, though without the derivation.
Bernoulli called this number b, probably after his name, since he had an immense ego.

No one paid much attention to this curious number until about 1740, when the incomparable Leonard Euler discovered that it was of fundamental importance in several areas of mathematics. It was Euler who named the constant “e”.  In contrast to Bernoulli, Euler didn’t choose the letter because his name started with it – he used e because it is the first letter of “exponential”.

e and Exponential Functions

Here is a graph of the exponential function 3^2x

As with any exponential function, it passes through the point (0,1). Note that an exponential function can be rewritten using a different base, and it will be the same function. For example, the above function can be written as 5^1.3652x. (see note below).

Now, let’s examine an important characteristic of exponentials – the rate at which they change. The graph below shows a tangent line touching our exponential curve at the point (0.5, 3).

The slope of this tangent line is a measure of how quickly the function is changing at the point x = 0.5. For an exponential function, the slope of this tangent line has a very interesting property:  the slope at any point is proportional to the value of the function at the point. Stated a bit differently, an exponential function’s rate of change is proportional to it’s size.  This property makes the exponential exactly what we need to model many situations, such as the size of a colony of bacteria, where the rate of increase is some constant times the colony size.

So what is this proportionality constant? Calculus provides the answer. For our example function, the rate of change (tangent slope) at a given x is

\[slope\: =\: 2\ln \left ( 3 \right )\, 3^{2x}\]

Thus, the proportionality constant is 2 ln(3), where the 2 is the coefficient of x in the exponent, and ln(3) is the natural logarithm of the base. Now remember that the base can be anything we like. Therefore, we can decide that the base is to be e. In that case, ln(e) is 1, and the expression for slope above becomes simpler. This is the main reason exponential functions are usually written with a base of e. That is, the rate of change at a given x is just the value of the function times the coefficient of x

Note on change of base

Suppose we want to make two exponential functions equal when they have different bases, a and c:

\[a^{bx}\: =\:c^{dx}\]

Take the log of both sides:

\[\log a^{bx}\: =\:\log c^{dx}\]

or

\[bx\, \log\, a\: =\:dx\, log\, c\]

Solving for d:

\[ d\: =\:b\frac{\log\, a }{\log c} \]

In other words, the new exponential function, with base c, just needs to have the coefficient of x changed according to the formula above.

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This entry was posted on Monday, March 8th, 2010 at 9:34 pm and is filed under Mathematics History, Trig/PreCalculus. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.