My 2019 Mathematics A To Z: Abacus


Today’s A To Z term is the Abacus. It was suggested by aajohannas, on Twitter as @aajohannas. Particularly asked for was how to use an abacus. The abacus has been used by a great many cultures over thousands of years. So it’s hard to say that there is any one right way to use it. I’m going to get into a way to use it to compute, any more than there is a right way to use a hammer. There are many hammers, and many things to hammer. But there are similarities between all hammers, and the ways to use them as hammers are similar. So learning one kind, and one way to use that kind, can be a useful start.

Cartoony banner illustration of a coati, a raccoon-like animal, flying a kite in the clear autumn sky. A skywriting plane has written 'MATHEMATIC A TO Z'; the kite, with the letter 'S' on it to make the word 'MATHEMATICS'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Abacus.

I taught at the National University of Singapore in the first half of the 2000s. At the student union was this sheltered overhang formed by a stairwell. Underneath it, partly exposed to the elements (a common building style there) was a convenience store. Up front were the things with high turnover, snacks and pop and daily newspapers, that sort of thing. In the back, beyond the register, in the areas that the rain, the only non-gentle element, couldn’t reach even whipped by wind, were other things. Miscellaneous things. Exam bluebooks faded with age and dust. Good-luck cat statues colonized by spiderwebs. Unlabelled power cables for obsolete electronics. Once when browsing through this I encountered two things that I bought as badges of office.

One was a slide rule, a proper twelve-inch one. I’d had one already, a $2 six-inch-long one I’d gotten as an undergraduate from a convenience store the university had already decided to evict. The NUS one was a slide rule you could do actual work on. Another was a soroban, a compact Japanese abacus, in a patterned cardboard box a half-inch too short to hold it. I got both. For the novelty, yes. Also, I taught Computational Science. I felt it appropriate to have these iconic human computing devices.

But do I use them? Other than for decoration? … No, not really. I have too many calculators to need them. Also I am annoyed that while I can lay my hands on the slide rule I have put the soroban somewhere so logical and safe I can’t find it. A couple photographs would improve this essay. Too bad.

Do I know how to use them? If I find them? The slide rule, sure. If you know that a slide rule works via logarithms, and you play with it a little? You know how to use a slide rule. At least a little, after a bit of experimentation and playing with the three times table.

The abacus, though? How do you use that?

In childhood I heard about abacuses. That there’s a series of parallel rods, each with beads on them. Four placed below a center beam, one placed above. Sometimes two placed above. That the lower beads on a rod represent one each. That the upper bead represents five. That some people can do arithmetic on that faster than others can an electric calculator. And that was about all I got, or at least retained. How to do this arithmetic never penetrated my brain. I imagined, well, addition must be easy. Say you wanted to do three plus six, well, move three lower beads up to the center bar. Then slide one lower and one upper bead, for six, to the center bar, and read that off. Right?

The bizarre thing is my naive childhood idea is right. At least in the big picture. Let each rod represent one of the numbers in base-ten style. It’s anachronistic to the abacus’s origins to speak of a ones rod, a tens rod, a hundreds rod, and so on. So what? We’re using this tool today. We can use the ideas of base ten to make our understanding easier.

Pick a row of beads that you want to represent the ones. The row to the left of that represents tens. To the left of that, hundreds. To the right of the ones is the one-tenths, and the one-hundredths, and so on. This goes on to however far your need and however big your abacus is.

Move beads to the center to represent numbers you want. If you want ’21’, slide two lower beads up in the tens column and one lower bead in the ones column. If you want ’38’, slide three lower beads up in the tends column and one upper and three lower beads in the ones column.

To add two numbers, slide more beads representing those numbers toward the center bar. To subtract, slide beads away. Multiplication and division were beyond my young imagination. I’ll let them wait a bit.

There are complications. The complications are for good reason. When you master them, they make computation swifter. But you pay for that later speed with more time spent learning. This is a trade we make, and keep making, in computational mathematics. We make a process more reliable, more speedy, by making it less obvious.

Some of this isn’t too difficult. Like, work in one direction so far as possible. It’s easy to suppose this is better than jumping around from, say, the thousands digit to the tens to the hundreds to the ones. The advice I’ve read says work from the left to the right, that is, from the highest place to the lowest. Arithmetic as I learned it works from the ones to the tens to the hundreds, but this seems wiser. The most significant digits get calculated first this way. It’s usually more important to know the answer is closer to 2,000 than to 3,000 than to know that the answer ends in an 8 rather than a 6.

Some of this is subtle. This is to cope with practical problems. Suppose you want to add 5 to 6? There aren’t that many beads on any row. A Chinese abacus, which has two beads on the upper part, could cope with this particular problem. It’s going to be in trouble when you want to add 8 to 9, though. That’s not unique to an abacus. Any numerical computing technique can be broken by some problem. This is why it’s never enough to calculate; we still have to think. For example, thinking will let us handle this five plus six difficulty.

Consider this: four plus one is five. So four and one are “complementary numbers”, with respect to five. Similarly, three and two are five’s complementary numbers. So if we need to add four to a number, that’s equivalent to adding five and subtracting one. If we need to add two, that’s equivalent to adding five and subtracting three. This will get us through some shortages in bead count.

And consider this: four plus six is ten. So four and six are ten-complementary numbers. Similarly, three and seven are ten’s complementary numbers. Two and eight. One and nine. This gets us through much of the rest of the shortage.

So here’s how this works. Suppose we have 35, and wish to add 6 to it. There aren’t the beads to add six to the ones column. So? Instead subtract the complement of six. That is, add ten and subtract four. In moving across the rows, when you reach the tens, slide one lower bead up, making the abacus represent 45. Then from the ones column subtract four. that is, slide the upper bead away from the center bar, and slide the complement to four, one bead, up to the center. And now the abacus represents 41, just like it ought.

If you’re experienced enough you can reduce some of these operations, sliding the beads above and below the center bar at once. Or sliding a bead in the tens and another in the ones column at once. Don’t fret doing this. Worry about making correct steps. You’ll speed up with practice. I remember advice from a typesetting manual I collected once: “strive for consistent, regular keystrokes. Speed comes with practice. Errors are time-consuming to correct”. This is, mutatis mutandis, always good advice.

Subtraction works like addition. This should surprise few. If you have the beads in place, just remove them: four minus two takes no particular insight. If there aren’t enough beads? Fall back on complements. If you wish to do 35 minus 6? Set up 35, and calculate 35 minus 10 plus 4. When you get to the tens rod, slide one bead down; this leaves you with 25. Then in the ones column, slide four beads up. This leaves you with 29. I’m so glad these seem to be working out.

Doing longer additions and subtractions takes more rows, but not actually more work. 35.2 plus 6.4 is the same work as 35 plus 6 and 2 plus 4, each of which you, in principle, know how to do. 35.2 minus 6.4 is a bit more fuss. When you get to the 2 minus 4 bit you have to do that addition-of-complements stuff. But that’s not any new work.

Where the decimal point goes is something you have to keep track of. As with the slide rule, the magnitude of these numbers is notional. Your fingers move the same way to add 352 and 64 as they will 0.352 and 0.064. That’s convenient.

Multiplication gets more tedious. It demands paying attention to where the decimal point is. Just like the slide rule demands, come to think of it. You’ll need columns on the abacus for both the multiplicands and the product. And you’ll do a lot of adding up. But at heart? 2038 times 3.7 amounts to doing eight multiplications. 8 times 7, 3 times 7, 0 times 7 (OK, that one’s easy), 2 times 7, 3 times 7, 3 times 3, 0 times 3 (again, easy), and 2 times 3. And then add up these results in the correct columns. This may be tedious, but it’s not hard. It does mean the abacus doesn’t spare you having to know some times tables. I mean, you could use the abacus to work out 8 times 7 by adding seven to itself over and over, but that’s time-consuming. You can save time, and calculation steps, by memorization. By knowing some answers ahead of time.

Totton Heffelfinger and Gary Flom’s page, from which I’m drawing almost all my practical advice, offers a good notation of lettering the rods of the abacus, A, B, C, D, and so on. To multiply, say, 352 by 64 start by putting the 64 on rods BC. Set the 352 on rods EFG. We’ll get the answer with rod K as the ones column. 2 times 4 is 8; put that on rod K. 5 times 4 is 20; add that to rods IJ. 3 times 4 is 12; add that to rods HI. 2 times 6 is 12; add that to rods IJ. 5 times 6 is 30; add that to rods HI. 3 times 6 is 18; add that to rods GH. All going well this should add up to 22,528, spread out along rods GHIJK. I can see right away at least the 8 is correct.

You would do the same physical steps to multiply, oh, 3.52 by 0.0064. You have to take care of the decimal place yourself, though.

I see you, in the back there, growing suspicious. I’ll come around to this. Don’t worry.

Division is … oh, I have to fess up. Division is not something I feel comfortable with. I can read the instructions and repeat the examples given. I haven’t done it enough to have that flash where I understand the point of things. I recognize what’s happening. It’s the work of division as done by hand. You know, 821 divided by 56 worked out by, well, 56 goes into 82 once with a remainder of 26. Then drop down the 1 to make this 261. 56 goes into 261 … oh, it would be so nice if it went five times, but it doesn’t. It goes in four times, with a remainder of 37. I can walk you through the steps but all I am truly doing is trying to keep up with Totton Heffelfinger and Gary Flom’s instructions here.

There are, I read, also schemes to calculate square roots on the abacus. I don’t know that there are cube-root schemes also. I would bet on there being such, though.

Never mind, though. The suspicious thing I expect you’ve noticed is the steps being done. They’re represented as sliding beads along rods, yes. But the meaning of these steps? They’re the same steps you would do by doing arithmetic on paper. Sliding two beads and then two more beads up to the center bar isn’t any different from looking at 2 + 2 and representing that as 4. All this ten’s-complement stuff to subtract one number from another is just … well, I learned it as subtraction by “borrowing”. I don’t know the present techniques but I’m sure they’re at heart the same. But the work is eerily like what you would do on paper, using Arabic numerals.

The slide rule uses a logarithm-based ruler. This makes the addition of distances along the slides match the multiplication of the values of the rulers. What does the abacus do to help us compute?

Why use an abacus?

What an abacus gives us is memory. It stores numbers. It lets us break a big problem into a series of small problems. It lets us keep a partial computation while we work through those steps. We don’t add 35.2 to 6.4. We add 3 to 0 and 5 to 6 and 2 to 4. We don’t multiply 2038 by 3.7. We multiply 8 by 7, and 8 by 3, and 3 by 7, and 3 by 3, and so on.

And this is most of numerical computing, even today. We describe what we want to do as these high-level operations. But the computation is a lot of calculations, each one of them simple. We use some memory to hold partially completed results. Memory, the ability to store results, lets us change hard problems into long strings of simple ones.

We do more things the way the abacus encourages. We even use those complementary numbers. Not five’s or ten’s complements, not with binary arithmetic computers. Two’s complement arithmetic makes it possible to subtract, or write negative numbers, in ways that are easy to calculate. That there are a set number of rods even has its parallel in modern computing. When representing a real number on the computer we have only so many decimal places. (Yes, yes, binary digit places.) At least unless we use a weird data structure. This affects our calculations. There are numbers we can’t represent perfectly, such as one-third. We need to think about whether this affects what we are using our calculation for.

There are major differences between a digital computer and a person using the abacus. But the processes are similar. This may help us to understand why computational science works the way it does. It may at least help us understand those contests in the 1950s where the abacus user was faster than the calculator user.

But no, I confess, I only use mine for decoration, or will when I find it again.


Thank you for reading. All the Fall 2019 A To Z posts should be at this link. Furthermore, both this year’s and all past A To Z sequences should be at this link. And I am still soliciting subjects for the first third of the alphabet.

How November 2016 Treated My Mathematics Blog


I didn’t forget about reviewing my last month’s readership statistics. I just ran short on time to gather and publish results is all. But now there’s an hour or so free to review that WordPress says my readership was like in November and I can see what was going on.

Well.

So, that was a bit disappointing. The start of an A To Z Glossary usually sees a pretty good bump in my readership. The steady publishing of a diverse set of articles usually helps. My busiest months have always been ones with an A To Z series going on. This November, though, there were 923 page views around here, from 575 distinct visitors. That’s up from October, with 907 page views and 536 distinct visitors. But it’s the same as September’s 922 page views from 575 distinct visitors. I blame the US presidential election. I don’t think it’s just that everyone I can still speak to was depressed by it. My weekly readership the two weeks after the election were about three-quarters that of the week before or the last two weeks of November. I’d be curious what other people saw. My humor blog didn’t see as severe a crash the week of the 14th, though.

Well, the people who were around liked what they saw. There were 157 pages liked in November, up from 115 in September and October. That’s lower than what June and July, with Theorem Thursdays posts, had, and below what the A To Z in March and April drew. But it’s up still. Comments were similarly up, to 35 in November from October’s 24 and September’s 20. That’s up to around what Theorem Thursdays attracted.

December starts with my mathematics blog having had 43,145 page views from a reported 18,022 distinct viewers. And it had 636 WordPress.com followers. You can be among them by clicking the “Follow” button on the upper right corner. It’s up from the 626 WordPress.com followers I had at the start of November. That’s not too bad, considering.

I had a couple of perennial favorites among the most popular articles in November:

This is the first time I can remember that a Reading The Comics post didn’t make the top five.

Sundays are the most popular days for reading posts here. 18 percent of page views come that day. I suppose that’s because I have settled on Sunday as a day to reliably post Reading the Comics essays. The most popular hour is 6 pm, which drew 11 percent of page views. In October Sundays were the most popular day, with 18 percent of page views. 6 pm as the most popular hour, but then it drew 14 percent of page views. Same as September. I don’t know why 6 pm is so special.

As ever there wasn’t any search term poetry. But there were some good searches, including:

  • how many different ways can you draw a trapizium
  • comics back ground of the big bang nucleosynthesis
  • why cramer’s rule sucks (well, it kinda does)
  • oliver twist comic strip digarm
  • work standard approach sample comics
  • what is big bang nucleusynthesis comics strip

I don’t understand the Oliver Twist or the nucleosynthesis stuff.

And now the roster of countries and their readership, which for some reason is always popular:

Country Page Views
United States 534
United Kingdom 78
India 36
Canada 33
Philippines 22
Germany 21
Austria 18
Puerto Rico 17
Slovenia 14
Singapore 13
France 12
Sweden 8
Spain 8
New Zealand 7
Australia 6
Israel 6
Pakistan 5
Hong Kong SAR China 4
Portugal 4
Belgium 3
Colombia 3
Netherlands 3
Norway 3
Serbia 3
Thailand 3
Brazil 2
Croatia 2
Finland 2
Malaysia 2
Poland 2
Switzerland 2
Argentina 1
Bulgaria 1
Cameroon 1
Cyprus 1
Czech Republic 1 (***)
Denmark 1
Japan 1 (*)
Lithuania 1
Macedonia 1
Mexico 1 (*)
Russia 1
Saudi Arabia 1 (*)
South Africa 1 (*)
United Arab Emirates 1 (*)
Vietnam 1

That’s 46 countries, the same as last month. 15 of them were single-reader countries; there were 20 single-reader countries in October. Japan, Mexico, Saudi Arabia, South Africa, and the United Arab Emirates have been single-reader countries for two months running. Czech has been one for four months.

Always happy to see Singapore reading me (I taught there for several years). The “European Union” listing seems to have vanished, here and on my humor blog. I’m sure that doesn’t signal anything ominous at all.

June 2014 In Mathematics Blogging


And with the start of July I look over how well the mathematics blog did in June and see what I can learn from that. It seems more people are willing to read when I post stuff, which is worth knowing, I guess. After May’s near-record of 751 views and 315 visitors I expected a fall, and, yeah, it came. The number of pages viewed dropped to 492, which is … well, the fourth-highest this year at least? And the number of unique visitors fell to 194, which is actually the lowest of this year. The silver lining is this means the views per visitor, 2.54, was the second-highest since WordPress started sharing those statistics with me, so, people who come around find themselves interested. I start the month at 16,174 views total and won’t cross 17,000 at that rate come July, but we’ll see what I can do. And between WordPress and Twitter I’m (as of this writing) at exactly 400 followers, which isn’t worldshaking but is a nice big round number. I admit thinking how cool it would be if that were 400 million but I’d probably get stage fright if it were.

If one thing defined June it was “good grief but there’s a lot of mathematics comics”, which I attributed to Comic Strip Master Command ordering cartoonists to clear the subject out before summer vacation. It does mean the top five posts for June are almost comically lopsided, though:

Now, that really is something neat about triangles in the post linked above so please do read it. What I’m not clear about is why the June 16th comics post was so extremely popular; it’s nearly twice as viewed as the runner-up. If I were sure what keyword is making it so popular I’d do more with that.

Now on to the international portion of this contest: what countries are sending me the most visitors? Of course the United States comes in first, at 336 views. Denmark finished second with 17, and there was a three-way tie for third as Australia, Austria, and the United Kingdom sent sixteen each. (Singapore and Canada came in next with nine each.) I had a pretty nice crop of single-reader countries this month: Argentina, Bosnia and Herzegovina, Cambodia, Egypt, Ghana, Hong Kong, Indonesia, Japan, Paraguay, Saudi Arabia, Switzerland, and Thailand. Hong Kong, Japan, and Switzerland are repeats from last month and nobody’s got a three-month streak going.

Among the interesting search terms to bring people to me:

  • names for big numbers octillion [ happy to help? ]
  • everything to need to know about trapezoids [ I’m going to be the world’s authority on trapezoids! ]
  • what does the fact that two trapezoids make a parallelogram say about tth midline [ I have some ideas but don’t want to commit to anything particular ]
  • latching onto you 80 version [ I … think I’m being asked for lyrics? ]
  • planet nebus [ I feel vaguely snarked upon, somehow ]
  • origin is the gateway to your entire gaming universe [ … thank you? ]
  • nebus student job for uae [ Um … I guess I can figure out a consulting fee or something if you ask? ]

Autocorrected Monkeys and Pulled Tea


The Twop Twips account on Twitter — I’m not sure how to characterize what it is exactly, but friends retweet it often enough — had the above advice about the infinite monkeys problem, and what seems to me correct advice that turning on autocorrect will get them to write the works of Shakespeare more quickly. And then John Kovaleski’s monkey-featuring comic strip Bo Nanas featured the infinite monkey problem today, so obviously I have to spend more time thinking of it.

It seems fair that monkeys with autocorrect will be more likely to hit a word than a monkey without will be. Let’s try something simpler than Shakespeare and just consider the chance of typing the word “the”, and to keep the numbers friendly let’s imagine that the keyboard has just the letters and a space bar. We’ll not care about punctuation or numbers; that’s what copy editors would be for, if anyone had been employed as a copy editor since 1996, when someone in the budgeting office discovered there was autocorrect.

Anyway, there’s 27 characters on this truncated keyboard, and if the monkeys were equally likely to hit any one of them, then, there’d be 27 times 27 times 27 — that is, 19,683 — different three-character strings they might hit. Exactly one of them is the desired word “the”. So, roughly, we would expect the monkey to get the word right one time in each 19,683 attempts at a three-character string. (We wouldn’t have to wait quite so long if we’ll accept the monkey as writing continuously and pluck out three characters in a row wherever they appear, but that’s more work than I feel like doing, and I doubt it would significantly change the qualitative results, of how much faster it’d be if autocorrect were on.)

But how many tries would be needed to hit a word that gets autocorrected to “the”? And here we get into the mysteries of the English language. I’d be surprised by a spell checker that couldn’t figure out “teh” probably means “the”. Similarly “hte” should get back to “the”. So we can suppose the five other permutations of the letters in “the” will be autocorrected. So there’s six different strings of the 19,683 possibilities that will get fixed to “the”. The monkey has one chance in 3280.5 of getting one of them and so, on average, the monkey can be expected to be right once in every 3281 attempts.

But there’s other typos possible: “thw” is probably just my finger slipping, and “ghe” isn’t too implausible either. At least my spell checker recognizes both as most likely meant to be “the”. Let’s suppose that a spell checker can get to the right word if any one letter is mistaken. This means that there are some 78 other three-character strings that would get fixed to “the”, for a total of 84 possible three-character strings which are either “the” or would get autocorrected to “the”. With that many, there’s one chance in a touch more than 234 that a three-character string will get corrected to “the”, and we have to wait, considering, not very long at all.

It gets better if two-character errors are allowed, but I can’t make myself believe that the spell check will turn “yje” into “the”, and that’s something which might be typed if you just had the right hand on the wrong keys. My checker hasn’t got any idea what “yje” is supposed to be anyway, so, one wrong letter is probably the limit.

Except. “tie” is one character wrong for “the” and no spell checker will protest “tie”. Similarly “she” and “thy” and a couple of other words. And it’d be a bit much to expect “t e” or “ he” to be turned back into “the” even though both are just the one keystroke off. And a spell checker would probably suppose that “tht” is a typo for “that”. It’s hard to guess how many of the one-character-off words will not actually be caught. Let’s say that maybe half the one-character-off words will be corrected to “the”; that’s still a pretty good 39 one-character misspellings, plus five permutations, plus the correct spelling or 45 candidate three-character strings for autocorrect to get. So our monkey has something like one chance in 450 of getting “the” in banging on the keyboard three times.

For four-letter words there are many more combinations — 531,441, if we just list the strings of our 27 allowed characters — but then there are more strings which would get autocorrected. Let’s say we want the string “thus”; there are 23 ways to arrange those letters in addition to the correct one. And there are 104 one-character-off strings; supposing that half of them will get us to “thus”, then, there’s 76 strings that get one to the desired “thus”. That’s a pretty dismal one chance in about 7,000 of typing one of them, unfortunately. Things get a little better if we suppose that some two-character errors are going to be corrected, although I can’t find one which my spell checker will accept right now, and if a single error and a transposition are viable.

With longer words yet there’s more chances for spell checker forgiveness: you can get pretty far off “accommodate” or “aneurysm” and still be saved by the spell checker, which is good for me as I last spelled “accommodate” correctly sometime in 1992, and I thought it looked wrong then.

So the conclusion has to be: you’ll get a bit of an improvement in speed by turning on autocorrect, for the obvious reason that you’re more likely to get one right out of 450 than you are to get one right out of 19,000. But it’s not going to help you very much; the number of ways to spell things so completely wrong that not even spell check can find you just grows far too rapidly to be helped. If I get a little bored I might work out the chance of getting a permutation-or-one-off for strings of different lengths.

And your monkey might be ill-served by autocorrect anyway. When I lived in Singapore I’d occasionally have teh tarik (“pulled tea”), black tea with sugar and milk tossed back and forth until it’s nice and frothy. It’s a fine drink but hard to write back home about because even if you get past the spell checker, the reader assumes the “teh” is a typo and mentally corrects for it. When this came up I’d include a ritual emphasis that I actually meant what I wrote, but you see the problem. Fortunately Shakespeare wrote relatively little about southeast Asian teas, but if you wanted to expand the infinite monkey problem to the problem of guiding tourists through Singapore, you’d have to turn the autocorrect off to have any hope of success.

February 2014’s Mathematics Blog Statistics


And so to the monthly data-tracking report. I’m sad to say that the total number of viewers dropped compared to January, although I have to admit given the way the month went — with only eight posts, one of them a statistics one — I can’t blame folks for not coming around. The number of individual viewers dropped from 498 to 423, and the number of unique visitors collapsed from 283 to 209. But as ever there’s a silver lining: the pages per viewer rose from 1.76 to 2.02, so, I like to think people are finding this more choice.

As usual the country sending me the most readers was the United States (235), with Canada in second (31) and Denmark, surprising to me, in third place (30). I suppose that’s a bit unreasonable on my part, since why shouldn’t Danes be interested in mathematics-themed comic strips, but, I’m used to the United Kingdom being there. Fourth place went to Austria (17) and I was again surprised by fifth place, Singapore (14), but happy to see someone from there reading, as I used to work there and miss the place, especially in the pits of winter. Sending me just a single reader each were: Albania, Argentina, Ecuador, Estonia, Ethiopia, Greece, Hungary, New Zealand, Peru, Saudia Arabia, South Korea, Thailand, United Arab Emirates, Uruguay, and Venezuela. Greece and South Korea are the only repeats from January 2013.

The most popular articles the past thirty days were:

  1. Reading The Comics, February 1, 2014, my bread-and-butter subject for the blog.
  2. How Many Trapezoids I Can Draw, which will be my immortal legacy.
  3. Reading The Comics, February 11, 2014: Running Out Pi Edition, see above, although now I’m trying out something in putting particular titles on things.
  4. The Liquefaction of Gases, Part I, referring to a real statistical mechanics post by CarnotCycle.
  5. I Know Nothing Of John Venn’s Diagram Work, my confession of ignorance, or at least of casualness in thought, in the use of a valuable tool.

The most interesting search terms bringing people to me the past month were “comics strip about classical and modern physics”, “1,898,600,000,000,000,000,000,000,000 in words”, and “how much could a contestant win on the $64.00 question”, which you’d superficially think would be a question you didn’t have to look up. (Of course, in the movie Take It Or Leave It, based on the radio quiz program, the amount of the gran jackpot is raised to a thousand dollars, for dramatic value. This is presumably not what the questioner was looking for.)

Something I Didn’t Know About Trapezoids


I have a little iPad app for keeping track of how this blog is doing, and I’m even able to use it to compose new entries and make comments. (The entry about the lottery was one of them.) Mostly it provides a way for me to watch the count of unique visits per day, so I can grow neurotic wondering why it’s not higher. But it also provides supplementary data, such as, what search queries have brought people to the site. The “Trapezoid Week” flurry of posts has proved to be very good at bringing in search referrals, with topics like “picture of a trapezoid” or “how do I draw a trapezoid” or “similar triangles trapezoid” bringing literally several people right to me.

Continue reading “Something I Didn’t Know About Trapezoids”