At the tail end of last month’s column I mentioned the mystical nature of mathematics. While I have no problem believing in that, I suspect that my audience might have a qualm or two. Before these parallels are explored further, I should present an example.
Let me warn that this month’s column emphasizes mathematics more than future columns will. The mathematician Paul Erdos used to visit people- announcing his presence by saying, “Open your mind,” and discuss mathematics until… well until he was kicked out of the house. Hopefully, this won’t be quite that bad.
When I was an undergraduate at Bard, I had a bad habit. I would write on the walls with magic markers. Whenever I was listening to music or reading or thinking and a line or an idea inspired me, I’d write it up there for later inspiration. Every year I would start out by writing the same thing. It wasn’t a song quote or a statement of identity. It was this:
pi * i
e + 1 = 0
Right now I can see people removing their links to Phish Stats, calling Tela a spy, and otherwise giving up on me. For those of you who decided to give me a chance, let me try to explain why that is so impressive.
There are five constants in that equation. I’m pretty sure that most of you know what 1 and 0 are, but what are the other three? Just to make life difficult, we have to start with the most confusing one- e.
For the sake of completeness, the technical definition of e is “The constant e is the root of the natural logarithm.” There we go. Now that we have that cleared up, we can move on to pi, right? Well maybe another word or two of explanation might be called for.
A logarithm is the answer to the question, “To what power do we have to raise our root to get a number?” To what power must we raise 2 to to get 8? Two cubed is 8, so log base 2 of 8 is 3. Of course then you have to ask why anyone would care about that. As tempted as I am to claim that logs were invented to scare algebra students, they actually had an important historical purpose. One of those rules that everyone hates remembering, is that if a * b = c, log a + log b = log c. So in those bleak days before calculators, if you had to multiply 13131 by 24291, you could look up those numbers in a big book of logs, add the result together, and look up the sum in that book.
OK that’s a log, what’s a natural log? Well like I glossed over before, you always need a base to compute a logarithm. For the most part it doesn’t matter what number you choose, just as long as you’re consistent. The Natural Log is 2.7182818245… and it was chosen because mathematicians just thought that was a natural number to pick, you know. Ok that’s a lie. It was chosen because for some reason the function e^x turns out to be extremely useful for modeling natural phenomena, such as population growth.
There’s one more question that has to be answered. Why e of all letters? Well usually I wouldn’t care about that but this is a great story. The constant e was discovered by the famous mathematician Euler (pronounced “Oiler”). In the paper that introduced e, there were five constants- a, b, c, d, and e. The first four were completely and utterly trivial. They’re so trivial that people suspect that they were just thrown into the paper so the really cool one could be named e for Euler.
Ok that’s e. Stop reading now, go to your stereo and put in Live Dead. Listen to the Dark Star. I’ll wait, I promise.
Great, you’re back. You listened to the St. Stephen too, didn’t you? I guess I’ll forgive you. Now where were we? Oh right, pi.
Of all of the mathematical constants throughout history, only one is so famous that it has managed to be the title of a movie. No, it wasn’t the square root of 2, or the sine of 30 degrees; it was pi. Pi definitely has a certain mystique to it. Almost every mathematically inclined person has memorized pi to an insane amount of places; my personal best was 78 places in one really boring high school summer vacation. There are books about Pi (A Brief History of Pi, and The Joy of Pi). A quick search on pi found pages that convert the digits of pi to music and let you search the digits of pi looking for your birthday.
So why the pi obsession? Not only is pi what I call a “scary number” in that it seems to pop up when you least expect it, but pi was an early example of a really freaky number. Because it comes up naturally, almost every even remotely advanced civilization had an estimation of pi to 2 or 3 decimal places. Since people use this number all the time, what they would really like is for pi to be expressible in the form of a fraction. It would be nice if we new it was 66039557/21021044 or something. Alas, it turned out that that is impossible. Not only is pi an irrational number, it turns out that it (along with e) is a transcendental number. As much as I’d like to claim that such numbers are used for transcendental meditation, what that means is that no polynomial equation (like x^2-5x+2=0 or 98(x^184)-17(x^4)+91=0) could have x=pi as a solution, no matter how weird the equation is. In short, like I said, pi is just freaky.
Finally we get to the third number, “i”. The number “i” was invented to solve a problem about square roots. As you may recall, only positive numbers can have square roots because whenever you square a number you get a positive result. So “i” was just defined out of whole cloth to be the square root of -1. I see the creation of “i” as one of the major turning points in modern mathematics. At its founding, mathematics tried to model reality. Numbers like 6 or the square root of 2 or pi were discovered naturally based on examining the real world. On the other hand, “i” was created out of thin air to be the solution to a problem. This is the way modern math tends to be done; the focus is not on what is true for the real world, rather people wonder what can be said about a system. The menu may not be the meal, but sometimes it is more interesting.
So there you have it. Three numbers: e, pi, and “i”- all extremely important constants, all completely unrelated to each other, all extremely weird; e and pi are transcendental and “i” can’t even really be considered to be a number per se. Yet somehow, against all expectations, in this equation all of their weirdness manages to cancel out. It’s like typing in two random numbers on a calculator, multiplying them together, and getting 12. The odds of this happening are so low that it has to mean something or so it would seem. So I study it. I play around with it (If you solve that equation for “i”, you get that “i” equals the natural log of -1 divided by pi, which would be really cool if taking logs of negative numbers made any sense.).
I write it on my wall and look at it.
pi * i
e + 1 = 0
When you first look at it, it can seem so trivial – random scribbles on a paper. The more you study it though, the more you can see a connection, the more you are drawn outside yourself into a different world, the more excited you can get about the possibilities it raises, the more you can feel that there is something out there – something extremely important but impossible to explain – that gives life greater meaning. If that isn’t the jamband experience, I don’t know what is.
_David Steinberg got his Masters Degree in mathematics from New Mexico State University in 1993. He first discovered the power of live music at the Capitol Centre in 1988 and never has been the same. His Phish stats website is at www.ihoz.com/PhishStats.html