When I wrote last weekend’s piece about how interesting a basketball tournament was, I let some terms slide without definition, mostly so I could explain what ideas I wanted to use and how they should relate. My love, for example, read the article and looked up and asked what exactly I meant by “interesting”, in the attempt to measure how interesting a set of games might be, even if the reasoning that brought me to a 63-game tournament having an interest level of 63 seemed to satisfy.

When I spoke about something being interesting, what I had meant was that it’s something whose outcome I would like to know. In mathematical terms this “something whose outcome I would like to know” is often termed an experiment’ to be performed or, even better, a message’ that presumably I wil receive; and the outcome is the “information” of that experiment or message. And information is, in this context, something you do not know but would like to.

So the information content of a foregone conclusion is low, or at least very low, because you already know what the result is going to be, or are pretty close to knowing. The information content of something you can’t predict is high, because you would like to know it but there’s no (accurately) guessing what it might be.

This seems like a straightforward idea of what information should mean, and it’s a very fruitful one; the field of “information theory” and a great deal of modern communication theory is based on them. This doesn’t mean there aren’t some curious philosophical implications, though; for example, technically speaking, this seems to imply that anything you already know is by definition not information, and therefore learning something destroys the information it had. This seems impish, at least. Claude Shannon, who’s largely responsible for information theory as we now know it, was renowned for jokes; I recall a Time Life science-series book mentioning how he had built a complex-looking contraption which, turned on, would churn to life, make a hand poke out of its innards, and turn itself off, which makes me smile to imagine. Still, this definition of information is a useful one, so maybe I’m imagining a prank where there’s not one intended.

And something I hadn’t brought up, but which was hanging awkwardly loose, last time was: granted that the outcome of a single game might have an interest level, or an information content, of 1 unit, what’s the unit? If we have units of mass and length and temperature and spiciness of chili sauce, don’t we have a unit of how informative something is?

We have. If we measure how interesting something is — how much information there is in its result — using base-two logarithms the way we did last time, then the unit of information is a bit. That is the same bit that somehow goes into bytes, which go on your computer into kilobytes and megabytes and gigabytes, and onto your hard drive or USB stick as somehow slightly fewer gigabytes than the label on the box says. A bit is, in this sense, the amount of information it takes to distinguish between two equally likely outcomes. Whether that’s a piece of information in a computer’s memory, where a 0 or a 1 is a priori equally likely, or whether it’s the outcome of a basketball game between two evenly matched teams, it’s the same quantity of information to have.

So a March Madness-style tournament has an information content of 63 bits, if all you’re interested in is which teams win. You could communicate the outcome of the whole string of matches by indicating whether the “home” team wins or loses for each of the 63 distinct games. You could do it with 63 flashes of light, or a string of dots and dashes on a telegraph, or checked boxes on a largely empty piece of graphing paper, coins arranged tails-up or heads-up, or chunks of memory on a USB stick. We’re quantifying how much of the message is independent of the medium.

## Reading the Comics, December 14, 2014: Pictures Gone Again? Edition

I’ve got enough comics to do a mathematics-comics roundup post again, but none of them are the King Features or Creators or other miscellaneous sources that demand they be included here in pictures. I could wait a little over three hours and give the King Features Syndicate comics another chance to say anything on point, or I could shrug and go with what I’ve got. It’s a tough call. Ah, what the heck; besides, it’s been over a week since I did the last one of these.

Bill Amend’s FoxTrot (December 7) bids to get posted on mathematics teachers’ walls with a bit of play on two common uses of the term “degree”. It’s also natural to wonder why the same word “degree” should be used to represent the units of temperature and the size of an angle, to the point that they even use the same symbol of a tiny circle elevated from the baseline as a shorthand representation. As best I can make out, the use of the word degree traces back to Old French, and “degré”, meaning a step, as in a stair. In Middle English this got expanded to the notion of one of a hierarchy of steps, and if you consider the temperature of a thing, or the width of an angle, as something that can be grown or shrunk then … I’m left wondering if the Middle English folks who extended “degree” to temperatures and angles thought there were discrete steps by which either quantity could change.

As for the little degree symbol, Florian Cajori notes in A History Of Mathematical Notations that while the symbol (and the ‘ and ” for minutes and seconds) can be found in Ptolemy (!), in describing Babylonian sexagesimal fractions, this doesn’t directly lead to the modern symbols. Medieval manuscripts and early printed books would use abbreviations of Latin words describing what the numbers represented. Cajori rates as the first modern appearance of the degree symbol an appendix, composed by one Jacques Peletier, to the 1569 edition of the text Arithmeticae practicae methods facilis by Gemma Frisius (you remember him; the guy who made triangulation into something that could be used for surveying territories). Peletier was describing astronomical fractions, and used the symbol to denote that the thing before it was a whole number. By 1571 Erasmus Reinhold (whom you remember from working out the “Prutenic Tables”, updated astronomical charts that helped convince people of the use of the Copernican model of the solar system and advance the cause of calendar reform) was using the little circle to represent degrees, and Tycho Brahe followed his example, and soon … well, it took a century or so of competing symbols, including “Grad” or “Gr” or “G” to represent degree, but the little circle eventually won out. (Assume the story is more complicated than this. It always is.)

Mark Litzer’s Joe Vanilla (December 7) uses a panel of calculus to suggest something particularly deep or intellectually challenging. As it happens, the problem isn’t quite defined well enough to solve, but if you make a reasonable assumption about what’s meant, then it becomes easy to say: this expression is “some infinitely large number”. Here’s why.

The numerator is the integral $\int_{0}^{\infty} e^{\pi} + \sin^2\left(x\right) dx$. You can think of the integral of a positive-valued expression as the area underneath that expression and between the lines marked by, on the left, $x = 0$ (the number on the bottom of the integral sign), and on the right, $x = \infty$ (the number on the top of the integral sign). (You know that it’s x because the integral symbol ends with “dx”; if it ended “dy” then the integral would tell you the left and the right bounds for the variable y instead.) Now, $e^{\pi} + \sin^2\left(x\right)$ is a number that depends on x, yes, but which is never smaller than $e^{\pi}$ (about 23.14) nor bigger than $e^{\pi} + 1$ (about 24.14). So the area underneath this expression has to be at least as big as the area within a rectangle that’s got a bottom edge at y = 0, a top edge at y = 23, a left edge at x = 0, and a right edge at x infinitely far off to the right. That rectangle’s got an infinitely large area. The area underneath this expression has to be no smaller than that.

Just because the numerator’s infinitely large doesn’t mean that the fraction is, though. It’s imaginable that the denominator is also infinitely large, and more wondrously, is large in a way that makes the ratio some more familiar number like “3”. Spoiler: it isn’t.

Actually, as it is, the denominator isn’t quite much of anything. It’s a summation; that’s what the capital sigma designates there. By convention, the summation symbol means to evaluate whatever expression there is to the right of it — in this case, it’s $x^{\frac{1}{e}} + \cos\left(x\right)$ — for each of a series of values of some index variable. That variable is normally identified underneath the sigma, with a line such as x = 1, and (again by convention) for x = 2, x = 3, x = 4, and so on, until x equals whatever the number on top of the sigma is. In this case, the bottom doesn’t actually say what the index should be, although since “x” is the only thing that makes sense as a variable within the expression — “cos” means the cosine function, and “e” means the number that’s about 2.71828 unless it’s otherwise made explicit — we can suppose that this is a normal bit of shorthand like you use when context is clear.

With that assumption about what’s meant, then, we know the denominator is whatever number is represented by $\left(1^{\frac{1}{e}} + \cos\left(1\right)\right) + \left(2^{\frac{1}{e}} + \cos\left(2\right)\right) + \left(3^{\frac{1}{e}} + \cos\left(3\right)\right) + \left(4^{\frac{1}{e}} + \cos\left(4\right)\right) + \cdots + \left(10^{\frac{1}{e}} + \cos\left(10\right)\right)$ (and 1/e is about 0.368). That’s a number about 16.549, which falls short of being infinitely large by an infinitely large amount.

So, the original fraction shown represents an infinitely large number.

Mark Tatulli’s Lio (December 7) is another “anthropomorphic numbers” genre comic, and since it’s Lio the numbers naturally act a bit mischievously.

Greg Evans’s Luann Againn (December 7, I suppose technically a rerun) only has a bit of mathematical content, as it’s really playing more on short- and long-term memories. Normal people, it seems, have a buffer of something around eight numbers that they can remember without losing track of them, and it’s surprisingly easy to overload that. I recall reading, I think in Joseph T Hallinan’s Why We Make Mistakes: How We Look Without Seeing, Forget Things In Seconds, And Are All Pretty Sure We are Way Above Average, and don’t think I’m not aware of how funny it would be if I were getting this source wrong, that it’s possible to cheat a little bit on the size of one’s number-buffer.

Hallinan (?) gave the example of a runner who was able to remember strings of dozens of numbers, well past the norm, but apparently by the trick of parsing numbers into plausible running times. That is, the person would remember “834126120820” perfectly because it could be expressed as four numbers, “8:34, 1:26, 1:20, 8:20”, that might be credible running times for something or other and the runner was used to remembering such times. Supporting the idea that this trick was based on turning a lot of digits into a few small numbers was that the runner would be lost if the digits could not be parsed into a meaningful time, like, “489162693077”. So, in short, people are really weird in how they remember and don’t remember things.