A Hundred, And Other Things

The other day my humor blog featured a little table of things for which a “hundred” of them isn’t necessarily 100 of them. It’s just a little bit of wonder I found on skimming the “Index to Units and Systems of Units” page, one of those simple reference sites that is just compelling in how much trivia there is to enjoy. The page offers examples of various units, from those that are common today (acres, meters, gallons), to those of local or historic use (the grosses tausend, the farthingdale), to those of specialized application (the seger cone, used by potters to measure the maximum temperature of kiln). It’s just a wonder of things that can be measured.

There’s a wonderful diversity of commodities for which a “hundred” is not 100 units, though. Many — skins, nails, eggs, herring — have a “hundred” that consists of 120. That seems to defy the definition of a “hundred”, but I like to think that serves as a reminder that units are creations of humans to organize the way we think about things, and it’s convenient to have a unit that is “an awful lot of, but not unimaginably lot of” whatever we’re talking about, and a “hundred” seems to serve that role pretty well. The “hundreds” which are actually 120 probably come about from wanting to have a count of things that’s both an awful lot of the thing and is also an amount that can be subdivided into equal parts very well. 120 of a thing can be divided evenly into two, three, four, five, six, eight, ten, twelve, and so on equal shares; 100 is relatively impoverished for equal subdivisions.

I do not know the story behind some of the more curious hundreds, such as the counting of 106 sheep or lambs as a hundred in Roxburghshire and Selkirkshire (counties in the southeast of Scotland), or the counting of 160 dried fish as a hundred, but it likely reflects the people working with such things finding these to be slightly more convenient numbers than a plain old 100 for the “big but not unimaginably lot of” a thing. The 225 making up a hundred of onions and garlic, for example, seems particularly exotic, but it’s less so when you notice that’s 15 times 15. One of the citations of this “hundred” describes it as “15 ropes and every rope each with 15 heads”. Suddenly this hundred is a reasonable number of things that are themselves reasonable numbers of things.

Of course if they hadn’t called it a “hundred” then I wouldn’t have had a pretty easy comic bit to build from it, but how were they to know the meaning of “hundred” in everyday speech would settle down to an unimaginative solitary value?

Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?

Well, I thought it’d be unlikely to get too many more mathematics comics before the end of the year, but Comic Strip Master Command apparently sent out orders to clear out the backlog before the new calendar year starts. I think Dark Side of the Horse is my favorite of the strips, blending a good joke with appealing artwork, although The Buckets gives me the most to talk about.

Greg Cravens’s The Buckets (December 28) is about what might seem only loosely a mathematical topic: that the calendar is really a pretty screwy creation. And it is, as anyone who’s tried to program a computer to show dates has realized. The core problem, I suppose, is that the calendar tries to meet several goals simultaneously: it’s supposed to use our 24-hour days to keep track of the astronomical year, which is an approximation to the cycle of seasons of the year, and there’s not a whole number of days in a year. It’s also supposed to be used to track short-term events (weeks) and medium-term events (months and seasons). The number of days that best approximate the year, 365 and 366, aren’t numbers that lend themselves to many useful arrangements. The months try to divide that 365 or 366 reasonably uniformly, with historial artifacts that can be traced back to the Roman calendar was just an unspeakable mess; and, something rarely appreciated, the calendar also has to make sure that the date of Easter is something reasonable. And, of course, any reforming of the calendar has to be done with the agreement of a wide swath of the world simultaneously. Given all these constraints it’s probably remarkable that it’s only as messed up as it is.

To the best of my knowledge, January starts the New Year because Tarquin Priscus, King of Rome from 616 – 579 BC, found that convenient after he did some calendar-rejiggering (particularly, swapping the order of February and January), though I don’t know why he thought that particularly convenient. New Years have appeared all over the calendar year, though, with the start of January, the start of September, Christmas Day, and the 25th of March being popular options, and if you think it’s messed up to have a new year start midweek, think about having a new year start in the middle of late March. It all could be worse.

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Reading the Comics, May 26, 2014: Definitions Edition

The most recent bunch of mathematics-themed comics left me feeling stumped for a theme. There’s no reason they have to have one, of course; cartoonists, as far as I know, don’t actually take orders from Comic Strip Master Command regarding what to write about, but often they seem to. Some of them seem to touch on definitions, at least, including of such ideas as the value of a quantity and how long it is between two events. I’ll take that.

Jef Mallet’s Frazz (May 23) does the kid-resisting-the-question sort of joke (not a word problem, for a change of pace), although I admit I didn’t care for the joke. I needed too long to figure out how the meaning of “value” for a variable might be ambiguous. Caulfield kind of has a point about mathematics needing to use precise words, but the process of making a word precise is a great and neglected part of mathematical history. Consider, for example: contemporary (English-language, at least) mathematicians define a prime number to be a counting number (1, 2, 3, et cetera) with exactly two factors. Why exactly two factors, except to rule out 1 as a prime number? But then why rule that 1 can’t be a prime number? As an idea gets used and explored we get a better idea of what’s interesting about it, and what it’s useful for, and can start seeing whether some things should be ruled out as not fitting a concept we want to describe, or be accepted as fitting because the concept is too useful otherwise and there’s no clear way to divide what we want from what we don’t.

I still can’t buy Caulfield’s proposition there, though.

Steve Boreman’s Little Dog Lost (May 25) circles around a bunch of mathematical concepts without quite landing on any of them. The obvious thing is the counting ability of animals: the crow asserts that crows can only count as high as nine, for example, and the animals try to work out ways to deal with the very large number of 2,615. The vulture asserts he’s been waiting for 2,615 days for the Little Dog to cross the road, and wonders how many years that’s been. The first installment of the strip, from the 26th of March, 2007, did indeed feature Vulture waiting for Little Dog to cross the road, although as I make it out there’s 2,617 days between those events.

At a guess, either Boreman was not counting the first and the last days of the interval between March 26, 2007, and May 25, 2014, or maybe he forgot the leap days. Finding how long it is between dates is a couple of kinds of messes, first because it isn’t necessarily clear whether to include the end dates, and second because the Gregorian calendar is a mess of months of varying lengths plus the fun of leap years, which include an exception for century years and an exception to the exception, making it all the harder. My preferred route for finding intervals is to not even try working the time out by myself, and instead converting every date to the Julian date, a simple serial count of the number of dates since noon Universal Time on the 1st of January, 4713 BC, on the Julian calendar. Let the Navy deal with leap days. I have better things to worry about.

Samson’s Dark Side Of The Horse (May 26) sees Horace trying to count sheep to get himself to sleep; different ways of denoting numbers confound him. I’m not sure if it’s known why counting sheep, or any task like that, is useful in getting to sleep. My guess would be that it just falls into the sort of activity that can be done without a natural endpoint and without demanding too much attention to keep one awake, while demanding enough attention that one isn’t thinking about the bank account or the noise inside the walls or the way the car lurches two lanes to the right every time one taps the brake at highway speeds. That’s a guess, though.

Tom Horacek’s Foolish Mortals (May 26) uses the “on a scale of one to ten” standard for something that’s not usually described so vaguely, and I like the way it teases the idea of how to measure things. The “scale of one to ten” is logically flawed, since we have no idea what the units are, how little of something one represents or how much the ten does, or even whether it’s a linear scale — the difference between “two” and “three” is the same as that between “three” and “four”, the way lengths and weight work — or a logarithmic one — the ratio between “two” and “three” equals that between “three” and “four”, the way stellar magnitudes, decibel sound readings, and Richter scale earthquake intensity measure work — or, for that matter, what normal ought to be. And yet there’s something useful in making the assessment, surely because the first step towards usefully quantifying a thing is to make a clumsy and imprecise quantification of it.

Dave Blazek’s Loose Parts (May 26) kind of piles together a couple references so a character can identify himself as a double major in mathematics and theology. Of course, the generic biography for a European mathematician, between about the end of the Western Roman Empire and the Industrial Revolution, is that he (males most often had the chance to do original mathematics) studied mathematics alongside theology and philosophy, and possibly astronomy, although that reflects more how the subjects were seen as rather intertwined, and education wasn’t as specialized and differentiated as it’s now become. (The other generic mathematician would be the shopkeeper or the exchequer, but nobody tells jokes about their mathematics.)

And, finally, Doug Savage’s Savage Chickens (May 28) brings up the famous typing monkeys (here just the one of them), and what really has to be counted as a bit of success for the project.

Some Facts For The Day

I’d just wanted to note the creation of another fact-of-the-day Twitter feed from the indefatigable John D Cook. This one is dubbed Unit Facts, and it’s aiming at providing information about where various units of measure come from. The first few days have begun with, naturally enough, the base units of the Metric System (can you name all seven?), and has stretched out already to things like what a knot is, how picas and inches are related, and what are ems and fortnights besides useful to know for crossword puzzles, or how something might be measured, as in the marshmallow tweet above.

Cook offers a number of interesting fact-of-the-day style feeds, which I believe are all linked to one another through their “Following” pages. These include algebra, topology, probability, and analysis facts of the day, as well as Unix tool tips, RegExp and TeX/LaTeX trivia, symbols (including a lot of Unicode and HTML entities), and the like. If you’re of the sort to get interested in neatly delivered bits of science- and math- and computer-related trivia, well, good luck with your imminent archive-binge.

Feynman Online Physics

Likely everybody in the world has already spotted this before, but what the heck: CalTech and the Feynman Lectures Website have put online an edition of volume one of The Feynman Lectures on Physics. This is an HTML 5 edition, so older web browsers might not be able to read it sensibly.

The Feynman Lectures are generally regarded as one of the best expositions of basic physics; they started as part of an introduction to physics class that spiralled out of control and that got nearly all the freshmen who were trying to take it lost. I know the sense of being lost; when I was taking introductory physics I turned to them on the theory they might help me understand what the instructor was going on about. It didn’t help me.

This isn’t because Feynman wasn’t explaining well what was going on. It’s just that he approached things with a much deeper, much broader perspective than were really needed for me to figure out my problems in — oh, I’m not sure, probably something like how long a block needs to slide down a track or something like that. Here’s a fine example, excerpted from Chapter 5-2, “Time”:

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How Big Was West Jersey?

A map of New Jersey's counties, with municipal boundaries added.

A book I’d read about the history of New Jersey mentioned something usable for a real-world-based problem in fraction manipulation, for a class which was trying to get students back up to speed on arithmetic on their way into algebra. It required some setup to be usable, though. The point is a property sale from the 17th century, from George Hutcheson to Anthony Woodhouse, transferring “1/32 of 3/90 of 90/100 shares” of land in the province of West Jersey. There were a hundred shares in the province, so, the natural question to build is: how much land was transferred?

The obvious question, to people who failed to pay attention to John T Cunningham’s This Is New Jersey in fourth grade, or who spent fourth grade not in New Jersey, or who didn’t encounter that one Isaac Asimov puzzle mystery (I won’t say which lest it spoil you), is: what’s West Jersey? That takes some historical context.

Continue reading “How Big Was West Jersey?”