Posted June 10, 2001
I was over at Kathy's house a few days ago, and she was studying for an exam in some business course she's taking. She ran across something in her textbook that seemed wrong, and she asked me about it. To my delight, not only was it wrong, but it was wrong about math. As if that weren't enough, it was also ironic.
This textbook, whose title and author I fortunately didn't catch, was trying to tell my friend something totally ridiculous about the exponential function. It quoted someone, probably Al Bartlett, as saying, "The greatest shortcoming of the human race is our inability to understand the exponential function." True, the exponential function is important, it's all around us, and most people don't know what it is. But to write in a textbook that something is the greatest shortcoming of the human race, you have to give some sort of evidence. The strategy used by the author was to show that that he doesn't understand it himself. There's a little sidebar -- the kind that's indicates that you should remember this if you're going to remember anything on this page -- and it says that the exponential function always increases. Which is true enough, but as an explanation, it's like explaining that an automobile is something that has a door. It doesn't really convey the importance.
So what is the exponential function? In short, it's something like 2^x or 10^x or e^x. When you add 1 to the input, the output is multiplied by some constant factor. My personal favourite, of course, is 2^x, because 2^x is how many x-bit binary numbers there are. Compound interest works exponentially: as each month passes, your bank balance is multiplied by 1.02, or 1.03 or whatever passes for an interest rate at the local Bank of Midas. And, as the financial planners will tell you, just about any interest rate is enough to make you a million bucks, if you can just manage to put in some money every month, and not take it out for 40 years. Of course, in 40 years, a million bucks might only buy you half a pair of Nike shoes, but at least you'll be a millionaire. See, you have to watch out because inflation also works exponentially. Financial planners often forget to mention that.
There are numerous interesting properties of the exponential function. Never mind that it always increases. Lots of functions always increase. This one is special. Picture a graph of e^x. If you don't know what e is, it's one of those special numbers like pi, and it's about 2.7. I'll tell you how to calculate the exact value later. On the far left, at e^-10, it's just a little above the X axis, and it's pointing nearly exactly horizontal. It gets a little steeper as it approaches the Y axis, which it intersects at Y=1. This should be no surprise because at the Y axis, X=0 and e^0=1, just like anything else to the power of 0. On the right hand side of the Y axis, it gets really big really fast. The further you look to the right, the bigger it is, and the faster it grows. At X=10, it's around 22000. At X=20, it's about half a billion. At X=100, well, I don't know a handy abbreviation for 44-digit numbers so I can't tell you.
If you followed that, you'll agree that e^x is quite a dramatic function. It grows too fast for normal graph paper, so if you plan to graph it, you would do well to compress the Y axis so you can get a better view. Let's try compressing the Y axis by 22000, essentially graphing e^x/22000, and see what shape we get. Well, starting at the far left with X=-10, we're even closer than before to the X axis, but you wouldn't know the difference by looking. At the Y axis where X=0, we've got e^x=1, but if we divide by 22000, we'll still be pretty darn close to the X axis. On the right at X=10, we'll get up to one square of graph paper. This is more reasonable than last time, where we were off the page by several miles at X=10. So far so good, but if we try X=20, we get 22000. And if X=30, we get half a billion again. So all we achieved by compressing the Y axis by 22000 was to move the problem 10 squares to the right. But the graph is still exactly the same shape. This is one of the handy things about the exponential function: it's exactly the same shape no matter how much you compress the Y axis.
Another cool thing about the exponential function, which is only cool if you think calculus is cool -- and in case you're wondering, yes, I think calculus is cool -- is that it's equal to its first derivative. And its second derivative, and its third derivative, and all the rest of them too. A derivative is a function that describes how fast another function changes. Speed describes how fast your position is changing, so speed is the derivative of position. Acceleration describes how fast your speed is changing, so acceleration is the derivative of speed. Acceleration is the derivative of the derivative of position; friends call it the second derivative. Case in point: e^x describes how fast e^x changes. When e^x is 1, it's got a slope of 1. When e^x is 22000, it's got a slope of 22000. This is a pretty good definition of e, because there's only one number that works out that way.
e^x is not the only function which is it's own derivative; there is one other important but boring function which is equal to its first, second, third, and all the rest of its derivatives. That function would be zero. The rate at which zero changes is zero.
So I'm going to say that the fundamental failing of the human race is that we can't understand zero. What part of zero don't you understand?
I said earlier that I would tell you how to calculate e exactly. Calculus fans will be pleased to know that this involves using the Taylor series. The Taylor series is a formula that approximates the value of a function F at X by starting with a nearby point near X, where the value of the function is known. It adds smaller and smaller refinements to the approximation by taking the first, second, third, and subsequent derivatives of F at A, and dividing by 1 factorial, 2 factorial, 3 factorial, and so on. If you take an infinite number of approximations, you will get infinitely close to the exact value of F at X, without ever actually calculating F at X, or any of the derivatives of F at X. Like many clever math tricks with names like Taylor Series, this may sound futile, since you'd think you could just calculate F at X instead of A in the first place, and then you could forego all the factorials and derivatives.
But if it were futile, I certainly wouldn't be doing it on this show, now would I? The point here is that F is e^x and we don't know what e is so we can't calculate any value of F. Except one. We do know that F at 0 is 1, because whatever e happens to be, everyone knows that if you raise it to the power of 0, the answer is 1. We also know that the first, second, and third derivatives, and all the rest of the derivatives of F, are all just e^x. And since we only need the derivatives at X=0, we can just say they're all 1. So the part of the Taylor Series that uses the nth derivative will just be 1. So if we want to know what e is, we can just calculate e^1 using the Taylor series. This is probably the easiest application of the Taylor series ever invented; almost everything in the whole Taylor formula reduces to 1 for one reason or another, and all we're left with is 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...
You can try that on a calculator. As you add more inverses of factorials, you'll get closer to the actual value of e. By the time you add 1/4! you'll have 2.7; if you get to 12 terms, you'll see something like 2.71828182828, which may lead you to think there's some kind of pattern. But don't let that thought carry you away because soon enough you'll get to 2.71828182845905 which will start to look just as unfathomable as 3.14159265358979. The patterns come and go but those numbers are definitely irrational. That means that there is no pair of integers you can divide in order to get either e or pi.
Oh, and the factorial of X? Just multiply X by all the natural numbers below X. 3 factorial is 3 x 2. 4 factorial is 4 x 3 x 2, also known as 24. 1 factorial is 1. 150 factorial is about 260 digits long. Your calculator might even have a button on it that calculates factorials for you; if so, it'll say x with an exclamation point. Like it's really excited. X!
And if you want to check your Taylor approximation, any calculator with a factorial button, and for that matter any calculator that deserves to exist at all, has a button labelled e^x. So you can press 1, then press the e^x button, and you'll get as much of 2.71828182845905 as can fit on the screen.