We come now almost to the end of the Summer 2017 A To Z. Possibly also the end of all these A To Z sequences. Gaurish of, For the love of Mathematics, proposed that I talk about the obvious logical choice. The last promising thing I hadn’t talked about. I have no idea what to do for future A To Z’s, if they’re even possible anymore. But that’s a problem for some later time.
Some good advice that I don’t always take. When starting a new problem, make a list of all the things that seem likely to be relevant. Problems that are worth doing are usually about things. They’ll be quantities like the radius or volume of some interesting surface. The amount of a quantity under consideration. The speed at which something is moving. The rate at which that speed is changing. The length something has to travel. The number of nodes something must go across. Whatever. This all sounds like stuff from story problems. But most interesting mathematics is from a story problem; we want to know what this property is like. Even if we stick to a purely mathematical problem, there’s usually a couple of things that we’re interested in and that we describe. If we’re attacking the four-color map theorem, we have the number of territories to color. We have, for each territory, the number of territories that touch it.
Next, select a name for each of these quantities. Write it down, in the table, next to the term. The volume of the tank is ‘V’. The radius of the tank is ‘r’. The height of the tank is ‘h’. The fluid is flowing in at a rate ‘r’. The fluid is flowing out at a rate, oh, let’s say ‘s’. And so on. You might take a moment to go through and think out which of these variables are connected to which other ones, and how. Volume, for example, is surely something to do with the radius times something to do with the height. It’s nice to have that stuff written down. You may not know the thing you set out to solve, but you at least know you’ve got this under control.
I recommend this. It’s a good way to organize your thoughts. It establishes what things you expect you could know, or could want to know, about the problem. It gives you some hint how these things relate to each other. It sets you up to think about what kinds of relationships you figure to study when you solve the problem. It gives you a lifeline, when you’re lost in a sea of calculation. It’s reassurance that these symbols do mean something. Better, it shows what those things are.
I don’t always do it. I have my excuses. If I’m doing a problem that’s very like one I’ve already recently done, the things affecting it are probably the same. The names to give these variables are probably going to be about the same. Maybe I’ll make a quick sketch to show how the parts of the problem relate. If it seems like less work to recreate my thoughts than to write them down, I skip writing them down. Not always good practice. I tell myself I can always go back and do things the fully right way if I do get lost. So far that’s been true.
So, the names. Suppose I am interested in, say, the length of the longest rod that will fit around this hallway corridor. Then I am in a freshman calculus book, yes. Fine. Suppose I am interested in whether this pinball machine can be angled up the flight of stairs that has a turn in it Then I will measure things like the width of the pinball machine. And the width of the stairs, and of the landing. I will measure this carefully. Pinball machines are heavy and there are many hilarious sad stories of people wedging them into hallways and stairwells four and a half stories up from the street. But: once I have identified, say, ‘width of pinball machine’ as a quantity of interest, why would I ever refer to it as anything but?
This is no dumb question. It is always dangerous to lose the link between the thing we calculate and the thing we are interested in. Without that link we are less able to notice mistakes in either our calculations or the thing we mean to calculate. Without that link we can’t do a sanity check, that reassurance that it’s not plausible we just might fit something 96 feet long around the corner. Or that we estimated that we could fit something of six square feet around the corner. It is common advice in programming computers to always give variables meaningful names. Don’t write ‘T’ when ‘Total’ or, better, ‘Total_Value_Of_Purchase’ is available. Why do we disregard this in mathematics, and switch to ‘T’ instead?
First reason is, well, try writing this stuff out. Your hand (h) will fall off (foff) in about fifteen minutes, twenty seconds. (15′ 20”). If you’re writing a program, the programming environment you have will auto-complete the variable after one or two letters in. Or you can copy and paste the whole name. It’s still good practice to leave a comment about what the variable should represent, if the name leaves any reasonable ambiguity.
Another reason is that sure, we do specific problems for specific cases. But a mathematician is naturally drawn to thinking of general problems, in abstract cases. We see something in common between the problem “a length and a quarter of the length is fifteen feet; what is the length?” and the problem “a volume plus a quarter of the volume is fifteen gallons; what is the volume?”. That one is about lengths and the other about volumes doesn’t concern us. We see a saving in effort by separating the quantity of a thing from the kind of the thing. This restores danger. We must think, after we are done calculating, about whether the answer could make sense. But we can minimize that, we hope. At the least we can check once we’re done to see if our answer makes sense. Maybe even whether it’s right.
For centuries, as the things we now recognize as algebra developed, we would use words. We would talk about the “thing” or the “quantity” or “it”. Some impersonal name, or convenient pronoun. This would often get shortened because anything you write often you write shorter. “Re”, perhaps. In the late 16th century we start to see the “New Algebra”. Here mathematics starts looking like … you know … mathematics. We start to see stuff like “addition” represented with the + symbol instead of an abbreviation for “addition” or a p with a squiggle over it or some other shorthand. We get equals signs. You start to see decimals and exponents. And we start to see letters used in place of numbers whose value we don’t know.
There are a couple kinds of “numbers whose value we don’t know”. One is the number whose value we don’t know, but hope to learn. This is the classic variable we want to solve for. Another kind is the number whose value we don’t know because we don’t care. I mean, it has some value, and presumably it doesn’t change over the course of our problem. But it’s not like our work will be so different if, say, the tank is two feet high rather than four.
Is there a problem? If we pick our letters to fit a specific problem, no. Presumably all the things we want to describe have some clear name, and some letter that best represents the name. It’s annoying when we have to consider, say, the pinball machine width and the corridor width. But we can work something out.
But what about general problems?
Is an easy problem to solve?
If we want to figure what ‘m’ is, yes. Similarly ‘y’. If we want to know what ‘b’ is, it’s tedious, but we can do that. If we want to know what ‘e’ is? Run and hide, that stuff is crazy. If you have to, do it numerically and accept an estimate. Don’t try figuring what that is.
And so we’ve developed conventions. There are some letters that, except in weird circumstances, are coefficients. They’re numbers whose value we don’t know, but either don’t care about or could look up. And there are some that, by default, are variables. They’re the ones whose value we want to know.
These conventions started forming, as mentioned, in the late 16th century. François Viète here made a name that lasts to mathematics historians at least. His texts described how to do algebra problems in the sort of procedural methods that we would recognize as algebra today. And he had a great idea for these letters. Use the whole alphabet, if needed. Use the consonants to represent the coefficients, the numbers we know but don’t care what they are. Use the vowels to represent the variables, whose values we want to learn. So he would look at that equation and see right away: it’s a terrible mess. (I exaggerate. He doesn’t seem to have known the = sign, and I don’t know offhand when ‘log’ and ‘cos’ became common. But suppose the rest of the equation were translated into his terminology.)
It’s not a bad approach. Besides the mnemonic value of consonant-coefficient, vowel-variable, it’s true that we usually have fewer variables than anything else. The more variables in a problem the harder it is. If someone expects you to solve an equation with ten variables in it, you’re excused for refusing. So five or maybe six or possibly seven choices for variables is plenty.
But it’s not what we settled on. René Descartes had a better idea. He had a lot of them, but here’s one. Use the letters at the end of the alphabet for the unknowns. Use the letters at the start of the alphabet for coefficients. And that is, roughly, what we’ve settled on. In my example nightmare equation, we’d suppose ‘y’ to probably be the variable we want to solve for.
And so, and finally, x. It is almost the variable. It says “mathematics” in only two strokes. Even π takes more writing. Descartes used it. We follow him. It’s way off at the end of the alphabet. It starts few words, very few things, almost nothing we would want to measure. (Xylem … mass? Flow? What thing is the xylem anyway?) Even mathematical dictionaries don’t have much to say about it. The letter transports almost no connotations, no messy specific problems to it. If it suggests anything, it suggests the horizontal coordinate in a Cartesian system. It almost is mathematics. It signifies nothing in itself, but long use has given it an identity as the thing we hope to learn by study.
And pirate treasure maps. I don’t know when ‘X’ became the symbol of where to look for buried treasure. My casual reading suggests “never”. Treasure maps don’t really exist. Maps in general don’t work that way. Or at least didn’t before cartoons. X marking the spot seems to be the work of Robert Louis Stevenson, renowned for creating a fanciful map and then putting together a book to justify publishing it. (I jest. But according to Simon Garfield’s On The Map: A Mind-Expanding Exploration of the Way The World Looks, his map did get lost on the way to the publisher, and he had to re-create it from studying the text of Treasure Island. This delights me to no end.) It makes me wonder if Stevenson was thinking of x’s service in mathematics. But the advantages of x as a symbol are hard to ignore. It highlights a point clearly. It’s fast to write. Its use might be coincidence.
But it is a letter that does a needed job really well.