The Summer 2017 Mathematics A To Z: X

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.

Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

X.

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 (f_off) 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.

Reading the Comics, July 29, 2015: Not Entirely Reruns Edition

Zach Weinersmith’s Saturday Morning Breakfast Cereal (July 25) gets its scheduled appearance here with a properly formed Venn Diagram joke. I’m unqualified to speak for rap musicians. When mathematicians speak of something being “for reals” they mean they’re speaking about a variable that might be any of the real numbers. This is as opposed to limiting the variable to being some rational or irrational number, or being a whole number. It’s also as opposed to letting the variable be some complex-valued number, or some more exotic kind of number. It’s a way of saying what kind of thing we want to find true statements about.

I don’t know when the Saturday Morning Breakfast Cereal first ran, but I know I’ve seen it appear in my Twitter feed. I believe all the Gocomics.com postings of this strip are reruns, but I haven’t read the strip long enough to say.

Steve Sicula’s Home And Away (July 26) is built on the joke of kids wise to mathematics during summer vacation. I don’t think this is a rerun, although we’ve seen the joke this summer before.

Daniel Beyer’s Offbeat Comics for the 27th of July, 2015.

Daniel Beyer’s Offbeat Comics (July 27) depicts an angel with a square halo because “I was good².” The association between squaring a number and squares goes back a long time. Well, it’s right there in the name, isn’t it? Florian Cajori’s A History Of Mathematical Notations cites the term “latus” and the abbreviation “l” to represent the side of a square being used by the Roman surveyor Junius Nipsus in the second century; for centuries this would be as good a term as anyone had for the thing to be calculated. (Res, meaning “thing”, was also popular.) Once you’ve taken the idea of calculating based on the length of a square, the jump to “square” for “length times itself” seems like a tiny one. But Cajori doesn’t seem to have examples of that being written until the 16th century.

The square of the quantity you’re interested in might be written q, for quadratus. The cube would be c, for cubus. The fourth power would be b or bq, for biquadratus, and so on. This is tolerable if you only have to work with a single unknown quantity, but the notation turns into gibberish the moment you want two variables in the mix. So it collapsed in the 17th century, replaced by the familiar x² and x³ and so on. Many authors developed notations close to this: James Hume would write xⁱⁱ or xⁱⁱⁱ; Pierre Hérigone x2 or x3, all in one line. Rene Descartes would write x² or x³ or so, and many, many followed him. Still, quite a few people — including Rene Descartes, Isaac Newton, and even as late a figure as Carl Gauss, in the early 19th century — would resist “x²”. They’d prefer “xx”. Gauss defended this on the grounds that “x²” takes up just as much space as “xx” and so fails the biggest point of having notation.

Corey Pandolph’s Toby, Robot Satan (July 27, rerun) uses sudoku as an example of the logic and reasoning problems that one would expect a robot should be able to do. It is weird to encounter one that’s helpless before them.

Cory Thomas’s Watch Your Head (July 27, rerun from 2007) mentions “Chebyshev grids” and “infinite boundaries” as things someone doing mathematics on the computer would do. And it does so correctly. Differential equations describe how things change on some domain over space and time. They can be very hard to solve exactly, but can be put on the computer very well. For this, we pick a representative set of points which we call a mesh. And we find an approximate representation of the original differential question, which we call a discretization or a difference equation. We can then solve this difference equation on the mesh, and if we’ve done our work right, this approximation will let us get a good estimate for the solution to the original problem over the whole original domain.

A Chebyshev grid is a particular arrangement of mesh points. It’s not uniform; it tends to clump up, becoming more common near the ends of the boundary. This is useful if you have reason to expect that the boundaries are more interesting than the middle of the domain. There’s no sense wasting good computing power calculating boring stuff. The mesh is named for Pafnuty Chebyshev, a 19th Century Russian mathematician whose name is all over mathematics. Unfortunately since he was a 19th Century Russian mathematician, his name is transcribed into English all sorts of ways. Chebyshev seems to be most common today, though Tchebychev used to be quite popular, which is why polynomials of his might be abbreviated as T. There are many alternatives.

Ah, but how do you represent infinite boundaries with the finitely many points of any calculatable mesh? There are many approaches. One is to just draw a really wide mesh and trust that all the action is happening near the center so omitting the very farthest things doesn’t hurt too much. Or you might figure what the average of things far away is, and make a finite boundary that has whatever that value is. Another approach is to make the boundaries repeating: go far enough to the right and you loop back around to the left, go far enough up and you loop back around to down. Another approach is to create a mesh that is bundled up tight around the center, but that has points which do represent going off very, very far, maybe in principle infinitely far away. You’re allowed to create meshes that don’t space points uniformly, and that even move points as you compute. That’s harder work, but it’s legitimate numerical mathematics.

So, the mathematical work being described here is — so far as described — legitimate. I’m not competent to speak about the monkey side of the research.

Greg Evans’s Luann Againn (July 29; rerun from July 29, 1987) name-drops the Law of Averages. There are actually multiple Laws of Averages, with slightly different assumptions and implications, but they all come to about the same meaning. You can expect that if some experiment is run repeatedly, the average value of the experiments will be close to the true value of whatever you’re measuring. An important step in proving this law was done by Pafnuty Chebyshev.

A Summer 2015 Mathematics A To Z: y-axis

y-axis.

It’s easy to tell where you are on a line. At least it is if you have a couple tools. One is a reference point. Another is the ability to say how far away things are. Then if you say something is a specific distance from the reference point you can pin down its location to one of at most two points. If we add to the distance some idea of direction we can pin that down to at most one point. Real numbers give us a good sense of distance. Positive and negative numbers fit the idea of orientation pretty well.

To tell where you are on a plane, though, that gets tricky. A reference point and a sense of how far things are help. Knowing something is a set distance from the reference point tells you something about its position. But there’s still an infinite number of possible places the thing could be, unless it’s at the reference point.

The classic way to solve this is to divide space into a couple directions. René Descartes made his name for himself — well, with many things. But one of them, in mathematics, was to describe the positions of things by components. One component describes how far something is in one direction from the reference point. The next component describes how far the thing is in another direction.

This sort of scheme we see as laying down axes. One, conventionally taken to be the horizontal or left-right axis, we call the x-axis. The other direction — one perpendicular, or orthogonal, to the x-axis — we call the y-axis. Usually this gets drawn as the vertical axis, the one running up and down the sheet of paper. That’s not required; it’s just convention.

We surely call it the x-axis in echo of the use of x as the name for a number whose value we don’t know right away. (That, too, is a convention Descartes gave us.) x carries with it connotations of the unknown, the sought-after, the mysterious thing to be understood. The next axis we name y because … well, that’s a letter near x and we don’t much need it for anything else, I suppose. If we need another direction yet, if we want something in space rather than a plane, then the third axis we dub the z-axis. It’s perpendicular to the x- and the y-axis directions.

These aren’t the only names for these directions, though. It’s common and often convenient to describe positions of things using vector notation. A vector describes the relative distance and orientation of things. It’s compact symbolically. It lets one think of the position of things as a single variable, a single concept. Then we can talk about a position being a certain distance in the direction of the x-axis plus a certain distance in the direction of the y-axis. And, if need be, plus some distance in the direction of the z-axis.

The direction of the x-axis is often written as , and the direction of the y-axis as . The direction of the z-axis if needed gets written . The circumflex there indicates two things. First is that the thing underneath it is a vector. Second is that it’s a vector one unit long. A vector might have any length, including zero. It’s convenient to make some mention when it’s a nice one unit long.

Another popular notation is to write the direction of the x-axis as the vector , and the y-axis as the vector , and so on. This method offers several advantages. One is that we can talk about the vector , that is, some particular direction without pinning down just which one. That’s the equivalent of writing “x” or “y” for a number we don’t want to commit ourselves to just yet. Another is that we can talk about axes going off in two, or three, or four, or more directions without having to pin down how many there are. And then we don’t have to think of what to call them. x- and y- and z-axes make sense. w-axis sounds a little odd but some might accept it. v-axis? u-axis? Nobody wants that, trust me.

Sometimes people start the numbering from so that the y-axis is the direction . Usually it’s either clear from context or else it doesn’t matter.

Reading the Comics, April 20, 2015: History of Mathematics Edition

This is a bit of a broad claim, but it seems Comic Strip Master Command was thinking of all mathematics one lead-time ago. There’s a comic about the original invention of mathematics, and another showing off 20th century physics equations. This seems as much of the history of mathematics as one could reasonably expect from the comics page.

Mark Anderson’s Andertoons gets its traditional appearance around here with the April 17th strip. It features a bit of arithmetic that is indeed lovely but wrong.

Why Call The Intercept b

Just because there are in principle uncountably many possible equations for any line doesn’t mean we ever actually see any of them. Actually, we just about always pick one of a handful of representations. They’re just the convenient ones. I’m going to say there’s four patterns that actually get used, because I can only think of three that turn up, as long as we’re sticking to Cartesian coordinate systems and aren’t doing something weird like parametric descriptions, and I want to leave some hedge room for when I realize I overlooked the obvious. The first one — that I want to talk about, anyway, and just about the first one anyone encounters — is called the slope-intercept form, and it’s probably what someone means if they do talk about “the” equation for a line.

Why A Line Doesn’t Have An Equation

[ To resume after some interruptions — it’s been quite a busy few weeks — the linear interpolations that I had been talking about, I will need equations describing a line. ]

To say something is the equation representing a line is to lie in the article. It’s little one, of the same order as pretending there’s just one answer to the question, “Who are you?” Who you are depends on context: you’re the person with this first-middle-last name combination. You’re the person with this first name. You’re the person with this nickname. You’re the third person in the phone queue for tech support. You’re the person with this taxpayer identification number. You’re the world’s fourth-leading expert on the Marvel “New Universe” line of comic books, and sorry for that. You’re the person who ordered two large-size fries at Five Guys Burgers And Fries and will soon learn you’ll never live long enough to eat them all. You’re the person who knows how to get the sink in the break room at work to stop dripping. These may all be correct, but depending on the context some of these answers are irrelevant, and maybe one or two of them is useful, or at least convenient. So it is with equations for a line: there are many possible equations. Some of them are just more useful, or even convenient.

What We Mean By x

[ Oh, wow. Yesterday’s entry had way fewer hits than average. I also put an equation out right up front where everyone could see it. I wonder if this might be a test of Stephen Hawking’s dictum about equations and sales. Or maybe I was just boring yesterday. I’d ask, but apparently, nobody found me interesting enough yesterday to know for comparison. ]

It shouldn’t be too hard to translate the the idea “I want to know the population of Charlotte at some particular time” into a polynomial. The polynomial ought to look something like y equals some pile of numbers times x’s raised to powers, and x somehow has to do with the particular time, and y has something to do with the population. And it’s not hard to do that translating, but I want to talk about some deeper issues. It’s probably better explaining them on the simple problem, where we know what we want things to mean, than it would be explaining them for a complicated problem.

Descartes and the Terror of the Negative

When René Descartes first described the system we’ve turned into Cartesian coordinates he didn’t put it forth in quite the way we build them these days. This shouldn’t be too surprising; he lived about four centuries ago, and we have experience with the idea of matching every point on the plane to some ordered pair of numbers that he couldn’t have. The idea has been expanded on, and improved, and logical rigor I only pretend to understand laid underneath the concept. But the core remains: we put somewhere on our surface an origin point — usually this gets labelled O, mnemonic for “origin” and also suggesting the zeroes which fill its coordinates — and we pick some direction to be the x-coordinate and some direction to be the y-coordinate, and the ordered pair for a point are how far in the x-direction and how far in the y-direction one must go from the origin to get there.

The most obvious difference between Cartesian coordinates as Descartes set them up and Cartesian coordinates as we use them is that Descartes would fill a plane with four chips, one quadrant each in the plane. The first quadrant is the points to the right of and above the origin. The second quadrant is to the left of and still above the origin. The third quadrant is to the left of and below the origin, and the fourth is to the right of the origin but below it. This division of the plane into quadrants, and even their identification as quadrants I, II, III, and IV respectively, still exists, one of those minor points on which prealgebra and algebra students briefly trip on their way to tripping over the trigonometric identities.

Descartes had, from his perspective, excellent reason to divide the plane up this way. It’s a reason difficult to imagine today. By separating the plane like this he avoided dealing with something mathematicians of the day were still uncomfortable with. It’s easy enough to describe a point in the first quadrant as being so far to the right and so far above the origin. But a point in the second quadrant is … not any distance to the right. It’s to the left. How far to the right is something that’s to the left?

Descartes’ Flies

There are a healthy number of legends about René Descartes. Some of them may be true. I know the one I like is the story that this superlative mathematician, philosopher, and theologian (fields not so sharply differentiated in his time as they are today; for that matter, fields still not perfectly sharply differentiated) was so insistent on sleeping late and sufficiently ingenious in forming arguments that while a student at the Jesuit Collè Royal Henry-Le-Grand he convinced his schoolmasters to let him sleep until 11 am. Supposedly he kept to this rather civilized rising hour until he last months of his life, when he needed to tutor Queen Christina of Sweden in the earliest hours of the winter morning.

I suppose this may be true; it’s certainly repeated often enough, and comes to mind often when I do have to wake to the alarm clock. I haven’t studied Descartes’ biography well enough to know whether to believe it, although as it makes for a charming and humanizing touch probably the whole idea is bunk and we’re fools to believe it. I’m comfortable being a little foolish. (I’ve read just the one book which might be described as even loosely biographic of Descartes — Russell Shorto’s Descartes’ Bones — and so, though I have no particular reason to doubt Shorto’s research and no question with his narrative style, suppose I am marginally worse-informed than if I were completely ignorant. It takes a cluster of books on a subject to know it.)

Place the name “Descartes” into the conversation and a few things pop immediately into mind. Those things are mostly “I think, therefore I am”, and some attempts to compose a joke about being “before the horse”. Running up sometime after that is something called “Cartesian coordinates”, which are about the most famous kind of coordinates and the easiest way to get into the problem of describing just where something is in two- or three-dimensional space.