My Little 2021 Mathematics A-to-Z: Hyperbola

John Golden, author of the Math Hombre blog, had several great ideas for the letter H in this little A-to-Z for the year. Here’s one of them.

Hyperbola.

The hyperbola is where advanced mathematics begins. It’s a family of shapes, some of the pieces you get by slicing a cone. You can make an approximate one shining a flashlight on a wall. Other conic sections are familiar, everyday things, though. Circles we see everywhere. Ellipses we see everywhere we look at a circle in perspective. Parabolas we learn, in approximation, watching something tossed, or squirting water into the air. The hyperbola should be as accessible. Hold your flashlight parallel to the wall and look at the outline of light it casts. But the difference between this and a parabola isn’t obvious. And it’s harder to see parabolas in nature. It’s the path a space probe swinging past a planet makes? Great guide for all us who’ve launched space probes past Jupiter.

When we learn of hyperbolas, somewhere in high school algebra or in precalculus, they seem designed to break the rules we had inferred. We’ve learned functions like lines and quadradics (parabolas) and cubics. They’re nice, simple, connected shapes. The hyperbola comes in two pieces. We’ve learned that the graph of a function crosses any given vertical line at most once. Now, we can expect to see it twice. We learn to sketch functions by finding a few interesting points — roots, y-intercepts, things like that. Hyperbolas, we’re taught to draw this little central box and then two asymptotes. Also, we have asymptotes, a simpler curve that the actual curve almost equals.

We’re trained to see functions having the couple odd points where they’re not defined. Nobody expects to mean anything when is zero. But we learn these as weird, isolated points. Now there’s this interval of x-values that don’t fit anything on the graph. Half the time, anyway, because we see two classes of hyperbolas. There’s ones that open like cups, pointing up and down. Those have definitions for every value of x. There’s ones that open like ears, pointing left and right. Those have a box in the center where no y satisfies the x’s. They seem like they’re taught just to be mean.

They’re not, of course. The only mathematical thing we teach just to be mean is integration by trigonometric substitution. The things which seem weird or new in hyperbolas are, largely, things we didn’t notice before. A vertical line put across a circle or ellipse crosses the curve twice, most points. There are two huge intervals, to the left and to the right of the circle, where no value of y makes the equation true. Circles are familiar, though. Ellipses don’t seem intimidating. We know we can’t turn (a typical circle) into a function without some work. We have to write either or , breaking the circle into two halves. The same happens for hyperbolas, though, with (a typical hyperbola) turning into or .

Even the definitions seem weird. The ellipse we can draw by taking a set distance and two focus points. If the distance from the first focus to a point plus the distance from the point to the second focus is that set distance, the point’s on the ellipse. We can use two thumbtacks and a piece of string to draw the ellipse. The hyperbola has a simliar rule, but weirder. You have your two focus points, yes. And a set distance. But the locus of points of the hyperbola is everything where the distance from the point to one focus minus the distance from the point to the other focus is that set distance. Good luck doing that with thumbtacks and string.

Yet hyperbolas are ready for us. Consider playing with a decent calculator, hitting the reciprocal button for different numbers. 1 turns to 1, yes. 2 turns into 0.5. -0.125 turns into -8. It’s the simplest iterative game to do on the calculator. If you sketch this, though, all the points (x, y) where one coordinate is the reciprocal of the other? It’s two curves. They approach without ever touching the x- and y-axes. Get far enough from the origin and there’s no telling this curve from the axes. It’s a hyperbola, one that obeys that vertical-line rule again. It has only the one value of x that can’t be allowed. We write it as or even . But it’s the shape we see when we draw , rotated. Or a rotation of one we see when we draw . The equations of rotated shapes are annoying. We do enough of them for ellipses and parabolas and hyperbolas to meet the course requirement. But they point out how the hyperbola is a more normal construct than we fear.

And let me look at that construct again. An equation describing a hyperbola that opens horizontally or vertically looks like for some constant numbers a, b, and c. (If a, b, and c are all positive, this is a hyperbola opening horizontally. If a and b are positive and c negative, this is a hyperbola opening vertically.) An equation describing an ellipse, similarly with its axes horizontal or vertical looks like . (These are shapes centered on the origin. They can have other centers, which make the equations harder but not more enlightening.) The equations have very similar shapes. Mathematics trains us to suspect things with similar shapes have similar properties. That change from a plus to a minus seems too important to ignore, and yet …

I bet you assumed x and y are real numbers. This is convention, the safe bet. If someone wants complex-valued numbers they usually say so. If they don’t want to be explicit, they use z and w as variables instead of x and y. But what if y is an imaginary number? Suppose , for some real number t, where . You haven’t missed a step; I’m summoning this from nowhere. (Let’s not think about how to draw a point with an imaginary coordinate.) Then is which is . And despite the weird letters, that’s a circle. By the same supposition we could go from , which we’d taken to be a circle, and get , a hyperbola.

Fine stuff inspiring the question “so?” I made up a case and showed how that made two dissimilar things look alike. All right. But consider trigonometry, built on the cosine and sine functions. One good way to see the cosine and sine of an angle is as the x- and y-coordinates of a point on the unit circle, where . (The angle is the one from the point to the origin to the point (1, 0).)

There exists, in parallel to the familiar trig functions, the “hyperbolic trigonometric functions”. These have imaginative names like the hyperbolic sine and hyperbolic cosine. (And onward. We can speak of the “inverse hyperbolic cosecant”, if we wish no one to speak to us again.) Usually these get introduced in calculus, to give the instructor a tiny break. Their derivatives, and integrals, look much like those of the normal trigonometric functions, but aren’t the exact same problems over and over. And these functions, too, have a compelling meaning. The hyperbolic cosine of an angle and hyperbolic sine of an angle have something to do with points on a unit hyperbola, .

Thinking back on the flashlight. We get a circle by holding the light perpendicular to the wall. We get a hyperbola holding the light parallel. We get a circle by drawing with x and y real numbers. We get a hyperbola by (somehow) drawing with x real and y imaginary. We remember something about representing complex-valued numbers with a real axis and an orthogonal imaginary axis.

One almost feels the connection. I can’t promise that pondering this will make hyperbolas be as familiar as circles or at least ellipses. But often a problem that brings us to hyperbolas has an alternate phrasing that’s ellipses, a nd vice-versa. But the common traits of these conic slices can guide you into a new understanding of mathematics.

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Thank you for reading. I hope to have another piece next week at this time. This and all of this year’s Little Mathematics A to Z essays should be at this link. And the A-to-Z essays for every year should be at this link.

My All 2020 Mathematics A to Z: Extraneous Solutions

Iva Sallay, the kind author of the Find the Factors recreational mathematics puzzle, suggested this topic for the letter X. It’s a fun chance to look at some of the basics of (high school) algebra again.

Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Extraneous Solutions.

When developing general relativity, Albert Einstein created a convention. He’s not unique in that. All mathematicians create conventions. They use shorthand for an idea that’s complicated or common. Relatively unique is that other people adopted his convention, because it expressed an idea compactly. This was in working with tensors, which look somewhat like matrixes and have a lot of indexes. In the equations of general relativity you need to take sums over many combinations of values of these indexes. What indexes there are are the same in most every problem. The possible values of the indexes is constant, problem to problem, too.

So Einstein saved himself writing, and his publishers from typesetting, a lot of redundant writing. This by writing out the conditions which implied “take the sums over these indexes on this range”. This is good for people doing general relativity, and certain kinds of geometry. It’s a problem only when an expression escapes its context. When it’s shown to a student or someone who doesn’t know this is a differential-geometry problem. Then the problem becomes confusing, and they can’t work on it.

This is not to fault the Einstein Summation Convention. It puts common necessary scaffolding out of the way and highlighting the interesting unique parts of a problem. Most conventions aim for that. We have the hazard, though, that we may not notice something breaking the convention.

And this is how we create extraneous solutions. And, as a bonus, to have missing solutions. We encounter them with the start of (high school) algebra, when we get used to manipulating equations. When we solve an equation what we always want is something clear, like

But it never starts that way. It always starts with something like

or worse. We learn how to handle this. We know that we can do six things that do not alter the truth of an equation. We can regroup terms in the equation. We can add the same number to both sides of the equation. We can multiply both sides of the equation by some number besides zero. We can add zero to one side of the equation. We can multiply one side of the equation by 1. We can replace one quantity with another that has the same value. That doesn’t sound like a lot. It covers more than it seems. Multiplying by 1, for example, is the same as multiplying by . If x isn’t zero, then we can multiply both sides of the equation by that x. And x can’t be zero, or else would not be 1.

So with my example there, start off by multiplying the right side by 1, in the guise . Then multiply both sides by that same non-zero x. At this point the right-hand side simplifies to being 6. Add a -6 to both sides. And then with a lot of shuffling around you work out that the equation is the same as

And that can only be true when x equals 2.

It should be easy to catch spurious solutions creeping in. They must result from breaking a rule. The obvious problem is multiplying — or dividing — by zero. We expect those to be trouble. Wikipedia has a fine example:

The obvious step is to multiply this whole mess by , which turns our work into a linear equation. Very soon we find the solution must be . Which would make at least two of the denominators in the original equation zero. We know not to want that.

The problems can be subtler, though. Consider:

That’s not hard to solve. Multiply both sides by . Although, before working out substitute that with something equal to it. We know one thing is equal to it, . Then we have

It’s a quadratic equation. A little bit of work shows the roots are 9 and 16. One of those answers is correct and the other spurious. At no point did we divide anything, by zero or anything else.

So what is happening and what is the necessary rhetorical link to the Einstein Summation Convention?

There are many ways to look at equations. One that’s common is to look at them as functions. This is so common that we’ll elide between an equation and a function representation. This confuses the prealgebra student who wants to know why sometimes we look at

and sometimes we look at

and sometimes at

The advantage of looking at the function which shadows any equation is we have different tools for studying functions. Sometimes that makes solving the equation easier. In this form, we’re looking for what in the domain matches with something particular in the range.

And now we’ve reached the convention. When we write down something lke we’re implicitly defining a function. A function has three pieces. It has a set called the domain, from which we draw the independent variable. It has a set called the range. It has a rule matching elements in the domain to an element in the range. We’ve only given the rule. What are the domain and what’s the range for ?

And here are the conventions. If we haven’t said otherwise, the domain and range are usually either the real numbers or the complex numbers. If we used x or y or t as the independent variable, we mean the real numbers. If we used z as the independent variable, and haven’t already put x and y in, we mean the complex numbers. Sometimes we call in s or w or another letter; never mind that. The range can be the whole set of real or complex numbers. It does us no harm to have too large a range.

The domain, though. We do insist that everything in the domain match to something in the range. And, like, ? That can’t mean anything if x equals 2.

So we take an implicit definition of the domain: it’s all the real numbers for which the function’s rule is meaningful. So, would have a domain “real numbers other than 2”. would have a domain “real numbers other than 2 and -2”.

We create extraneous solutions — or we lose some — when our convention changes the domain. An extraneous solution is one that existed outside the original problem’s domain. A missing solution is one that existed in an excised part of the domain. To go from to by dividing out x is to cut out of the space of possible solutions.

A complaint you might raise. What is the domain for ? Rewrite that as a function. would seem to have a domain “x greater than or equal to 0”. The extraneous solution is , a number which rumor has it is greater than or equal to 0. What happened?

We have to take that equation-handling more slowly. We had started out with

The domain has to be “x is greater than or equal to 0” here. All right. The next step was multiplying both sides by the same quantity, . So:

The domain is still “x is greater than or equal to 0”. The next step, though, was a substitution. I wanted to replace the on the right with . We know, from the original equation, that those are equal. At least, they’re equal wherever the original equation is true. What happens when , though?

We start to see the catch. 9 – 12 is -3. And while it’s true that -3 squared will be 9, it’s false that -3 is the square root of 9. The equation can only be true, for real numbers, if is nonnegative. We can make this rigorous with two supplementary functions. Let me call and .

has an implicit domain of “x greater than or equal to 0”. What’s the domain of ? If , like we said it does, then they have to agree for every x in either’s domain. So can’t have in its domain any x for which isn’t defined. So the domain of has to be “x for which x – 12 is greater than or equal to 0”. And that’s “x greater than or equal to 12”.

So the domain for the original equation is “x greater than or equal to 12”. When we keep that domain in mind, the extraneous nature of is clear, and we avoid trouble.

Not all extraneous solutions come from algebraic manipulations. Sometimes there are constraints on the problem, rather than the numbers, that make a solution absurd. There is a betting strategy called the martingale. This amounts to doubling the bet every time one loses. This makes the first win balance out all the losses leading to it. This solution fails because the player has a finite wallet, and after a few losses any player hasn’t got the money to continue.

Or consider a case that may be legend. It concerns the Apollo Guidance Computer. It was designed to take the Lunar Module to a spot at zero altitude above the moon’s surface, with zero velocity. The story is that in early test runs, the computer would not avoid trajectories that dropped to a negative altitude along the way to the surface. One imagines the scene after the first Apollo subway trip. (I have not found a date when such a test run was done, or corrections to the code ordered. If someone knows, I’d appreciate learning specifics.)

The convention, that we trust the domain is “everything which makes sense”, is not to blame here. It’s normally a good convention. Explicitly noting the domain at every step is tedious and, most of the time, unenlightening. It belongs in the background. We also must check our possible solutions, and that they represent things that make sense. We can try to concentrate our thinking on the obvious interesting parts, but must spend some time on the rest also.

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I am surprised to be so near the end of the 2020 A-to-Z, and to 2020, I hope. This and all the other glossary essays for the year should be at this link. All the essays from every A-to-Z series should be at this link. Thank you for reading.

Reading the Comics, November 26, 2016: What is Pre-Algebra Edition

Here I’m just closing out last week’s mathematically-themed comics. The new week seems to be bringing some more in at a good pace, too. Should have stuff to talk about come Sunday.

Darrin Bell and Theron Heir’s Rudy Park for the 24th brings out the ancient question, why do people need to do mathematics when we have calculators? As befitting a comic strip (and Sadie’s character) the question goes unanswered. But it shows off the understandable confusion people have between mathematics and calculation. Calculation is a fine and necessary thing. And it’s fun to do, within limits. And someone who doesn’t like to calculate probably won’t be a good mathematician. (Or will become one of those master mathematicians who sees ways to avoid calculations in getting to an answer!) But put aside the obviou that we need mathematics to know what calculations to do, or to tell whether a calculation done makes sense. Much of what’s interesting about mathematics isn’t a calculation. Geometry, for an example that people in primary education will know, doesn’t need more than slight bits of calculation. Group theory swipes a few nice ideas from arithmetic and builds its own structure. Knot theory uses polynomials — everything does — but more as a way of naming structures. There aren’t things to do that a calculator would recognize.

Richard Thompson’s Poor Richard’s Almanac for the 25th I include because I’m a fan, and on the grounds that the Summer Reading includes the names of shapes. And I’ve started to notice how often “rhomboid” is used as a funny word. Those who search for the evolution and development of jokes, take heed.

John Atkinson’s Wrong Hands for the 25th is the awaited anthropomorphic-numerals and symbols joke for this past week. I enjoy the first commenter’s suggestion tha they should have stayed in unknown territory.

Rick Kirkman and Jerry Scott’s Baby Blues for the 26th of November, 2016. I suppose Kirkman and Scott know their characters better than I do but isn’t Zoe like nine or ten? Isn’t pre-algebra more a 7th or 8th grade thing? I can’t argue Grandma being post-algebra but I feel like the punch line was written and then retrofitted onto the characters.

Rick Kirkman and Jerry Scott’s Baby Blues for the 26th does a little wordplay built on pre-algebra. I’m not sure that Zoe is quite old enough to take pre-algebra. But I also admit not being quite sure what pre-algebra is. The central idea of (primary school) algebra — that you can do calculations with a number without knowing what the number is — certainly can use some preparatory work. It’s a dazzling idea and needs plenty of introduction. But my dim recollection of taking it was that it was a bit of a subject heap, with some arithmetic, some number theory, some variables, some geometry. It’s all stuff you’ll need once algebra starts. But it is hard to say quickly what belongs in pre-algebra and what doesn’t.

Art Sansom and Chip Sansom’s The Born Loser for the 26th uses two ancient staples of jokes, probabilities and weather forecasting. It’s a hard joke not to make. The prediction for something is that it’s very unlikely, and it happens anyway? We all laugh at people being wrong, which might be our whistling past the graveyard of knowing we will be wrong ourselves. It’s hard to prove that a probability is wrong, though. A fairly tossed die may have only one chance in six of turning up a ‘4’. But there’s no reason to think it won’t, and nothing inherently suspicious in it turning up ‘4’ four times in a row.

We could do it, though. If the die turned up ‘4’ four hundred times in a row we would no longer call it fair. (This even if examination proved the die really was fair after all!) Or if it just turned up a ‘4’ significantly more often than it should; if it turned up two hundred times out of four hundred rolls, say. But one or two events won’t tell us much of anything. Even the unlikely happens sometimes.

Even the impossibly unlikely happens if given enough attempts. If we do not understand that instinctively, we realize it when we ponder that someone wins the lottery most weeks. Presumably the comic’s weather forecaster supposed the chance of snow was so small it could be safely rounded down to zero. But even something with literally zero percent chance of happening might.

Imagine tossing a fair coin. Imagine tossing it infinitely many times. Imagine it coming up tails every single one of those infinitely many times. Impossible: the chance that at least one toss of a fair coin will turn up heads, eventually, is 1. 100 percent. The chance heads never comes up is zero. But why could it not happen? What law of physics or logic would it defy? It challenges our understanding of ideas like “zero” and “probability” and “infinity”. But we’re well-served to test those ideas. They hold surprises for us.

Everything I Learned In Eighth-Grade Math

My title is an exaggeration. In eighth grade Prealgebra I learned many things, but I confess that I didn’t learn well from that particular teacher that particular year. What I most clearly remember learning I picked up from a substitute who filled in a few weeks. It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation. But it’s fun to do, and I liked it a lot, and I’ve never seen it described as a way to factor quadratic expressions. So let me put it on the web and do what I can to preserve its legacy, and get hundreds of people telling me what it actually is and how everybody but the people I know went through a phase of using it.

It’s a method which looks at first like it’s going to be a magic square, but it’s not, and I’m at a loss what to call it. I don’t remember the substitute teacher’s name, so I can’t use that. I do remember the regular teacher’s name, but it wasn’t, as far as I know, part of his lesson plan, and it’d not be fair to him to let his legacy be defined by one student who just didn’t get him.