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InfinitesimalPosted September 30, 2001
Hello and welcome to the 21st century. You're listening to CJLY
93.5fm in Nelson, Kootenay Coop Radio. Radiating So if you've ever wondered what happens after forever, or how many fractions exist, or just how thin a two-dimensional surface is, well... you'll probably still have to wonder. But if you thought math was only concerned with numbers, or if you've wondered what a number 8 would look like if you turned it on its side, then maybe I can help you. But first we'll listen to some music. As you listen, you can imagine that I know something you don't, and that you will learn something from me when I come back. Have fun! --- You're listening to Mostly Mozart on Kootenay Coop Radio. Comfort and Joy is happy to sponsor Mostly Mozart. For reasons which I hope are not obvious, today's topic is infinity. Someone accosted me at the Be Good Tanyas concert many months ago, accused me of having a radio show about science (which I don't by the way, it's actually called Mostly Mozart), and he asked me something like, "What's all this about infinity?" I pretended not to hear him, which seemed appropriate at the time, but later I thought maybe that would be an interesting thing to talk about on my Mozart show. Infinity is not a number. You can't have something that's equal to infinity. You can't add one to infinity, or divide it by three. You can get it out of a formula, but you can't put it in. For example, you could say that three divided by zero is infinity. But zero times infinity is not three. Which makes lots of sense if I replace the word "infinity" with the phrase "not a number," like this: three divided by zero is not a number; but zero times (not a number) is not three. In fact, zero times (not a number) is not a number. For example, laughter is not a number. So three times laughter is also not a number.
Well, that's no big surprise. If you turn the number 1 on its side,
that's not a number. If you turn 6 on its side, that's not a number
either. So if you turn 8 on its side, it should be no big surprise
that Even though it's not a number, infinity does have some purpose in life. Intuitively, it's something that would be a number if it weren't so darn big. Mathematically, it's something that numbers can get closer to but never quite reach. If you can't quite reach it, you can't quite type it into a calculator or do long division on it, so you can't have much of an idea of what would happen to it if you did. But fortunately for physicists, and unfortunately for high school students, Newton and Liebnitz invented calculus. Calculus lets you figure out what the answer would be if the question were possible. For example, the intuitive definition of a derivative is the equation that gives the slope, or steepness, of a function at any given point. It's easy to calculate the slope of a straight line; you just pick two points, and divide the vertical difference by the horizontal difference. The answer is how much vertical distance is covered in one unit of horizontal distance. The trouble comes when you try to calculate the slope of a curve. If you just pick two points and divide vertical difference by horizontal difference, you'll get some number. But that number just describes the slope of the straight line passing through the two points. That's not the same as the slope of the curve; it can't be, first of all because the slope of the curve is probably different at one point than at the other point, and you only get one number if you use the Y divided by X formula. You can bring the two points really close together, so that the slope is almost exactly the same at both points, and the straight line between the two points is almost the same as the part of the curve that you're trying to measure. But no matter how close together the points are, the curve will always be slightly different from the straight line, and the slope will always be slightly different at one point than it is at the other. If you bring the points so close together that they're both in the same place, then you solve the problem of the straight line not being the same as the curve -- but you have a new problem, which is that the vertical difference divided by the horizontal difference is always 0 divided by 0. Which tells you absolutely nothing about the slope of the curve at that point. 18th century engineers were probably quite satisfied with this state of affairs, because they could just choose how precise they needed their measurements to be. They could bring the two points sufficiently close together that the difference between the straight line and the curve was negligible, and use the slope of the straight line as if it were the slope of the curve. But mathematicians don't use the word "negligible." Nothing is negligible to a mathematician. Just because it's small doesn't mean it's unimportant. In math you're just not allowed to decide that anything smaller than a thousandth of an inch doesn't matter. So along came Newton and Liebnitz, or maybe I should say along came Newton, and, somewhere far away, along came Liebnitz, neither one knowing what the other was doing. They both came up with the same idea anyway, which was the mathematical equivalent of negligible. The difference between negligible and infinitesimal is that negligible differences are small enough not to care about, while infinitesimals are smaller than any positive number. They can be arbitrarily small, which just means that if you tell me any really tiny positive number, I can just say, "oh, my infinitesimal can be even smaller than that, and my equations will still work". The neat trick of derivatives is to use an infinitesimal difference between the two points on the curve. You can't know exactly how big the difference is; the whole point is that you can make it smaller and smaller without disturbing your calculations. So let's call the horizontal difference delta-X and the vertical distance delta-Y, so the slope is delta-Y divided by delta-X. If we can give Y as a function of X, then we can also give delta-Y as a function of delta-X, and that means we can also give (delta-Y over delta-X), also known as the slope, as a function of delta-X. This should come as no big surprise. We already established that the slope you calculate by drawing a straight line between two points on the curve depends on how close together the two points are. As the two points get closer together, the answer gets more and more precise.
What we'd really like is the answer when delta-X is zero. But the
best we can do is called a limit. If the answers start to converge on
one number as delta-X gets closer and closer to zero, then it's pretty
obvious that |