Welcome, all, to the start of my 2018 Mathematics A To Z. Twice each week for the rest of the year I hope to have a short essay explaining a term from mathematics. These are fun and exciting for me to do, since I mostly take requests for the words, and I always think I’m going to be
father farther ahead of deadline than I actually am.
Today’s word comes from longtime friend of my blog Iva Sallay, whose Find the Factors page offers a nice daily recreational logic puzzle. Also trivia about each whole number, in turn.
You know how everything feels messy and complicated right now? But you also feel that, at least in the distant past, they were simpler and easier to understand? And how you hope that, sometime in the future, all our current woes will have faded and things will be simple again? Hold that thought.
There is no one thing that every mathematician does, apart from insist to friends that they can’t do arithmetic well. But there are things many mathematicians do. One of those is to work with functions. A function is this abstract concept. It’s a triplet of things. One is a domain, a set of things that we draw the independent variables from. One is a range, a set of things that we draw the dependent variables from. And last thing is a rule, something that matches each thing in the domain to one thing in the range.
The domain and range can be the same thing. They’re often things like “the real numbers”. They don’t have to be. The rule can be almost anything. It can be simple. It can be complicated. Usually, if it’s interesting, there’s at least something complicated about it.
The asymptote, then, is an expression of our hope that we have to work with something that’s truly simple, but has some temporary complicated stuff messing it up just now. Outside some local embarrassment, our function is close enough to this simpler asymptote. The past and the future are these simpler things. It’s only the present, the local area, that’s messy and confusing.
We can make this precise. Start off with some function we both agree is interesting. Reach deep into the imagination to call it ‘f’. Suppose that there is an asymptote. That’s also a function, with the same domain and range as ‘f’. Let me call it ‘g’, because that’s a letter very near ‘f’.
You give me some tolerance for error. This number mathematicians usually call ‘ε’. We usually think of it as a small thing. But all we need is that it’s larger than zero. Anyway, you give me that ε. Then I can give you, for that ε, some bounded region in the domain. Everywhere outside that region, the difference between ‘f’ and ‘g’ is smaller than ε. That is, our complicated original function ‘f’ and the asymptote ‘g’ are indistinguishable enough. At least everywhere except this little patch of the domain. There’s different regions for different ε values, unless something weird is going on. The smaller then ε the bigger the region of exceptions. But if the domain is something like the real numbers, well, big deal. Our function and our asymptote are indistinguishable roughly everywhere.
If there is an asymptote. We’re not guaranteed there is one. But if there is, we know some nice things. We know what our function looks like, at least outside this local range of extra complication. If the domain represents something like time or space, and it often does, then the asymptote represents the big picture. What things look like in deep time. What things look like globally. When studying a function we can divide it into the easy part of the asymptote and the local part that’s “function minus the asymptote”.
Usually we meet asymptotes in high school algebra. They’re a pair of crossed lines that hang around hyperbolas. They help you sketch out the hyperbola. Find equations for the asymptotes. Draw these crossed lines. Figure whether the hyperbola should go above-and-below or left-and-right of the crossed lines. Draw discs accordingly. Then match them up to the crossed lines. Asymptotes don’t seem to do much else there. A parabola, the other exotic shape you meet about the same time, doesn’t have any asymptote that’s any simpler than itself. A circle or an ellipse, which you met before but now have equations to deal with, doesn’t have an asymptote at all. They aren’t big enough to have any. So at first introduction asymptotes seem like a lot of mechanism for a slight problem. We don’t need accurate hand-drawn graphs of hyperbolas that much.
In more complicated mathematics they get useful again. In dynamical systems we look at descriptions of how something behaves in time. Often its behavior will have an asymptote. Not always, but it’s nice to see when it does. When we study operations, how long it takes to do a task, we see asymptotes all over the place. How long it takes to perform a task depends on how big a problem it is we’re trying to solve. The relationship between how big the thing is and how long it takes to do is some function. The asymptote appears when thinking about solving huge examples of the problem. What rule most dominates how hard the biggest problems are? That’s the asymptote, in this case.
Not everything has an asymptote. Some functions are always as complicated as they started. Oscillations, for example, if they don’t dampen out. A sine wave isn’t complicated. Not if you’re the kind of person who’ll write things like “a sine wave isn’t complicated”. But if the size of the oscillations doesn’t decrease, then there can’t be an asymptote. Functions might be chaotic, with values that vary along some truly complicated system, and so never have an asymptote.
But often we can find a simpler function that looks enough like the function we care about. Everywhere except some local little embarrassment. We can enjoy the promise that things were understandable at one point, and maybe will be again.