I remember part of why I stopped doing Reading the Comics posts regularly was their volume. I read a lot of comics and it felt like everyone wanted to do a word problem joke. Since I started easing back into these posts it’s seemed like they’ve disappeared. When I put together this week’s collection, I only had three interesting ones. And one was Andertoons for the 10th of April. Andertoons is a stalwart here, but this particular strip was one I already talked about, back in 2019.
Another was the Archie repeat for the 10th of April. And that only lists mathematics as a school subject. It would be the same joke if it were English lit. Saying “differential calculus” gives it the advantage of specificity. It also suggests Archie is at least a good enough student to be taking calculus in high school, which isn’t bad. Differential calculus is where calculus usually starts, with the study of instantaneous changes. A person can, and should, ask how a change can be instantaneous. Part of what makes differential calculus is learning how to find something that matches our intuition about what it should be. And that never requires us to do something appalling like divide zero by zero. Our current definition took a couple centuries of wrangling to find a scheme that makes sense. It’s a bit much to expect high school students to pick it up in two months.
The problem sounds like it shouldn’t be hard. The two-by-two version of this is easy. So is three-by-three and four-by-four and even five-by-five. Oddly, seven-by-seven is, too. It looks like some form of magic square, and seems not far off being a sudoku problem either. So it seems weird that six-by-six should be particularly hard, but sometimes it happens like that. In fact, this happens to be impossible; a paper by Gaston Terry in 1901 proved there were none.
The solution discussed by Ripley’s is of a slightly different problem. So I’m not saying to not believe it, just, that you need to believe it with reservations. The modified problem casts this as a quantum-entanglement, in which the rank and regiment of an officer in one position is connected to that of their neighbors. I admit I’m not sure I understand this well enough to explain; I’m not confident I can give a clear answer why a solution of the entangled problem can’t be used for the classical problem.
The problem, at this point, isn’t about organizing officers anymore. It never was, since that started as an idle pastime. Legend has it that it started as a challenge about organizing cards; if you look at the paper you’ll see it presenting states as card suits and values. But the problem emerged from idle curiosity into practicality. These turn out to be applicable to quantum error detection codes. I’m not certain I can explain how myself. You might be able to convince yourself of this by thinking how you know that someone who tells you the sum of six odd numbers is itself an odd number made a mistake somewhere, and you can then look for what went wrong.
All right, I can be a little more clear. By the inverse I mean subtraction is the name the name we give to adding the additive inverse of something. It’s what lets addition be a group action. That is, we write to mean we find whatever number, added to b, gives us 0. Then we add that to a. We do this pretty often, so it’s convenient to have a name for it. The word “subtraction” appears in English from about 1400. It grew from the Latin for “taking away”. By about 1425 the word has its mathematical meaning. I imagine this wasn’t too radical a linguistic evolution.
All right, so some other thoughts. What’s so interesting about subtraction that it’s worth a name? We don’t have a particular word for reversing, say, a permutation. But don’t go very far in school not thinking about inverting an addition. Must come down to subtraction’s practical use in finding differences between things. Often in figuring out change. Debts at least. Nobody needs the inverse of a permutation unless they’re putting a deck of cards back in order.
Subtraction has other roles, though. Not so much in mathematics, but in teaching us how to learn about mathematics. For example, subtraction gives us a good reason to notice zero. Zero, the additive identity, is implicit to addition. But if you’re learning addition, and you think of it as “put these two piles of things together into one larger pile”? What good does an empty pile do you there? It’s easy to not notice there’s a concept there. But subtraction, taking stuff away from a pile? You can imagine taking everything away, and wanting a word for that. This isn’t the only way to notice zero is worth some attention. It’s a good way, though.
There’s more, though. Learning subtraction teaches us limits of what we can do, mathematically. We can add 3 to 7 or, if it’s more convenient, 7 to 3. But we learn from the start that while we can subtract 3 from 7, there’s no subtracting 7 from 3. This is true when we’re learning arithmetic and numbers are all positive. Some time later we ask, what happens if we go ahead and do this anyway? And figure out a number that makes sense as the answer to “what do you get subtracting 7 from 3”? This introduces us to the negative numbers. It’s a richer idea of what it is to have numbers. We can start to see addition and subtraction as expressions of the same operation.
But we also notice they’re not quite the same. As mentioned, addition can be done in any order. If I need to do 7 + 4 + 3 + 6 I can decide I’d rather do 4 + 6 + 7 + 3 and make that 10 + 10 before getting to 20. This all simplifies my calculating. If I need to do 7 – 4 – 3 – 6 I get into a lot of trouble if I simplify my work by writing 4 – 6 – 7 – 3 instead. Even if I decide I’d rather take the 3 – 6 and turn that into a negative 3 first, I’ve made a mess of things.
The first property this teaches us to notice we call “commutativity”. Most mathematical operations don’t have that. But a lot of the ones we find useful do. The second property this points out is “associativity”, which more of the operations we find useful have. It’s not essential that someone learning how to calculate know this is a way to categorize mathematics operations. (I’ve read that before the New Math educational reforms of the 1960s, American elementary school mathematics textbooks never mentioned commutativity or associativity.) But I suspect it is essential that someone learning mathematics learn the things you can do come in families.
So let me mention division, the inverse of multiplication. (And that my chosen theme won’t let me get to in sequence.) Like subtraction, division refuses to be commutative or associative. Subtraction prompts us to treat the negative numbers as something useful. In parallel, division prompts us to accept fractions as numbers. (We accepted fractions as numbers long before we accepted negative numbers, mind. Anyone with a pie and three friends has an interest in “one-quarter” that they may not have with “negative four”.) When we start learning about numbers raised to powers, or exponentials, we have questions ready to ask. How do the operations behave? Do they encourage us to find other kinds of number?
And we also think of how to patch up subtraction’s problems. If we want subtraction to be a kind of addition, we have to get precise about what that little subtraction sign means. What we’ve settled on is that is shorthand for , where is the additive inverse of .
Once we do that all subtraction’s problems with commutativity and associativity go away. 7 – 4 – 3 – 6 becomes 7 + (-4) + (-3) + (-6), and that we can shuffle around however convenient. Say, to 7 + (-3) + (-4) + (-6), then to 7 + (-3) + (-10), then to 4 + (-10), and so -6. Thus do we domesticate a useful, wild operation like subtraction.
Any individual subtraction has one right answer. There are many ways to get there, though. I had learned, for example, to do a problem such as 738 minus 451 by subtracting one column of numbers at a time. Right to left, so, subtracting 8 minus 1, and then 3 minus 5, and after the borrowing then 6 minus 4. I remember several elementary school textbooks explaining borrowing as unwrapping rolls of dimes. It was a model well-suited to me.
We don’t need to, though. We can go from the left to the right, doing 7 minus 4 first and 8 minus 1 last. We can go through and figure out all the possible carries before doing any work. There’s a slick method called partial differences which skips all the carrying. But it demands writing out several more intermediate terms. This uses more paper, but if there isn’t a paper shortage, so what?
There are more ways to calculate. If we turn things over to a computer, we’re likely to do subtraction using a complements technique. When I say computer you likely think electronic computer, or did right up to the adjective there. But mechanical computers were a thing too. Blaise Pascal’s computing device of the 1650s used nines’ complements to subtract on the gears that did addition. Explaining the trick would take me farther afield than I want to go now. But, you know how, like, 6 plus 3 is 9? So you can turn a subtraction of 6 into an addition of 3. Or a subtraction of 3 into an addition of 6. Plus some bookkeeping.
A digital computer is likely to use ones’ complements, representing every number as a string of 0’s and 1’s. This has great speed advantages. The complement of 0 is 1 and vice-versa, and it’s very quick for a computer to swap between 0 and 1. Subtraction by complements is different and, to my eye, takes more steps. But they might be steps you do better.
One more thought subtraction gives us, though. In a previous paragraph I wrote out 7 – 4, and also wrote 7 + (-4). We use the symbol – for two things. Do those two uses of – mean the same thing? You may think I’m being fussy here. After all, the value of -4 is the same as the value of 0 – 4. And even a fussy mathematician says whichever of “minus four” and “negative four” better fits the meter of the sentence. But our friends in the philosophy department would agree this is a fair question. Are we collapsing two related ideas together by using the same symbol for them?
My inclination is to say that the – of -4 is different from the – in 0 – 4, though. The – in -4 is a unary operation: it means “give me the inverse of the number on the right”. The – in 0 – 4 is a binary operation: it means “subtract the number on the right from the number on the left”. So I would say these are different things sharing a symbol. Unfortunately our friends in the philosophy department can’t answer the question for us. The university laid them off four years ago, part of society’s realignment away from questions like “how can we recognize when a thing is true?” and towards “how can we teach proto-laborers to use Excel macros?”. We have to use subtraction to expand our thinking on our own.
I have Dina Yagodich to thank for my inspiration this week. As will happen with these topics about something fundamental, this proved to be a hard topic to think about. I don’t know of any creative or professional projects Yagodich would like me to mention. I’ll pass them on if I learn of any.
In May 1962 Mercury astronaut Deke Slayton did not orbit the Earth. He had been grounded for (of course) a rare medical condition. Before his grounding he had selected his flight’s callsign and capsule name: Delta 7. His backup, Wally Schirra, who did not fly in Slayton’s place, named his capsule the Sigma 7. Schirra chose sigma for its mathematical and scientific meaning, representing the sum of (in principle) many parts. Slayton said he chose Delta only because he would have been the fourth American into space and Δ is the fourth letter of the Greek alphabet. I believe it, but do notice how D is so prominent a letter in Slayton’s name. And S, Σ, prominent in both Slayton and Schirra’s.
Δ is also a prominent mathematics and engineering symbol. It has several meanings, with several of the most useful ones escaping mathematics and becoming vaguely known things. They blur together, as ideas that are useful and related and not identical will do.
If “Δ” evokes anything mathematical to a person it is “change”. This probably owes to space in the popular imagination. Astronauts talking about the delta-vee needed to return to Earth is some of the most accessible technical talk of Apollo 13, to pick one movie. After that it’s easy to think of pumping the car’s breaks as shedding some delta-vee. It secondarily owes to school, high school algebra classes testing people on their ability to tell how steep a line is. This gets described as the change-in-y over the change-in-x, or the delta-y over delta-x.
Δ prepended to a variable like x or y or v we read as “the change in”. It fits the astronaut and the algebra uses well. The letter Δ by itself means as much as the words “the change in” do. It describes what we’re thinking about, but waits for a noun to complete. We say “the” rather than “a”, I’ve noticed. The change in velocity needed to reach Earth may be one thing. But “the” change in x and y coordinates to find the slope of a line? We can use infinitely many possible changes and get a good result. We must say “the” because we consider one at a time.
Used like this Δ acts like an operator. It means something like “a difference between two values of the variable ” and lets us fill in the blank. How to pick those two values? Sometimes there’s a compelling choice. We often want to study data sampled at some schedule. The Δ then is between one sample’s value and the next. Or between the last sample value and the current one. Which is correct? Ask someone who specializes in difference equations. These are the usually numeric approximations to differential equations. They turn up often in signal processing or in understanding the flows of fluids or the interactions of particles. We like those because computers can solve them.
Δ, as this operator, can even be applied to itself. You read ΔΔ x as “the change in the change in x”. The prose is stilted, but we can understand it. It’s how the change in x has itself changed. We can imagine being interested in this Δ2 x. We can see this as a numerical approximation to the second derivative of x, and this gets us back to differential equations. There are similar results for ΔΔΔ x even if we don’t wish to read it all out.
In principle, Δ x can be any number. In practice, at least for an independent variable, it’s a small number, usually real. Often we’re lured into thinking of it as positive, because a phrase like “x + Δ x” looks like we’re making a number a little bigger than x. When you’re a mathematician or a quality-control tester you remember to consider “what if Δ x is negative”. From testing that learn you wrote your computer code wrong. We’re less likely to assume this positive-ness for the dependent variable. By the time we do enough mathematics to have opinions we’ve seen too many decreasing functions to overlook that Δ y might be negative.
Notice that in that last paragraph I faithfully wrote Δ x and Δ y. Never Δ bare, unless I forgot and cannot find it in copy-editing. I’ve said that Δ means “the change in”; to write it without some variable is like writing √ by itself. We can understand wishing to talk about “the square root of”, as a concept. Still it means something else than √ x does.
We do write Δ by itself. Even professionals do. Written like this we don’t mean “the change in [ something ]”. We instead mean “a number”. In this role the symbol means the same thing as x or y or t might, a way to refer to a number whose value we might not know. We might not care about. The implication is that it’s small, at least if it’s something to add to the independent variable. We use it when we ponder how things would be different if there were a small change in something.
Small but not tiny. Here we step into mathematics as a language, which can be as quirky and ambiguous as English. Because sometimes we use the lower-case δ. And this also means “a small number”. It connotes a smaller number than Δ. Is 0.01 a suitable value for Δ? Or for δ? Maybe. My inclination would be to think of that as Δ, reserving δ for “a small number of value we don’t care to specify”. This may be my quirk. Others might see it different.
We will use this lowercase δ as an operator too, thinking of things like “x + δ x”. As you’d guess, δ x connotes a small change in x. Smaller than would earn the title Δ x. There is no declaring how much smaller. It’s contextual. As with δ bare, my tendency is to think that Δ x might be a specific number but that δ x is “a perturbation”, the general idea of a small number. We can understand many interesting problems as a small change from something we already understand. That small change often earns such a δ operator.
There are smaller changes than δ x. There are infinitesimal differences. This is our attempt to make sense of “a number as close to zero as you can get without being zero”. We forego the Greek letters for this and revert to Roman letters: dx and dy and dt and the other marks of differential calculus. These are difficult numbers to discuss. It took more than a century of mathematicians’ work to find a way our experience with Δ x could inform us about dx. (We do not use ‘d’ alone to mean an even smaller change than δ. Sometimes we will in analysis write d with a space beside it, waiting for a variable to have its differential taken. I feel unsettled when I see it.)
Much of the completion of work we can credit to Augustin Cauchy, who’s credited with about 800 publications. It’s an intimidating record, even before considering its importance. Cauchy is, per Florian Cajori’s History Mathematical Notations, one of the persons we can credit with the use of Δ as symbol for “the change in”. (Section 610.) He’s not the only one. Leonhardt Euler and Johann Bernoulli (section 640) used Δ to represent a finite difference, the difference between two values.
I’m not aware of an explicit statement why Δ got the pick, as opposed to other letters. It’s hard to imagine a reason besides “difference starts with d”. That an etymology seems obvious does not make it so. It does seem to have a more compelling explanation than the use of “m” for the slope of a line, or , though.
Slayton’s Mercury flight, performed by Scott Carpenter, did not involve any appreciable changes in orbit, a Δ v. No crewed spacecraft would until Gemini III. The Mercury flight did involve tests in orienting the spacecraft, in Δ θ and Δ φ on the angles of the spacecraft’s direction. These might have been in Slayton’s mind. He eventually flew into space on the Apollo-Soyuz Test Project, when an accident during landing exposed the crew to toxic gases. The investigation discovered a lesion on Slayton’s lung. A tiny thing, ultimately benign, which discovered earlier could have kicked him off the mission and altered his life so.
Jacob Siehler suggested the term for today’s A to Z essay. The letter V turned up a great crop of subjects: velocity, suggested by Dina Yagodich, and variable, from goldenoj, were also great suggestions. But Siehler offered something almost designed to appeal to me: an obscure function that shone in the days before electronic computers could do work for us. There was no chance of my resisting.
A story about the comeuppance of a know-it-all who was not me. It was in mathematics class in high school. The teacher was explaining logic, and showing off diagrams. These would compute propositions very interesting to logic-diagram-class connecting symbols. These symbols meant logical AND and OR and NOT and so on. One of the students pointed out, you know, the only symbol you actually need is NAND. The teacher nodded; this was so. By the clever arrangement of enough NAND operations you could get the result of all the standard logic operations. He said he’d wait while the know-it-all tried it for any realistic problem. If we are able to do NAND we can construct an XOR. But we will understand what we are trying to do more clearly if we have an XOR in the kit.
So the versine. It’s a (spherical) trigonometric function. The versine of an angle is the same value as 1 minus the cosine of the angle. This seems like a confused name; shouldn’t something called “versine” have, you know, a sine in its rule? Sure, and if you don’t like that 1 minus the cosine thing, you can instead use this. The versine of an angle is two times the square of the sine of half the angle. There is a vercosine, so you don’t need to worry about that. The vercosine is two times the square of the cosine of half the angle. That’s also equal to 1 plus the cosine of the angle.
This is all fine, but what’s the point? We can see why it might be easier to say “versine of θ” than to say “2 sin(1/2 θ)”. But how is “versine of θ” easier than “one minus cosine of θ”?
The strongest answer, at the risk of sounding old, is to ask back, you know we haven’t always done things the way we do them now, right?
We have, these days, settled on an idea of what the important trigonometric functions are. Start with Cartesian coordinates on some flat space. Draw a circle of radius 1 and with center at the origin. Draw a horizontal line starting at the origin and going off in the positive-x-direction. Draw another line from the center and making an angle with respect to the horizontal line. That line intersects the circle somewhere. The x-coordinate of that point is the cosine of the angle. The y-coordinate of that point is the sine of the angle. What could be more sensible?
That depends what you think sensible. We’re so used to drawing circles and making lines inside that we forget we can do other things. Here’s one.
Start with a circle. Again with radius 1. Now chop an arc out of it. Pick up that arc and drop it down on the ground. How far does this arc reach, left to right? How high does it reach, top to bottom?
Well, the arc you chopped out has some length. Let me call that length 2θ. That two makes it easier to put this in terms of familiar trig functions. How much space does this chopped and dropped arc cover, horizontally? That’s twice the sine of θ. How tall is this chopped and dropped arc? That’s the versine of θ.
We are accustomed to thinking of the relationships between pieces of a circle like this in terms of angles inside the circle. Or of the relationships of the legs of triangles. It seems obviously useful. We even know many formulas relating sines and cosines and other major functions to each other. But it’s no less valid to look at arcs plucked out of a circle and the length of that arc and its width and its height. This might be more convenient, especially if we are often thinking about the outsides of circular things. Indeed, the oldest tables we in the Western tradition have of trigonometric functions list sines and versines. Cosines would come later.
This partly answers why there should have ever been a versine. But we’ve had the cosine since Arabian mathematicians started thinking seriously about triangles. Why had we needed versine the last 1200 years? And why don’t we need it anymore?
One answer here is that mention about the oldest tables of trigonometric functions. Or of less-old tables. Until recently, as things go, anyone who wanted to do much computing needed tables of common functions at many different values. These tables might not have the since we really need of, say, 1.17 degrees. But if the table had 1.1 and 1.2 we could get pretty close.
But trigonometry will be needed. One of the great fields of practical mathematics has long been navigation. We locate points on the globe using latitude and longitude. To find the distance between points is a messy calculation. The calculation becomes less longwinded, and more clear, written using versines. (Properly, it uses the haversine, which is one-half times the versine. It will not surprise you that a 19th-century English mathematician coined that name.)
Having a neat formula is pleasant, but, you know. It’s navigators and surveyors using these formulas. They can deal with a lengthy formula. The typesetters publishing their books are already getting hazard pay. Why change a bunch of references to instead?
We get a difference when it comes time to calculate. Like, pencil on paper. The cosine (sine, versine, haversine, whatever) of 1.17 degrees is a transcendental number. We do not have the paper to write that number out. We’ll write down instead enough digits until we get tired. 0.99979, say. Maybe 0.9998. To take 1 minus that number? That’s 0.00021. Maybe 0.0002. What’s the difference?
It’s in the precision. 1.17 degrees is a measure that has three significant digits. 0.00021? That’s only two digits. 0.0002? That’s got only one digit. We’ve lost precision, and without even noticing it. Whatever calculations we’re making on this have grown error margins. Maybe we’re doing calculations for which this won’t matter. Do we know that, though?
This reflects what we call numerical instability. You may have made only a slight error. But your calculation might magnify that error until it overwhelms your calculation. There isn’t any one fix for numerical instability. But there are some good general practices. Like, don’t divide a large number by a small one. Don’t add a tiny number to a large one. And don’t subtract two very-nearly-equal numbers. Calculating 1 minus the cosines of a small angle is subtracting a number that’s quite close to 1 from a number that is 1. You’re not guaranteed danger, but you are at greater risk.
A table of versines, rather than one of cosines, can compensate for this. If you’re making a table of versines it’s because you know people need the versine of 1.17 degrees with some precision. You can list it as 2.08488 times 10-4, and trust them to use as much precision as they need. For the cosine table, 0.999792 will give cosine-users the same number of significant digits. But it shortchanges versine-users.
And here we see a reason for the versine to go from minor but useful function to obscure function. Any modern computer calculates with floating point numbers. You can get fifteen or thirty or, if you really need, sixty digits of precision for the cosine of 1.17 degrees. So you can get twelve or twenty-seven or fifty-seven digits for the versine of 1.17 degrees. We don’t need to look this up in a table constructed by someone working out formulas carefully.
This, I have to warn, doesn’t always work. Versine formulas for things like distance work pretty well with small angles. There are other angles for which they work badly. You have to calculate in different orders and maybe use other functions in that case. Part of numerical computing is selecting the way to compute the thing you want to do. Versines are for some kinds of problems a good way.
There are other advantages versines offered back when computing was difficult. In spherical trigonometry calculations they can let one skip steps demanding squares and square roots. If you do need to take a square root, you have the assurance that the versine will be non-negative. You don’t have to check that you aren’t slipping complex-valued numbers into your computation. If you need to take a logarithm, similarly, you know you don’t have to deal with the log of a negative number. (You still have to do something to avoid taking the logarithm of zero, but we can’t have everything.)
So this is what the versine offered. You could get precision that just using a cosine table wouldn’t necessarily offer. You could dodge numerical instabilities. You could save steps, in calculations and in thinking what to calculate. These are all good things. We can respect that. We enjoy now a computational abundance, which makes the things versine gave us seem like absurd penny-pinching. If computing were hard again, we might see the versine recovered from obscurity to, at least, having more special interest.
Wikipedia tells me that there are still specialized uses for the versine, or for the haversine. Recent decades, apparently, have found useful tools for calculating lunar distances, and sight reductions. The lunar distance is the angular separation between the Moon and some other body in the sky. Sight reduction is calculating positions from the apparent positions of reference objects. These are not problems that I work on often. But I would appreciate that we are still finding ways to do them well.
I mentioned that besides the versine there was a coversine and a haversine. There’s also a havercosine, and then some even more obscure functions with no less wonderful names like the exsecant. I cannot imagine needing a hacovercosine, except maybe to someday meet an obscure poetic meter, but I am happy to know such a thing is out there in case. A person on Wikipedia’s Talk page about the versine wished to know if we could define a vertangent and some other terms. We can, of course, but apparently no one has found a need for such a thing. If we find a problem that we would like to solve that they do well, this may change.
A friend sent me this video, after realizing that I had missed an earlier mention of it and thought it weird I never commented on it. And I wanted to pass it on, partly because it’s neat and partly because I haven’t done enough writing about topics besides the comics recently.
Particle Life: A Game Of Life Made Of Particles is, at least in video form, a fascinating little puzzle. The Game of Life referenced is one that anybody reading a pop mathematics blog is likely to know. But here goes. The Game of Life is this iterative process. We look at a grid of points, with each point having one of a small set of possible states. Traditionally, just two. At each iteration we go through every grid location. We might change that state. Whether we do depends on some simple rules. In the original Game of Life it’s (depending on your point of view) two or either three rules. A common variation is to include “mutations”, where a location’s state changes despite what the other rules would dictate. And the fascinating thing is that these very simple rules can yield incredibly complicated and beautiful patterns. It’s a neat mathematical refutation of the idea that life is so complicated that it must take a supernatural force to generate. It turns out that many things following simple rules can produce complicated patterns. We will often call them “unpredictable”, although (unless we do have mutations) they are literally perfectly predictable. They’re just chaotic, with tiny changes in the starting conditions often resulting in huge changes in behavior quickly.
This Particle Life problem is built on similar principles. The model is different. Instead of grid locations there are a cloud of particles. The rules are a handful of laws of attraction-or-repulsion. That is, that each particle exerts a force on all the other particles in the system. This is very like the real physics, of clouds of asteroids or of masses of electrically charged gasses or the like. But, like, a cloud of asteroids has everything following the same rule, everything attracts everything else with an intensity that depends on their distance apart. Masses of charged particles follow two rules, particles attracting or repelling each other with an intensity that depends on their distance apart.
This simulation gets more playful. There can be many kinds of particles. They can follow different and non-physically-realistic rules. Like, a red particle can be attracted to a blue, while a blue particle is repelled by a red. A green particle can be attracted to a red with twice the intensity that a red particle’s attracted to a green. Whatever; set different rules and you create different mock physics.
The result is, as the video shows, particles moving in “unpredictable” ways. Again, here, it’s “unpredictable” in the same way that I couldn’t predict when my birthday will next fall on a Tuesday. That is to say, it’s absolutely predictable; it’s just not obvious before you do the calculations. Still, it’s wonderful watching and tinkering with, if you have time to create some physics simulators. There’s source code for one in C++ that you might use. If you’re looking for little toy projects to write on your own, I suspect this would be a good little project to practice your Lua/LOVE coding, too.
My subject for today is another from Iva Sallay, longtime friend of the blog and creator of the Find the Factors recreational mathematics game. I think you’ll likely find something enjoyable at her site, whether it’s the puzzle or the neat bits of trivia as she works through all the counting numbers.
We don’t notice how unit fractions are around us. Likely there’s some in your pocket. Or there have been recently. Think of what you do when paying for a thing, when it’s not a whole number of dollars. (Pounds, euros, whatever the unit of currency is.) Suppose you have exact change. What do you give for the 38 cents?
Likely it’s something like a 25-cent piece and a 10-cent piece and three one-cent pieces. This is an American and Canadian solution. I know that 20-cent pieces are more common than 25-cent ones worldwide. It doesn’t make much difference; if you want it to be three 10-cent, one five-cent, and three one-cent pieces that’s as good. And granted, outside the United States it’s growing common to drop pennies altogether and round prices off to a five- or ten-cent value. Again, it doesn’t make much difference.
But look at the coins. The 25 cent piece is one-quarter of a dollar. It’s even called that, and stamped that on one side. I sometimes hear a dime called “a tenth of a dollar”, although mostly by carnival barkers in one-reel cartoons of the 1930s. A nickel is one-twentieth of a dollar. A penny is one-hundredth. A 20-cent piece is one-fifth of a dollar. And there are half-dollars out there, although not in the United States, not really anymore.
(Pre-decimalized currencies offered even more unit fractions. Using old British coins, for familiarity-to-me and great names, there were farthings, 1/960th of a pound; halfpennies, 1/480th; pennies, 1/240th; threepence, 1/80th of a pound; groats, 1/60th; sixpence, 1/40th; florins, 1/10th; half-crowns, 1/8th; crowns, 1/4th. And what seem to the modern wallet like impossibly tiny fractions like the half-, third-, and quarter-farthings used where 1/3840th of a pound might be a needed sum of money.)
Unit fractions get named and defined somewhere in elementary school arithmetic. They go on, becoming forgotten sometime after that. They might make a brief reappearance in calculus. There are some rational functions that get easier to integrate if you think of them as the sums of fractions, with constant numerators and polynomial denominators. These aren’t unit fractions. A unit fraction has a 1, the unit, in the numerator. But we see units along the way to integrating as an example. And see it in the promise that there are still more amazing integrals to learn how to do.
They get more attention if you take a history of computation class. Or read the subject on your own. Unit fractions stand out in history. We learn the Ancient Egyptians worked with fractions as sums of unit fractions. That is, had they dollars, they would not look at the we do. They would look at plus plus plus plus . When we count change we are using, without noticing it, a very old computing scheme.
This isn’t quite true. The Ancient Egyptians seemed to shun repeating a unit like that. To use once is fine; three times is suspicious. They would prefer something like plus plus . Or maybe some other combination. I just wrote out the first one I found.
But there are many ways we can make 38 cents using ordinary coins of the realm. There are infinitely many ways to make up any fraction using unit fractions. There’s surely a most “efficient”. Most efficient might be the one which uses the fewest number of terms. Most efficient might be the one that uses the smallest denominators. Choose what you like; no one knows a scheme that always turns up the most efficient, either way. We can always find some representation, though. It may not be “good”, but it will exist, which may be good enough. Leonardo of Pisa, or as he got named in the 19th century, Fibonacci, proved that was true.
We may ask why the Egyptians used unit fractions. They seem inefficient compared to the way we work with fractions. Or, better, decimals. I’m not sure the question can have a coherent answer. Why do we have a fashion for converting fractions to a “proper” form? Why do we use the number of decimal points we do for a given calculation? Sometimes a particular mode of expression is the fashion. It comes to seem natural because everyone uses it. We do it too.
And there is practicality to them. Even efficiency. If you need π, for example, you can write it as 3 plus plus and your answer is off by under one part in a thousand. Combine this with the Egyptian method of multiplication, where you would think of (say) “11 times π” as “1 times π plus 2 times π plus 8 times π”. And with tables they had worked up which tell you what and would be in a normal representation. You can get rather good calculations without having to do more than addition and looking up doublings. Represent π as 3 plus plus plus and you’re correct to within one part in 130 million. That isn’t bad for having to remember four whole numbers.
(The Ancient Egyptians, like many of us, were not absolutely consistent in only using unit fractions. They had symbols to represent and , probably due to these numbers coming up all the time. Human systems vary to make the commonest stuff we do easier.)
Enough practicality or efficiency, if this is that. Is there beauty? Is there wonder? Certainly. Much of it is in number theory. Number theory splits between astounding results and results that would be astounding if we had any idea how to prove them. Many of the astounding results are about unit fractions. Take, for example, the harmonic series . Truncate that series whenever you decide you’ve had enough. Different numbers of terms in this series will add up to different numbers. Eventually, infinitely many numbers. The numbers will grow ever-higher. There’s no number so big that it won’t, eventually, be surpassed by some long-enough truncated harmonic series. And yet, past the number 1, it’ll never touch a whole number again. Infinitely many partial sums. Partial sums differing from one another by one-googol-plex and smaller. And yet of the infinitely many whole numbers this series manages to miss them all, after its starting point. Worse, any set of consecutive terms, not even starting from 1, will never hit a whole number. I can understand a person who thinks mathematics is boring, but how can anyone not find it astonishing?
There are more strange, beautiful things. Consider heptagonal numbers, which Iva Sallay knows well. These are numbers like 1 and 7 and 18 and 34 and 55 and 1288. Take a heptagonal number of, oh, beads or dots or whatever, and you can lay them out to form a regular seven-sided figure. Add together the reciprocals of the heptagonal numbers. What do you get? It’s a weird number. It’s irrational, which you maybe would have guessed as more likely than not. But it’s also transcendental. Most real numbers are transcendental. But it’s often hard to prove any specific number is.
Unit fractions creep back into actual use. For example, in modular arithmetic, they offer a way to turn division back into multiplication. Division, in modular arithmetic, tends to be hard. Indeed, if you need an algorithm to make random-enough numbers, you often will do something with division in modular arithmetic. Suppose you want to divide by a number x, modulo y, and x and y are relatively prime, though. Then unit fractions tell us how to turn this into finding a greatest common denominator problem.
They teach us about our computers, too. Much of serious numerical mathematics involves matrix multiplication. Matrices are, for this purpose, tables of numbers. The Hilbert Matrix has elements that are entirely unit fractions. The Hilbert Matrix is really a family of square matrices. Pick any of the family you like. It can have two rows and two columns, or three rows and three columns, or ten rows and ten columns, or a million rows and a million columns. Your choice. The first row is made of the numbers and so on. The second row is made of the numbers and so on. The third row is made of the numbers and so on. You see how this is going.
Matrices can have inverses. It’s not guaranteed; matrices are like that. But the Hilbert Matrix does. It’s another matrix, of the same size. All the terms in it are integers. Multiply the Hilbert Matrix by its inverse and you get the Identity Matrix. This is a matrix, the same number of rows and columns as you started with. But nearly every element in the identity matrix is zero. The only exceptions are on the diagonal — first row, first column; second row, second column; third row, third column; and so on. There, the identity matrix has a 1. The identity matrix works, for matrix multiplication, much like the real number 1 works for normal multiplication.
Matrix multiplication is tedious. It’s not hard, but it involves a lot of multiplying and adding and it just takes forever. So set a computer to do this! And you get … uh ..
For a small Hilbert Matrix and its inverse, you get an identity matrix. That’s good. For a large Hilbert Matrix and its inverse? You get garbage. Large isn’t maybe very large. A 12 by 12 matrix gives you trouble. A 14 by 14 matrix gives you a mess. Well, on my computer it does. Cute little laptop I got when my former computer suddenly died. On a better computer? One designed for computation? … You could do a little better. Less good than you might imagine.
The trouble is that computers don’t really do mathematics. They do an approximation of it, numerical computing. Most use a scheme called floating point arithmetic. It mostly works well. There’s a bit of error in every calculation. For most calculations, though, the error stays small. At least relatively small. The Hilbert Matrix, built of unit fractions, doesn’t respect this. It and its inverse have a “numerical instability”. Some kinds of calculations make errors explode. They’ll overwhelm the meaningful calculation. It’s a bit of a mess.
Numerical instability is something anyone doing mathematics on the computer must learn. Must grow comfortable with. Must understand. The matrix multiplications, and inverses, that the Hilbert Matrix involves highlights those. A great and urgent example of a subtle danger of computerized mathematics waits for us in these unit fractions. And we’ve known and felt comfortable with them for thousands of years.
There’ll be some mathematical term with a name starting ‘V’ that, barring surprises, should be posted Friday. What’ll it be? I have an idea at least. It’ll be available at this link, as are the rest of these glossary posts.
The reruns of Donald Duck comics which appear at creators.com recently offered the above daily strip, featuring Ludwig von Drake and one of those computers of the kind movies and TV shows and comic strips had before anybody had computers of their own and, of course, the classic IBM motto that maybe they still have but I never hear anyone talking about it except as something from the distant and musty past. (Unfortunately, creators.com doesn’t note the date a strip originally ran, so all I can say is the strip first ran sometime after September of 1961 and before whenever Disney stopped having original daily strips drawn; I haven’t been able to find any hint of when that was other than not earlier than 1969 when cartoonist Al Taliaferro retired from it.)
[ Curious: one of the search engine terms which brought people here yesterday was “inner obnoxious”. I can think of when I’d used the words together, eg, in a phrase like “your inner obnoxious twelve-year-old”, the person who makes any kind of attempt at instruction difficult. But who’s searching for that? I find also that “the gil blog by norm feuti” and “heavenly nostrils” brought me visitors so, good for everyone, I think. ]
So polynomials have a number of really nice properties. They’re easy to work with, which is a big one. We might work with difficult mathematical objects, but, rather as with people, we’ll only work with the difficult if they offer something worthwhile in trade, such as solving problems we otherwise can’t hope to tackle. Polynomials are nice and friendly, uncomplaining, and as mathematical objects go, quite un-difficult. Polynomials can be used to approximate any function, which is another big one, as long as we don’t take that “any function” too literally. We still have to think about it some. But here’s an advantage so big it’s almost invisible: to evaluate a polynomial we take some number x and raise it to a variety of powers, which we get by multiplying x by itself over and over again. We take each of those powers and multiply them by a corresponding number, a coefficient. We then add up the products of those coefficients with those powers of x. In all that time we’ve done something great.