Pinball and Large Numbers


I had another little occasion to reflect on the ways of representing numbers, as well as the chance to feel a bit foolish, this past weekend so I’m naturally driven to share it. This came about on visiting the Silverball Museum, a pinball museum, or arcade, in Asbury Park, New Jersey. (I’m not sure the exact difference between a museum in which games are playable by visitors and an arcade, except for the signs affixed to nearly all the games.) Naturally I failed to bring my camera, so I can’t easily show what I had in mind; too bad.

Pinballs, at least once they got around to having electricity installed, need to show the scores. Since about the mid-1990s these have been shown by dot matrix displays, which are pretty easy to read — the current player’s score can be shown extremely large, for example — and make it easy for the game to go into different modes, where the scoring and objectives of play vary for a time. From about the mid-1970s to the mid-1990s eight-segment light-emitting diodes were preferred, for that “small alarm clock” look. And going before that were rotating number wheels, which are probably the iconic look to pinball score boards, to the extent anyone thinks of a classic pinball machine in that detail.

But there’s another score display, which I must admit offends my sense of order. In this, which I noticed mostly in the machines from the 1950s, with a few outliers in the early 60s (often used in conjunction with the rotating wheels), the parts of the number are broken apart, and the score is read by adding up the parts which are lit up. The machine I was looking at had one column of digits for the millions, another for hundreds of thousands, and then another with two-digit numbers.

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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?

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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.

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A Quick Impersonation Of Base Nine


I now resume the thread of spotting multiples of numbers easily. Thanks to the way positional notation lets us write out numbers as some multiple of our base, which is so nearly always ten it takes some effort to show where it’s not, it’s easy to spot whether a number is a multiple of that base, or some factor of the base, just by looking at the last digit. And if we’re interested in factors of some whole power of the base, of the ten squared which is a hundred, or the ten cubed which is a thousand, or so, we can find all we want to know just by looking at the last two or last three or last or-so digits.

Sadly, three and nine don’t go into ten, and never go into any power of ten either. Six and seven won’t either, although that exhausts the numbers below ten which don’t go into any power of ten. Of course, we also have the unpleasant point that eleven won’t go into a hundred or thousand or ten-thousand or more, and so won’t many other numbers we’d like.

If we didn’t have to use base ten, if we could use base nine, then we could get the benefits of instantly recognizing multiples of three or nine that we get for multiples of five or ten. If the digits of a number are some strand R finished off with an a, then the number written as Ra means the number gotten by multiplying nine by R and adding to that a. The whole strand will be divisible by nine whenever a is, which is to say when a is zero; and the whole strand will be divisible by three when a is, that is, when a is zero, three, or six.

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Illicitly Counted Coins


The past month I’ve had the joy of teaching a real, proper class again, after a hiatus of a few years. The hiatus has given me the chance to notice some things that I would do because that was the way I had done them, and made it easier to spot things that I could do differently.

To get a collection of data about which we could calculate statistics, I had everyone in the class flip a coin twenty times. Besides giving everyone something to do besides figure out which of my strange mutterings should be written down in case they turn out to be on the test, the result would give me a bunch of numbers, centered around ten, once they reported the number of heads which turned up. Counting the number of heads out of a set of coin flips is one of the traditional exercises to generate probability-and-statistics numbers.

Good examples are some of the most precious and needed things for teaching mathematics. It’s never enough to learn a formula; one needs to learn how to look at a problem, think of what one wants to know as a result of its posing, identify what one needs to get those results, and pick out which bits of information in the problem and which formulas allow the result to be found. It’s all the better if an example resembles something normal people would find to raise a plausible question. Here, we may not be all that interested in how many times a coin comes up heads or tails, but we can imagine being interested in how often something happens given a number of chances for it to happen, and how much that count of happenings can vary if we watch several different runs.

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Can A Ball Of Yarn Threaten Three-Dimensional Space?


All that talk about numbering spots on the New York Thruway had a goal, that of establishing how we could set up a coordinate system for the points on a line. It turns out just as easy to do this for a curve, even one a little bit complicated like a branch of the Thruway. About the only constraints we said anything about were that we shouldn’t have branches. Lurking unstated was the idea that we didn’t have loops. For the Thruway that’s nothing exceptional; if we had a traffic circle in the middle of a high-speed limited-access highway we wouldn’t very long have a high-speed highway. Worse, we’d have some point — where the loop crosses itself — that would have two numbers describing its position. We don’t want to face that. But we’ve got this satisfying little system where we can assign unique numbers to all the points on a single line, or even a curve.

The natural follow-up idea is whether we can set up a system where we can describe a point on a surface or even in all of space using the same sort of coordinates scheme. And there’s the obvious answer of how to do it, using two numbers to describe where something is on a surface, since that’s a two-dimensional thing; or three numbers to describe where it is in space, since that’s a three-dimensional thing. So I’m not going to talk about that just now. I want to do something more fun, the kind of thing that could do nicely in late-night conversations in the dorm lounge if undergraduates still have late-night conversations in the dorm lounge.

If we have a long enough thread, or a strand of yarn, or whatever the quite correct term is, we know this can be set up with a coordinate system by marking off distance along that thread. We imagined doing that, more or less, with the numbering system on the Thruway and imagining the straightening out and curving and other moving around of the highway’s center line. As long as we didn’t stretch or compress the strand any, we could spread it out in any shape we liked, and have coordinates for whatever path the strand traces out.

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How To Recognize Multiples Of 100 From Not So Far Away


MJ Howard last week answered my little demonstration that it was easy to tell multiples of two, five, and ten by looking at just the last digit of a whole number, but that there weren’t any ways to tell from just the last digit whether it was divisible by four. He pointed out we could look at the last two digits, and if those were divisible by four, then the entire number would be. This is perfectly true, and it’s only by asserting that I was looking for a rule based on the last digit alone that my forecast of doom about an instant check for divisibility-by-four could be sustained.

Remember the reasoning by which we wrote out a whole number as some string of digits which I call R followed by whatever goes in the units column, which I call a. (I had been thinking of R as in the “rest” of the number, but it struck me over the week that R is also the symbol used in organic chemistry to denote a chain of carbon atoms when one doesn’t really care how many of them are lined up. This interests me as I got on this thread with a set of numbers I called “alcoholic” due to their structural resemblance to organic chemistry’s idea of alcohols.) Since we’re writing in base ten, then, the number written as Ra is ten times R plus a. Ten times R can’t help being divisible by ten, or by any of the factors of ten, which are two and five (and one, which nobody cares about).

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Searching For Infinity On The New York Thruway


So with several examples I’ve managed to prove what nobody really questioned, that it’s possible to imagine a complicated curve like the route of the New York Thruway and assign to all the points on it, or at least to the center line of the road, a unique number that no other point on the road has. And, more, it’s possible to assign these unique numbers in many different ways, from any lower bound we like to any upper bound we like. It’s a nice system, particularly if we’re short on numbers to tell us when we approach Loudonville.

But I’m feeling ambitious right now and want to see how ridiculously huge, positive or negative, a number I can assign to some point on the road. Since we’d measured distances from a reference point by miles before and got a range of about 500, or by millimeters and got a range of about 800,000,000, obviously we could get to any number, however big or small, just by measuring distance using the appropriate unit: lay megaparsecs or angstroms down on the Thruway, or even use some awkward or contrived units. I want to shoot for infinitely big numbers. I’ll start by dividing the road in two.

After all, there are two halves to the Thruway, a northern and a southern end, both arranged like upside-down u’s across the state. Instead of thinking of the center line of the whole Thruway, then, think of the center lines of the northern road and of the southern. They’re both about the same 496-mile length, but, it’d be remarkable if they were exactly the same length. Let’s suppose the northern belt is 497 miles, and the southern 495. Pretty naturally the northern belt we can give numbers from 0 to 497, based on how far they are from the south-eastern end of the road; similarly, the southern belt gets numbers from 0 to 495, from the same reference point.

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One Goat Short


I like game shows. Liking game shows is not one of the more respectable hobbies, compared to, say, Crimean War pedantry, or laughing at goats. Game shows have a long history of being sneered at by people who can’t be bothered to learn enough about game shows to sneer at them for correct reasons. Lost somewhere within my archives is even an anthology of science fiction short stories about game shows, which if you take out the punch lines of “and the loser DIES!” or “and the host [ typically Chuck Woolery ] is SATAN!”, would leave nearly nothing, and considering that science fiction as a genre has spent most of its existence feeling picked-on as the “smelly, unemployed cousin of the entertainment industry” (Mike Nelson’s Movie Megacheese) that’s quite some sneering. Sneering at game shows even earned an episode of The Mary Tyler Moore show which managed to be not just bad but offensively illogical.

Nevertheless, I like them, and was a child at a great age for game shows on broadcast television: the late 1970s and early 1980s had an apparently endless menu of programs, testing people’s abilities to think of words, to spell words, to price household goods, and guess how other people answered surveys. We haven’t anything like that anymore; on network TV about the only game shows that survive are Jeopardy! (which nearly alone of the genre gets any respect), Wheel of Fortune, The Price Is Right, and, returned after decades away, Let’s Make A Deal. (I don’t regard reality shows as game shows, despite a common programming heritage. I can’t say what it is precisely other than location and sometimes scale that, say, Survivor or The Amazing Race do that Beat The Clock or Truth Or Consequences do not, but there’s something.) Now and then something new flutters into being, but it vanishes without leaving much of a trace, besides retreading jokes about the people who’d watch it.

All that is longwinded preliminary to one of those things that amuses mostly me. On the Thursday (27 October) episode of Let’s Make A Deal, they briefly looked like they might be playing The Monty Hall Problem.

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How To Recognize Multiples Of Ten From Quite A Long Way Away


I got so caught up last week talking about the different possible bases that I forgot to the interesting thing I had wanted to talk about those bases. I suppose that will happen as long as I write to passion rather than plan. It gives me something to speak about today, at least.

Here is one thing implied by having a consistent base for all these numbers in which position is relevant: a one in each column represents the base-number of units of whatever the next column over represents. That is, in base ten, a one in the tens column represents ten units of one; a one in the thousands column represents ten units of one hundred. I mention this obvious point because it is so familiar and simple as to pass into invisibility. (It also extends past the decimal point; a one in the hundredths column is equivalent to ten units of a thousandth. But I want to talk about divisibility, in the whole numbers, and so leave fractions for some later time.)

This is tidy, in a way that we don’t see in variable bases. It will give us one tool for neat little divisibility rules. That tool appears just by writing things in the appropriate way, which is the best sort of tool. It saves on time trying to prove it works.

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43 McNuggets Made Difficult


I mentioned in the last comments thread the McNuggets Problem, and realized belatedly that maybe not everybody knew just what that was. It’s a cute little one, which Wolfram’s Mathworld is able to date to 1991, or maybe 1990. There’s a reference to a March 1990 puzzle on Usenet newsgroup rec.puzzles, but to find it would require some Google-like search engine capable of finding postings on Usenet, and that technology is sadly beyond us.

Whether 1990 or 1991 seems late, since I’m certain the puzzle first appeared about the same time people first saw the original Chicken McNuggets menu options on sale, sometime in the mid-80s. In the original offerings, one could buy a pack of six, nine, or if Mom was feeling particularly flush with cash or you gave a credible impersonation of being willing to share with your siblings, twenty. The obvious question, then, is what’s the largest number of McNuggets which can’t be bought by some combination of these?

This can be studied rigorously, although I don’t know anyone who actually would. It’s more fun to play and see what can be constructed: 12, obviously; 15, as surely; 18 as well (and that by two different patterns, three packs of six or two packs of nine). 21, 24 (again by two paths), 26, 27, 29, 30 … it looks very much like we’re running out of numbers to buy, and some experimentation finds that 43 is the biggest number of McNuggets which can’t be bought. At least, we can find the formulas for 44, 45, 46, 47, 48, and 49, and obviously, any number above that you can get by buying enough six-packs on top of whatever one of those is.

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How I Make Myself Look Foolish


It seems to me that I need to factor numbers more often than most people do. I can’t even attribute this to my being a mathematician, since I don’t think along the lines of anything like mathematical work; I just find that I need to know, say, that 272,250 is what you get by multiplying 2 and 3 to the second power and 5 to the third power and 11 to the second power. And I reliably go to places I know will do calculations quickly, like the desktop Calculator application or what you get from typing mathematical expressions into Google, and find that since the last time I looked they still haven’t added a factorization tool. I have tools I can use, particularly Matlab or its open-source work-just-enough-alike-to-make-swapping-code-difficult replica Octave, which takes a long time to start up for one lousy number.

So I got to thinking: I’ve wanted to learn a bit about writing apps, and surely, writing a factorization app is both easy and quick and would prove I could write something. The routine is easy, too: take a number (272,250) as input; then divide by two as many times as you can (just one, giving 136,125), then divide by three as many times as you can (twice, giving 15,125), then by five as many times as you can (three times, reaching 121), then by seven (you can’t), then eleven (twice, reaching 1), until you’ve run the whole number down. You just need to divide repeatedly by the prime numbers, starting at two, and going up only to the square root of whatever your input number is.

Without bothering to program, then, I thought about how I could make this a more efficient routine. Figuring out more efficient ways to code is good practice, because if you think long enough about how to code efficiently, you can feel satisfied that you would have written a very good program and never bother to actually do it, which would only spoil the beauty of the code anyway. Here’s where the possible inefficiency sets in: how do you know what all the prime numbers up to the square root of whatever you’re interested in is?

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Bases For Comparison


Back to the theme of divisibility of numbers. Since we have the idea of writing numbers with a small set of digits, and with the place of those digits carrying information about how big the number is, we can think about what’s implied by that information.

In the number 222, the first two is matched to blocks (hundreds) that are ten times as large as those for the second two (tens), and the second two is matched to units (tens) which are ten times as large as those for the third two (units). It is now extremely rare to have the size of those blocks differ from one place to the next; that is, a number before the initial two here we take without needing it made explicit to represent ten times that hundreds unit, and a number after the final two (and therefore after the decimal point) would represent units which are one-tenth that of the final two’s size.

It has also become extremely rare for the relationship between blocks to be anything but a factor of ten, with two exceptions which I’ll mention next paragraph. The only block other than those with common use which comes to my mind is the sixty-to-one division of hours or degrees into minutes, and then of minutes into seconds. Even there the division of degrees of arc into minutes and seconds might be obsolete, as it’s so much easier on the computer to enter a latitude and longitude with decimals instead. So blocks of ten, decimals, it is, or in the way actual people speak of such things, a number written in base ten.

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Searching For 800,000,000 On The New York Thruway


So we’ve introduced, with maybe more words than strictly necessary, the idea that we can set up a match between the numbers from 0 to 496 and particular locations on the New York Thruway. There are a number of practical quibbles that can be brought against this scheme. For example: could we say for certain that the “outer” edge of this road, which has roughly the shape of an upside-down u, isn’t loger than the “inner” edge? We may need more numbers for the one side than the other. And the mile markers, which seemed like an acceptable scheme for noting where one was, are almost certainly only approximately located.

But these aren’t very important. We can imagine the existence of the “ideal” Thruway, some line which runs along the median of the whole extent of the highway, so there’s no difference in length running either direction, and we can imagine measuring it to arbitrarily great precision. The actual road approximates that idealized road. And this gives what I had really wanted, a kind of number line. All the numbers from zero to 496 (or so) match a point on this ideal Thruway line, and all the points on this Thruway match some number between zero and 496. That the line wriggles all over the place and changes direction over and over, well, do we really insist that a line has to be straight?

Well, we can at least imagine taking this “ideal” Thruway, lifting it off the globe and straightening it out, if we really want to. Here we invoke a host of assumptions even past the idea that we can move this curvy idealized road around. We assume that we can straighten it out without changing its length, for example. This isn’t too unreasonable if we imagine this curve as being something like a tangled bit of string and that we straighten it out without putting any particular tension on it; but if we imagined the idealized road as being a rubber band, held taut at the New York City and Ripley, New York, ends and pinned in place at the major turns we notice that isn’t actually guaranteed. Let’s assume we can do this straightening-out without distorting the lengths, though.

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Searching For e On The New York Thruway


To return to my introduction of e using the most roundabout method possible I’d like to imagine the problem of telling someone just where it is you’ve been stranded in a broken car on the New York Thruway. Actually, I’d rather imagine the problem of being stranded in a broken car on the New Jersey Turnpike, as it’s much closer to my home, but the Turnpike has a complexity I don’t want distracting this chat, so I place the action one state north. Either road will do.

There’s too much toll road to just tell someone to find you there, and the majority of their lengths are away from any distinctive scenery, like an airport or a rest area, which would pin a location down. A gradual turn with trees on both sides is hardly distinctive. What’s needed is some fixed reference point. Fortunately, the Thruway Authority has been generous and provided more than sixty of them. These are the toll plazas: if we report that we are somewhere between exits 23 and 24, we have narrowed down our location to a six-mile stretch, which over a 496-mile road is not doing badly. We can imagine having our contact search that.

But the toll both standard has many inconveniences. The biggest is that exits are not uniformly spaced. At the New York City end of the Thruway, before tolls start, exits can be under a mile apart; upstate, where major centers of population become sparse, they can spread out to nearly twenty miles apart. As we wait for rescue those twenty miles seem to get longer.

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What Are Numbers Made Of?


To return to my second major theme: my Dearly Beloved told me that I must explain that trick where one adds up the digits of a number and finds out from that whether it’s divisible by 9. I wanted to anyway, but a request like that is irresistible. The answer can be given quickly — and several of my hopefully faithful readers did, in comments, last Friday — but I’d like to take the long way around because I do that and because it lets a lot of other interesting divisibility properties show themselves.

We use ten numerals and the place where we write them to express all the counting numbers out there. We put one of the numerals, such as `2′, in a place which denotes whether we mean to say two tens, or two hundreds, or two millions. That’s a clever tool, and not one inherent to the idea of numbers. We could as easily use different symbols for different magnitudes; the only familiar example of this (in the west) is Roman numerals, where we use I, X, C, and M for increasing powers of ten, and then notice we aren’t really quite sure what to do past M.

The Romans were not very sure either, and individual variations developed when someone found they needed to express an M of M very often. The system has fewer numerals, symbols representing numbers, than ours does, with V and L and D the only additional numerals reasonably common. By the Middle Ages some symbols were improvised to allow for extremely large numbers such as the hundred thousands, and some extra symbols were pulled in for numbers such as 7 or 40, but they have faded to the point of obscurity. This is a numbering system which runs out when the numbers get too large, which seems impossibly limited at first glance. But we haven’t changed much from these times: while we have a numbering system that can, in principle, work with arbitrarily big or tiny numbers, in practice we only use a small range of them. When we turn over arithmetic to computers, in fact, we accept numbering systems which have limits on how big (positive or negative) a number may be, or how close to zero one may work. We accept those limits because of their convenience and are only sometimes annoyed to find, for example, that the spreadsheet trying to calculate a bill has decided we want 0.9999999 of a penny.

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The Person Who Named e


One of the personality traits which my Dearly Beloved most often tolerates in me is my tendency toward hyperbole, a rhetorical device employed successfully on the Internet by almost four people and recognized as such as recently as 1998. I’m not satisfied saying there was an enormous, slow-moving line for a roller coaster we rode last August; I have to say that fourteen months later we’re still on that line.

I mention this because I need to discuss one of those rare people who can be discussed accurately only in hyperbole: Leonhard Euler, 1703 – 1783. He wrote about essentially every field of mathematics it was possible to write about: calculus and geometry and physics and algebra and number theory and graph theory and logic, on music and the motions of the moon, on optics and the finding of longitude, on fluid dynamics and the frequency of prime numbers. After his death the Saint Petersburg Academy needed nearly fifty years to finish publishing his remaining work. If you ever need to fake being a mathematician, let someone else introduce the topic and then speak of how Euler’s Theorem is fundamental to it. There are several thousand Euler’s Theorems, although some of them share billing with another worthy, and most of them are fundamental to at least sixteen fields of mathematics each. I exaggerate; I must, but I note that a search for “Euler” on Wolfram Mathworld turns up 681 matches, as of this moment, out of 13,081 entries. It’s difficult to imagine other names taking up more than five percent of known mathematics. Even Karl Friedrich Gauss only matches 272 entries, and Isaac Newton a paltry 138.

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Something Cute Without 9’s and a 6


The cute little thing about a string of 9’s followed by a 6 being a number divisible by 6 inspired my Dearly Beloved, who spent some time looking for other patterns in this kind of number. I’m glad for that; this sort of pattern, while it may not be terribly important, is often fun to play with. And interesting things can be found in play.

I don’t know a good name for this kind of number, and admit it feels awkward to say just “this kind of number”. If I have to talk about them much longer some group name is probably worth devising. Unfortunately the only names which come to my mind come there through organic chemistry, where it’s reasonably common to have an arbitrarily long chain of carbon atoms terminated with some distinctly different group. For example, an alcohol is a string of carbons ending with an oxygen and hydrogen molecule. But an “alcoholic number”, while an imagination-capturing name, doesn’t quite fit. I suppose aldehydes, which end on a double-bond to an oxygen atom, preserves the metaphor, but no one knows the adjective form of aldehyde.

My Dearly Beloved’s experiments found no other numbers for which a repeated string, terminated by a 6, would produce a number divisible by 6. This overlooked the obvious case, though: a string of 6’s, followed by another 6, is itself divisible by 6. Obvious cases are like that, and many people would think of a uniform string of 6’s not part of the pattern “an arbitrary number of one digit, followed by a 6”.

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In Defense Of FOIL


I do sometimes read online forums of educators, particularly math educators, since it’s fun to have somewhere to talk shop, and the topics of conversation are constant enough you don’t have to spend much time getting the flavor of a particular group before participating. If you suppose the students are lazy, the administrators meddling, the community unsupportive, and the public irrationally terrified of mathematics you’ve covered most forum threads. I had no luck holding forth my view on one particular topic, though, so I’ll try fighting again here where I can easily squelch the opposition.

The argument, a subset of students-are-lazy (as they don’t wish to understand mathematics), was about a mnemonic technique called FOIL. It’s a tool to help people multiply binomials. Binomials are the sum (or difference) of two quantities, for example, (a + 2) or (b + 5). Here a and b are numbers whose value I don’t care about; I don’t care about the 2 or 5 either, but by picking specific values I avoid having too much abstraction in my paragraph. The product of (a + 2) with (b + 5) is the sum of all the pairs made by multiplying one term in the first binomial by one term in the second. There are four such pairs: a times b, and a times 5, and 2 times b, and 2 times 5. And therefore the product (a + 2) * (b + 5) will be a*b + a*5 + 2*b + 2*5. That would usually be cleaned up by writing 5*a instead of a*5, and by writing 10 instead of 2*5, so the sum would become a*b + 5*a + 2*b + 10.

FOIL is a way of making sure one has covered all the pairs. The letters stand for First, Outer, Inner, Last, and they mean: take the product of the First terms in each binomial, a and b; and those of the Outer terms, a and 5; and those of the Inner terms, 2 and b; and those of the Last terms, 2 and 5.

Here is my distinguished colleague’s objection to FOIL: Nobody needs it. This is true.

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Some Names Which e Doesn’t Have


I’ve outlined now some of the numbers which grew important enough to earn their own names. Most of them are counting numbers; the stragglers are a handful of irrational numbers which proved themselves useful, such as π (pi), or attractive, such as φ (phi), or physically important, such as the fine structure constant. Unnamed except in the list of categories is the number whose explanation I hope to be the first movement of this blog: e.

It’s an important number physically, and a convenient and practical number mathematically. For all that, it defies a simple explanation like π enjoys. The simplest description of which I’m aware is that it is the base of the natural logarithm, which perfectly clarifies things to people who know what logarithms are, know which one is the natural logarithm, and know what the significance of the base is. This I will explain, but not today. For now it’s enough to think of the base as a size of the measurement tool, and to know that switching between one base and another is akin to switching between measuring in centimeters and measuring in inches. What the logarithm is will also wait for explanation; for now, let me hold off on that by saying it’s, in a way, a measure of how many digits it takes to write down a number, so that “81” has a logarithm twice that of “9”, and “49” twice that of “7”, and please don’t take this description so literally as to think the logarithm of “81” is equal to that of “49”.

I agree it’s not clear why we should be interested in the natural logarithm when there are an infinity of possible logarithms, and we can convert a logarithm base e into a logarithm base 10 just by multiplying by the correct number. That, too, will come.

Another common explanation is to say that e describes how fast savings will grow under the influence of compound interest. A dollar invested at one-percent interest, compounded daily, for a year, will grow to just about e dollars. Compounded hourly it grows even closer; compounded by the second it grows closer still; compounded annually, it stays pretty far away. The comparison is probably perfectly clear to those who can invest in anything with interest compounded daily. For my part I note when I finally opened an individual retirement account I put a thousand dollars into an almost thoughtfully selected mutual fund, and within mere weeks had lost $15. That about finishes off compound interest to me.

Continue reading “Some Names Which e Doesn’t Have”

Something Cute With 9’s and a 6.


After the last few essays I’d like to take a moment for a distinct, cute little problem of no practical use but cute.

Write down as many 9’s as you like, and when finished with that place a 6 at the right end. The result is divisible by 6.

That is, whatever number you’ve written, divided by 6, produces a whole number. Divisibility is one of those things which turns up whenever you have a collection of things which can be multiplied, and one thing is divisible by the second if you can find something in your collection so that the second multiplied by your find equals the first. It’s most often used to talk about the integers — the positive counting numbers, their negative counterparts, and zero if we didn’t include that already — and if it isn’t said divisible-with-respect-to-what then integers are what is usually meant. Partly that’s because integers are the first thing where divisibility stands out: if we look at the real numbers, everything is divisible by everything else (as long as that “else” is not zero), and a property that’s (almost) always true is usually too dull to mention. The next topic where divisibility gets mentioned much is usually polynomials, with a few eccentrics holding out for the complex numbers where the real part and the imaginary part are both integers.

There are several ways to prove this string of 9’s followed by 6 is divisible by 6. Here’s a proof which I like.

Continue reading “Something Cute With 9’s and a 6.”

How Many Numbers Have We Named?


I want to talk about some numbers which have names, and to argue that surprisingly few of numbers do. To make that argument it would be useful to say what numbers I think have names, and which ones haven’t; perhaps if I say enough I will find out.

For example, “one” is certainly a name of a number. So are “two” and “three” and so on, and going up to “twenty”, and going down to “zero”. But is “twenty-one” the name of a number, or just a label for the number described by the formula “take the number called twenty and add to it the number called one”?

It feels to me more like a label. I note for support the former London-dialect preference for writing such numbers as one-and-twenty, two-and-twenty, and so on, a construction still remembered in Charles Dickens, in nursery rhymes about blackbirds baked in pies, in poetry about the ways of constructing tribal lays correctly. It tells you how to calculate the number based on a few named numbers and some operations.

None of these are negative numbers. I can’t think of a properly named negative number, just ones we specify by prepending “minus” or “negative” to the label given a positive number. But negative numbers are fairly new things, a concept we have found comfortable for only a few centuries. Perhaps we will find something that simply must be named.

That tips my attitude (for today) about these names, that I admit “thirty” and “forty” and so up to a “hundred” as names. After that we return to what feel like formulas: a hundred and one, a hundred and ten, two hundred and fifty. We name a number, to say how many hundreds there are, and then whatever is left over. In ruling “thirty” in as a name and “three hundred” out I am being inconsistent; fortunately, I am speaking of peculiarities of the English language, so no one will notice. My dictionary notes the “-ty” suffix, going back to old English, means “groups of ten”. This makes “thirty” just “three tens”, stuffed down a little, yet somehow I think of “thirty” as different from “three hundred”, possibly because the latter does not appear in my dictionary. Somehow the impression formed in my mind before I thought to look.
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What’s Remarkable About Naming Sixty?


Here’s the astounding thing Christopher Hibbert did with his estimate of how much prices in 18th century Britain had to be multiplied to get an estimate for their amount in modern times: he named it.

Superficially, I have no place calling this astounding. If Hibbert didn’t have an estimate for how to convert 1782 prices to 1998 ones he would have not mentioned the topic at all. But consider: the best fit for a conversion factor could be from any of, literally, infinitely many imaginable numbers. That it should happen to be a familiar, common number, one so ordinary it even has a name, is the astounding part.

Part of that is a rounding-off, certainly. Perhaps the best possible fit to convert those old prices to the modern was actually a slight bit under 62, or was 57 and three-eighteenths. But nobody knows what £200 times 57 and three-eighteenths would be, as evaluating it would require multiplying by sevens, which no one feels comfortable doing, and dividing by eighteen, which makes multiplying by seven seem comfortable, unless we remember where we left the calculator, and why would we dig out a calculator to read about King George III?
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Did King George III pay too little for astronomers or too much for tea?


In the opening pages of his 1998 biography George III: A Personal History, Christopher Hibbert tosses a remarkable statement into a footnote just after describing the allowance of Frederick, Prince of Wales, at George III’s birth:

Because of the fluctuating rate of inflation and other reasons it is not really practicable to translate eighteen-century sums into present-day equivalents. Multiplying the figures in this book by about sixty should give a very rough guide for the years before 1793. For the years of war between 1793 and 1815 the reader should multiply by about thirty, and thereafter by about forty.

“Not really practical” is wonderful understatement: it’s barely possible to compare the prices of things today to those of a half-century ago, and the modern economy at least existed in cartoon back then. I could conceivably have been paid for programming computers back then, but it would be harder for me to get into the field. To go back 250 years — before electricity, mass markets, public education, mass production, general incorporation laws, and nearly every form of transportation not muscle or wind-powered — and try to compare prices is nonsense. We may as well ask how many haikus it would take to tell Homer’s Odyssey, or how many limericks Ovid’s Metamorphoses would be.
Continue reading “Did King George III pay too little for astronomers or too much for tea?”