My July 2013 Statistics

As I’ve started keeping track of my blog statistics here where it’s all public information, let me continue.

WordPress says that in July 2013 I had 341 pages read, which is down rather catastrophically from the June score of 713. The number of distinct visitors also dropped, though less alarmingly, from 246 down to 156; this also implies the number of pages each visitor viewed dropped from 2.90 down to 2.19. That’s still the second-highest number of pages-per-visitor that I’ve had recorded since WordPress started sharing that information with me, so, I’m going to suppose that the combination of school letting out (so fewer people are looking for help about trapezoids) and my relatively fewer posts this month hit me. There are presently 215 people following the blog, if my Twitter followers are counted among them. They hear about new posts, anyway.

My most popular posts over the past 30 days have been:

  1. John Dee, the ‘Mathematicall Praeface’ and the English School of Mathematics, which is primarily a pointer to the excellent mathematics history blog The Renaissance Mathematicus, and about the really quite fascinating Doctor John Dee, advisor to England’s Queen Elizabeth I.
  2. Counting From 52 To 11,108, some further work from Professor Inder J Taneja on a lovely bit of recreational mathematics. (Professor Taneja even pops in for the comments.)
  3. Geometry The Old-Fashioned Way, pointing to a fun little web page in which you can work out geometric constructions using straightedge and compass live and direct on the web.
  4. Reading the Comics, July 5, 2013, and finally; I was wondering if people actually still liked these posts.
  5. On Exact And Inexact Differentials, another “reblog” style pointer, this time to Carnot Cycle, a thermodynamics-oriented blog.
  6. And The $64 Question Was, in which I learned something about a classic game show and started to think about how it might be used educationally.

My all-time most popular post remains How Many Trapezoids I Can Draw, because I think there are people out there who worry about how many different kinds of trapezoids there are. I hope I can bring a little peace to their minds. (I make the answer out at six.)

The countries sending me the most viewers the past month have been the United States (165), then Denmark (32), Australia (24), India (18), and the United Kingdom and Brazil (12 each). Sorry, Canada (11). Sending me a single viewer each were Estonia, Slovenia, South Africa, the Netherlands, Argentina, Pakistan, Angola, France, and Switzerland. Argentina and Slovenia did the same for me last month too.

Where Do Negative Numbers Come From?

Some time ago — and I forget when, I’m embarrassed to say, and can’t seem to find it because the search tool doesn’t work on comments — I was asked about how negative numbers got to be accepted. That’s a great question, particularly since while it seems like the idea of positive numbers is probably lost in prehistory, negative numbers definitely progressed in the past thousand years or so from something people might wildly speculate about to being a reasonably comfortable part of daily mathematics.

While searching for background information I ran across a doctoral thesis, Making Sense Of Negative Numbers, which is uncredited in the PDF I just linked to but appears to be by Dr Cecilia Kilhamn, of the University of Gothenburg, Sweden. Dr Kilhamn’s particular interest (here) is in how people learn to use negative numbers, so most of the thesis is about the conceptual difficulties people have when facing the minus sign (not least because it serves two roles, of marking a number as negative and of marking the subtraction operation), but the first chapters describe the historical process of developing the concept of negative numbers.

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Counting From 52 to 11,108

fluffy once again brings to my attention the work of Inder J Taneja, who got into the Annals of Improbable Research for a fun parlor-game sort of project a couple of months ago. This was for coming up with ways to (most of) the numbers from 44 up to 1,000 using the digits 1 through 9 in order (ascending and descending), in combinations of addition, multiplication, and exponentiation. Taneja got back in Improbable this weekend with a follow-up project, listing the numbers that can be formed all the way out to a pleasant 11,111.

Taneja’s paper, available at, is that rare mathematics paper that you don’t need to be a mathematician to read, although it isn’t going to strike anyone as very enlightening. The ingenuity involved in many of them is impressive, though, and Taneja lists some interesting things such as how many numbers in a given range can’t be made by the digits in ascending or descending order. (Remarkably, to me at least, everything from 1,001 to 2,000 can be done in ascending or descending order.)

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From the Venetian Quarter

In 1204 the Fourth Crusade, reaching the peak of its mission to undermine Christianity in the eastern Mediterranean, sacked Constantinople and established a Latin ruler in the remains of the Roman Empire, which we dub today the Byzantine Empire. This I mention because I’m reading John Julius Norwich’s A History of Venice, and it discusses one of the consequences. Venice had supported the expedition, in no small part to divert the Fourth Crusaders from attacking its trading partners in Egypt, and also to reduce Constantinople as a threat to Venice’s power. Venice got direct material rewards too, and Norwich mentions one of them:

When, on 5 August 1205, Sebastiano Ziani’s son Pietro was unanimously elected Doge of Venice, the first question that confronted him was one of identity. To the long list of sonorous but mostly empty titles which had gradually become attached to the ducal throne, there had now been added a new one which meant exactly what it said: Lord of a Quarter and Half a Quarter of the Roman Empire.

This I mention because the reward of three-eighths of the Byzantine Empire (the Byzantines considered themselves the Roman Empire, quite reasonably, and called themselves that) is phrased here in a way that just wouldn’t be said today. Why go to the circumlocution of “a quarter and half a quarter” instead of “three-eights”?

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Counting To 52

fluffy brought to my attention a cute, amusing little bit from the Annals of Improbable Research, itself passing on some work by one Inder J Taneja. Taneja worked out a paper, available from, which lists results to the sort of mathematical puzzle that’s open to anyone with some paper and a pencil and some desire to do some recreational stuff.

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Going Fishing In Pi

I have to imagine I’ve run across this before, but here’s a nice little page which allows one to search the (base ten) digits of π for any sequence of up to 120 digits that one wants. It searches the first 200 million digits of pi, which is enough digits that you can be reasonably sure that any string of six or seven digits you look for are there, and it’s not ridiculously unlikely that a string of ten digits in a row will turn up. The natural question is, why is this interesting?

People who’ve learned a bit about pi may have heard that it’s probably a “normal number”, that is, a number whose digits contain every possible finite string of digits within it somewhere. That suggests that finding any particular string of digits in pi is no more surprising than finding any particular word in a complete dictionary (if we imagine there’s a dictionary that ever did include all the words of a language). The story’s a little more complicated than that.

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