A Leap Day 2016 Mathematics A To Z: X-Intercept


Oh, x- and y-, why are you so poor in mathematics terms? I brave my way.

X-Intercept.

I did not get much out of my eighth-grade, pre-algebra, class. I didn’t connect with the teacher at all. There were a few little bits to get through my disinterest. One came in graphing. Not graph theory, of course, but the graphing we do in middle school and high school. That’s where we find points on the plane with coordinates that make some expression true. Two major terms kept coming up in drawing curves of lines. They’re the x-intercept and the y-intercept. They had this lovely, faintly technical, faintly science-y sound. I think the teacher emphasized a few times they were “intercepts”, not “intersects”. But it’s hard to explain to an eighth-grader why this is an important difference to make. I’m not sure I could explain it to myself.

An x-intercept is a point where the plot of a curve and the x-axis meet. So we’re assuming this is a Cartesian coordinate system, the kind marked off with a pair of lines meeting at right angles. It’s usually two-dimensional, sometimes three-dimensional. I don’t know anyone who’s worried about the x-intercept for a four-dimensional space. Even higher dimensions are right out. The thing that confused me the most, when learning this, is a small one. The x-axis is points that have a y-coordinate of zero. Not an x-coordinate of zero. So in a two-dimensional space it makes sense to describe the x-intercept as a single value. That’ll be the x-coordinate, and the point with the x-coordinate of that and the y-coordinate of zero is the intercept.

If you have an expression and you want to find an x-intercept, you need to find values of x which make the expression equal to zero. We get the idea from studying lines. There are a couple of typical representations of lines. They almost always use x for the horizontal coordinate, and y for the vertical coordinate. The names are only different if the author is making a point about the arbitrariness of variable names. Sigh at such an author and move on. An x-intercept has a y-coordinate of zero, so, set any appearance of ‘y’ in the expression equal to zero and find out what value or values of x make this true. If the expression is an equation for a line there’ll be just the one point, unless the line is horizontal. (If the line is horizontal, then either every point on the x-axis is an intercept, or else none of them are. The line is either “y equals zero”, or it is “y equals something other than zero”. )

There’s also a y-intercept. It is exactly what you’d imagine once you know that. It’s usually easier to find what the y-intercept is. The equation describing a curve is typically written in the form “y = f(x)”. That is, y is by itself on one side, and some complicated expression involving x’s is on the other. Working out what y is for a given x is straightforward. Working out what x is for a given y is … not hard, for a line. For more complicated shapes it can be difficult. There might not be a unique answer. That’s all right. There may be several x-intercepts.

There are a couple names for the x-intercepts. The one that turns up most often away from the pre-algebra and high school algebra study of lines is a “zero”. It’s one of those bits in which mathematicians seem to be trying to make it hard for students. A “zero” of the function f(x) is generally not what you get when you evaluate it for x equalling zero. Sorry about that. It’s the values of x for which f(x) equals zero. We also call them “roots”.

OK, but who cares?

Well, if you want to understand the shape of a curve, the way a function looks, it helps to plot it. Today, yeah, pull up Mathematica or Matlab or Octave or some other program and you get your plot. Fair enough. If you don’t have a computer that can plot like that, the way I did in middle school, you have to do it by hand. And then the intercepts are clues to how to sketch the function. They are, relatively, easy points which you can find, and which you know must be on the curve. We may form a very rough sketch of the curve. But that rough picture may be better than having nothing.

And we can learn about the behavior of functions even without plotting, or sketching a plot. Intercepts of expressions, or of parts of expressions, are points where the value might change from positive to negative. If the denominator of a part of the expression has an x-intercept, this could be a point where the function’s value is undefined. It may be a discontinuity in the function. The function’s values might jump wildly between one side and another. These are often the important things about understanding functions. Where are they positive? Where are they negative? Where are they continuous? Where are they not?

These are things we often want to know about functions. And we learn many of them by looking for the intercepts, x- and y-.

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A Leap Day 2016 Mathematics A To Z: Lagrangian


It’s another of my handful of free choice days today. I’ll step outside the abstract algebra focus I’ve somehow gotten lately to look instead at mechanics.

Lagrangian.

So, you likely know Newton’s Laws of Motion. At least you know of them. We build physics out of them. So a lot of applied mathematics relies on them. There’s a law about bodies at rest staying at rest. There’s one about bodies in motion continuing in a straight line. There’s one about the force on a body changing its momentum. Something about F equalling m a. There’s something about equal and opposite forces. That’s all good enough, and that’s all correct. We don’t use them anyway.

I’m overstating for the sake of a good hook. They’re all correct. And if the problem’s simple enough there’s not much reason to go past this F and m a stuff. It’s just that once you start looking at complicated problems this gets to be an awkward tool. Sometimes a system is just hard to describe using forces and accelerations. Sometimes it’s impossible to say even where to start.

For example, imagine you have one of those pricey showpiece globes. The kind that’s a big ball that spins on an axis, and whose axis in on a ring that can tip forward or back. And it’s an expensive showpiece globe. That axis is itself in another ring that rotates clockwise and counterclockwise. Give the globe a good solid spin so it won’t slow down anytime soon. Then nudge the frame, so both the horizontal ring and the ring the axis is on wobble some. The whole shape is going to wobble and move in some way. We ought to be able to model that. How? Force and mass and acceleration barely seem to even exist.

The Lagrangian we get from Joseph-Louis Lagrange, who in the 18th century saw a brilliant new way to understand physics. It doesn’t describe how things move in response to forces, at least not directly. It describes how things move using energy. In particular, it uses on potential energy and kinetic energy.

This is brilliant on many counts. The biggest is in switching from forces to energy. Forces are vectors; they carry information about their size and their direction. Energy is a scalar; it’s just a number. A number is almost always easier to work with than a number alongside a direction.

The second big brilliance is that the Lagrangian gives us freedom in choosing coordinate systems. We have to know where things are and how they’re changing. The first obvious guess for how to describe things is their position in space. And that works fine until we look at stuff such as this spinning, wobbling globe. That never quite moves, although the spinning and the wobbling is some kind of motion. The problem begs us to think of the globe’s rotation around three different axes. Newton doesn’t help us with that. The Lagrangian, though —

The Lagrangian lets us describe physics using “generalized coordinates”. By this we mean coordinates that make sense for the problem even if they don’t directly relate to where something or other is in space. Any pick of coordinates is good, as long as we can describe the potential energy and the kinetic energy of the system using them.

I’ve been writing about this as if the Lagrangian were the cure for all hard work ever. It’s not, alas. For example, we often want to study big bunches of particles that all attract (or repel) each other. That attraction (or repulsion) we represent as potential energy. This is easier to deal with than forces, granted. But that’s easier, which is not the same as easy.

Still, the Lagrangian is great. We can do all the physics we used to. And we have a new freedom to set up problems in convenient ways. And the perspective of looking at energy instead of forces gives us a fruitful view on physics problems.

Reading the Comics, December 13, 2015: More Nearly Like It Edition


This has got me closer to the number of comics I like for a Reading the Comics post. There’s two comics already in my file, for the 14th of December, but those can wait until later in the week.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of December has a mathematics topic. The quotes in the final answer are the hint that it’s a bit of wordplay. The mention of “subtraction” is a hint.

Words: 'SOLPI', 'NALST', 'BAVEHE', 'CANYLU'. Circled letters, O O - - O, O - - O -, - O - O - O, O - O - - -. The puzzle: To teach subtraction the teacher had a '- - - - - -' - - - -.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of December, 2015. The link will probably expire in mid-January 2016. Also somehow I’m writing about 2016 being in the imminent future.

Brian Kliban’s cartoon for the 11th of December (a rerun from who knows when) promises an Illegal Cube Den, and delivers. I’m just delighted by the silliness of it all.

Greg Evans’s Luann Againn for the 11th of December reprints the 1987 Luann. “Geometric principles of equitorial [sic] astronomical coordinate systems” gets mentioned as a math-or-physics-sounding complicated thing to do. The basic idea is to tell where things are in the sky, as we see them from the surface of the Earth. In an equatorial coordinate system we imagine — we project — where the plane of the equator is, and we can measure things as north or south of that plane. (North is on the same side that the Earth’s north pole is.) That celestial equator is functionally equivalent to longitude, although it’s called declination.

We also need something functionally equivalent to longitude; that’s called the right ascension. To define that, we need something that works like the prime meridian. Projecting the actual prime meridian out to the stars doesn’t work. The prime meridian is spinning every 24 hours and we can’t publish updated star charts that quickly. What we use as a reference meridian instead is spring. That is, it’s where the path of the sun in the sky crosses the celestial equator in March and the (northern hemisphere) spring.

There are catches and subtleties, which is why this makes for a good research project. The biggest one is that this crossing point changes over time. This is because the Earth’s orbit around the sun changes. So right ascensions of points change a little every year. So when we give coordinates, we have to say in which system, and which reference year. 2000 is a popular one these days, but its time will pass. 1950 and 1900 were popular in their generations. It’s boring but not hard to convert between these reference dates. And if you need this much precision, it’s not hard to convert between the reference year of 2000 and the present year. I understand many telescopes will do that automatically. I don’t know directly because I have little telescope experience, and I couldn’t even swear I had seen a meteor until 2013. In fairness, I grew up in New Jersey, so with the light pollution I was lucky to see night sky.

Peter Maresca’s Origins of the Sunday Comics for the 11th of December showcases a strip from 1914. That, Clare Victor Dwiggins’s District School for the 12th of April, 1914, is just a bunch of silly vignettes. It’s worth zooming in to look at. It’s got a student going “figger juggling” and that gives me an excuse to point out the strip to anyone who’ll listen.

Samson’s Dark Side of the Horse for the 13th of December enters another counting-sheep joke into the ranks. Tying it into angles is cute. It’s tricky to estimate angles by sight. I think people tend to over-estimate how big an angle is when it’s around fifteen or twenty degrees. 45 degrees is easy enough to tell by sight. But for angles smaller than that, I tend to estimate angles by taking the number I think it is and cutting it in half, and I get closer to correct. I’m sure other people use a similar trick.

Brian Anderson’s Dog Eat Doug for the 13th of December has the dog, Sophie, deploy a lot of fraction talk to confuse a cookie out of Doug. A lot of new fields of mathematics are like that the first time you encounter them. I am curious where Sophie’s reasoning would have led, if not interrupted. How much cookie might she have cadged by the judicious splitting of halves and quarters and, perhaps, eighths and such? I’m not sure where her patter was going.

Shannon Wheeler’s Too Much Coffee Man for the 13th of December uses the traditional blackboard full of symbols to denote a lot of deeply considered thinking. Did you spot the error?

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 \hat{i} , and the direction of the y-axis as \hat{j} . The direction of the z-axis if needed gets written \hat{k} . 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 \hat{e}_1 , and the y-axis as the vector \hat{e}_2 , and so on. This method offers several advantages. One is that we can talk about the vector \hat{e}_j , 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 \hat{e}_0 so that the y-axis is the direction \hat{e}_1 . Usually it’s either clear from context or else it doesn’t matter.