Reading the Comics, December 12, 2013

It’s a bit of a shame there weren’t quite enough comics to run my little roundup on the 11th of December, for that nice 11/12/13 sequence, but I’m not in charge of arranging these things. For this week’s gathering of mathematically themed comic strips there’s not any deeper theme than they mention mathematic points, but at least the first couple of them have some real meat to the subject matter. (It feels to me like if one of the gathered comics inspires an essay, it’s usually one of the first couple in a collection. That might indicate that I get tired while writing these out, or it might reflect a biased recollection of when I do break out an essay.)

John Allen’s Nest Heads (December 5) is built around a kid not understanding a probability distribution: how many days in a row does it take to get the chance of snow to be 100 percent? The big flaw here is the supposition that the chance of snow is a (uhm) cumulative thing, so that if the snow didn’t happen yesterday or the day before it’s the more likely to happen today or tomorrow. As we actually use weather forecasts, though, they’re … well, I’m not sure I’d say they’re independent, that yesterday’s 30 percent chance of snow has nothing to do with today’s 25 percent chance, since it seems to me plausible that whether it snowed yesterday affects whether it snows today. But they don’t just add up until we get a 100 percent chance of snow when things start to drop.

It is possible and sometimes sensible to talk about the cumulative probability of something happening, for example, the chance that there’ll be a snowstorm by no later than this date. That’s a probability that starts at zero, whenever you start the forecast from, and that cumulative probability either stays constant or rises until it reaches the point by which a storm is certain to happen. For example, if it’s certain that a storm will happen next week, but it might be any day of the week, this cumulative probability would start at 0 percent at the start of Sunday, rise to 1/7 (about 14 percent) at the start of Monday, to 2/7 (about 28 percent) at the start of Tuesday, and eventually to 100 percent by the end of the week. That’s not what’s being talked about here, though.

The numbers given in this Nest Heads don’t quite add up to 100 percent, by the way: tomorrow’s 20 percent chance, yesterday’s 30 percent, and the day before’s 25 percent adds up only to 75 percent. But today’s chance of snow wasn’t given and as long as it was more than 25 percent we get the needed total.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (December 6) build a strip around the fact, surprisingly controversial among non-mathematicians, that 0.999… is equal to 1. It’s a neat result, since you can go from it to show that any rational number that otherwise terminates can be written as an equal number that ends with an infinitely long sequence of 9’s past the decimal point (for example, 2 is also equal to 1.999…; 0.123 is equal to 0.122999…; -6.5 is equal to -6.4999…), although once you get past that it’s mostly a curiosity. It adds a little complication in some number theory proofs that depend on making lists of all the unique numbers that have some property, since these are two ways of writing the same non-repeating rational number, but that mostly means some extra words are added to the proofs to avoid that ambiguity.

The proof the Weingartens and Clark give, by the way, is one that I don’t like, because it feels to me that it depends on assuming two things that aren’t proved. First, it assumes that 1/9 actually is equal to the number 0.111…; and second, it assumes that you can just multiply 9 by 0.111… and know that it’s 0.999… at the end. Both of those can be defended, and I’m not sure which is actually the easier to argue (I suspect the second is), but see the commenter georgelcsmith on the page there, who insists that 0.111… is a little bit less than 1/9. It’s not. georgelcsmith appears to be imagining 0.111… to be a number that has some large but finite number of 1’s past the decimal point and then just stops, which would be a little less than 1/9, but that’s not what the number means. By convention, 0.111… means to take the sum of 1/10 and 1/100 and 1/1000 and 1/10,000 and 1/100,000 and on and on, without ceasing.

A better argument, I think, is to skip the 1/9 altogether: instead, look at how big the difference is between 0.999… and 1. That difference is fairly easy to show is smaller than any positive number; and since the difference between two numbers has to be greater than or equal to zero, that means the difference between 0.999… and 1 must be zero. And if two numbers aren’t different, then they have to be the same.

Barney and Clyde pop back in here December 8th with the structure of a classic calculus word problem — about a predator adjusting its path to chase a prey’s trajectory — as part of … it’s not quite a shaggy dog story, but I suppose that’s the best fit for what kind of joke this is.

Jenny Campbell’s Flo and Friends (December 7) does the 87th consecutive comic strip roundup in these parts where the joke is “I never need to use algebra”.

Steve Melcher’s That Is Priceless (December 7) — which takes classical paintings and gives them funny captions — shows a portrait from Nicholas Neufchatel, here of Nuremberg writing master Johann Neudörfer (1497 – 1563), the founder of German calligraphy if the web pages I’ve looked this up on are correct, and a student. They’re doing, obviously, some sort of mathematics given the work with wireframe Platonic solids. It seems probable to me the work is in some way tied to astronomy, since much of mathematics of the day was tied to the problems of understanding the motions of the planets, when it wasn’t tied to problems of finance.

John Zakour and Scott Roberts’s Working Daze (December 11) makes mention of the lineup of sequential dates that we got on the 11th, if you write the digits in one of the correct orders. It does sadden me a bit when I realize I missed an amusing lineup of dates, but I do tend to get over it quickly.

Scott Adams’s Dilbert (December 12) alludes to the famous infinite-monkeys theorem, which has come up here before. Of course, the trouble with an infinite number of monkeys typing away has to be finding the good results out of all that random hash. Offhand I’m aware of only two attempts to address that problem, from Bob Newhart (“To be or not to be, that is the gazellnikof”) and The Simpsons; someone wanting to fit Borel into their sketch comedy troupe might want to consider this.

And finally, dro-mo (December 12) runs the π pun.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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