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Math: Not-so Easy e

November 29th, 2009 by jason Leave a reply »

In mathematics, something of interest is the slope of a graphed equation at a give point on the line. The slope is defined as rise over run. For straight lines the slope is easy:

x = y

$f(x) = x$

The line is defined as $f(n) = x$ , which just means if we set x to 0 then the result is 0, x to 1 then the result is 1. This straight line has a slope of 1/1, or 1. If it rises one unit, it runs one unit. This is pretty straight forward, but how do we determine the slope for a curved line like the one below?

exp

$f(x) = a^x$

The equation $f(x) = a^x$ is generating the line where a is a constant raised by variable x. The important thing to take away is that we have a curve (in blue), and on a specific point we want to know the slope (in red). There are various ways to determine the slope. The simplest way in a simple case like the one illustrated is to just draw the curve on some graph paper, then guess the slope by drawing it like we see. This isn’t very fancy, and isn’t necessarily very accurate, but it does work well for simple situations. Luckily math comes to the rescue to give us a more accurate answer, and provides a method that works in more complex situations. Modern Calculus (invented independently by some guy named Issac Newton and another guy named Gottfried Wilhelm Leibniz) can find the slope using something called a derivative. I am not going to dig into derivatives here; I am going to cheat a little. I learned derivatives in high school calculus and used them in both high school and college math and science. I know I am blogging about a calculus class I am taking so that isn’t fair. For our discussion we only need to know that there is a transformation that will take an equation and (transformer sounds here) turn it into an equation for the slope at a given point. I will be providing that transformation shortly.

The point of this blog post really isn’t just about slope, its about a special constant known as e. This constant is one of the most important constants in math for a variety of reasons, almost as important as $\pi$ . The constant e has two very unique qualities. First, the derivative of e^x is e^x! What we reallycare about here is that for the general function $f(x) = a^x$ , the only value for a that produces the slope of 1 where x is 0 is e. That figure above is actually the graph of $f(x) = e^x$ . So, what the hell is e? Its a constant value that starts 2.71828182845904523536*. The next question we should ask is, how the hell do we get this number?

An option that is provided by the online textbook of my class is to try to create an infinite series to estimate this value. This didn’t jump out at me as the obvious way to discover the value of e. I thought, “Why the heck is the prof pointing towards this crazy solution?” There was no explanation, so I went searching for why one would use an infinite series to solve for anything, let alone this. I found this interesting web page explaining that infinite series expansions are a rather useful tool in math. What I discovered is that the prof was suggesting the use of something called a Maclaurin Series to find the value of e.

We start with $x^n$ , which we will use to generate our series. Using Maclaurin, the series is:

$e^x = a_0 + a_1x + a_2x^2 + (etc)$

We know that in the original formula $f(x) = e^x$ that that value will be 1 when x is 0. When you put this in context of the series, $a_0 = 1$ if we start with 1 as x value.

When we differentiate the series, we get:

$\dfrac{d e^x}{dx} = a_1 + 2a_2x + 3a_3x^2 + (etc)$

This sets up an interesting equality. The derivative of e^x can be proven to equal e^x as stated above. So we can match up these two equations and see that $a_0 = a_1, and 2a_2 = a_1, 3a_3 = a_2, a_1 = 1/2, etc$ . In general this boils down to each position being:

$\dfrac{x^i}{i!}$

So to sum the whole series we get:

$\sum_{i=0}^{\infty}\dfrac{x^i}{i!}$

Plugging 1 into x yields the sum to find e. You can see that is one of the common equations for e here.

Again, understanding a fundamental value of math requires some work. There are others ways to solver for e, but this is the one that seemed to make the most sense for me. I thought of going through the proof here for why $\dfrac{d e^x}{dx} = e^x$ , but it involved even more math external to the problem at hand and probably would have doubled the length of the post.