stuff I don't understand: the story of e

Pi (π) and e are numbers. Strange and ethereal numbers that cannot be quantified. Because the phrase "be quantified" denotes the perfect tense, which implies a perfected action. Done. Complete. Finito.

Well, actually, finito means finite. But it still captures the anomalous nature of the situation because these numbers (π and e), though less than 4 and 3 respectively, cannot be said ever to end. It makes one wonder how anyone ever computed π, since it's defined (in words) as the ratio of a circle's circumference to its diameter. A ratio is by definition rational, is it not? Someone measured the length of a circle's circumference and its diameter and divided the first by the second. What's irrational about that? A rational number, after all, is defined as one that can be expressed as a ratio of two integers. Yet π is nothing if not irrational.

But this blog isn't about π. That's not the book I read. This blog is about e.

Let's say you have a curve that if the value of the function that defines it is checked at any given point of x (that is, what's y equal to at any given x?), you find out that that value is the same as the slope of the curve at that point. And what if this is true no matter what point you pick on the curve? This represents an exponential function--but not just any exponential function. It's the exponential function. The one that's related to e. The one we'll have to come back to shortly.

Now, we all know that functions are usually expressed in terms of x. Such as x plus two (x+2) or x squared (x²) or x cubed (x³). Usually if you work out these formulas by putting in a numerical value for x, you'll get a numerical answer back. Since this answer changes for every value of x, we usually just call it y till we know what it is for sure. Example: x+2 = y. But exponential functions are different. They have the variable (the x-value) in the exponential position. Like this: three to the x (3^x) or four to the x (4^x) or anything to the x (a^x).

I wrote above about the slope of a curve at any given point (2 paragraphs up). It might seem strange to think of a curve having a slope at just one point, because the direction of the curve is always changing. (Actually, mathematicians usually call even straight lines curves, but let's think of the really curvaceous ones right now). Think of the slope as the general direction that the curve happens to be taking at that point. What if it stopped turning right then? Which way would it be headed? It's something like that. And it turns out that if you know a formula for a given curve, there's a way use it to find a new formula that will give you the slope at any time. (Remember, this e curve I described above is special. No other kind of curve would ever have given us the value of the slope at a point simply by finding out y. In fact, for a long time, no one even knew it existed.)

But exponential functions in general are kind of special too. When you follow the method for finding the slope-at-any-given-point (which math people call the derivative), you just end up with the function you began with multiplied by some constant. The value of that constant would depend on the function we started with in the first place. So someone brilliant thought, what if we could find that number that when it's at the base of the exponential function (as 3 is at the base of this exponential function: 3^x) the constant that comes back to multiply it in the derivative is 1? What if we could find that number? That would mean that we found a function whose derivative (slope-at-any-given-point) is itself (since anything times 1 is itself). This is a HUGE deal, because that would make it so much easier to work out a bunch of high-level mathematical things that I don't even know about yet.

It turns out that number does exist somewhere too. And since the constant that comes back to multiply 2^x is about 0.693 and the one that comes back with 3^x is about 1.099, then they knew the number they sought had to be between 2 and 3. Trying a whole bunch of different numbers in between brought back the answer 2.7182818284 (continuing forever), which they shortened for mortality sake to "e". So e^x is our special function whose derivative is e^x.

This was my first introduction to the number e.