Greg Evans’s Luann Againn for the 12th features some poor tutoring on Gunther’s part. Usually a person isn’t stuck for what the answer to a problem is; they’re stuck on how to do it correctly. Maybe on how to do it efficiently. But tutoring is itself a skill, and it’s a hard one to learn. We don’t get enough instruction in how to do it.
The problem Luann’s doing is one of simplifying an expression. I remember doing a lot of this, in middle school algebra like that. Simplifying expressions does not change their value; we don’t create new ideas by writing them. So why simplify?
Any grammatically correct expression for a concept may be as good as any other grammatically correct expression. This is as true in writing as it is in mathematics. So what is good writing? There are a thousand right answers. One trait that I think most good writing has is that it makes concepts feel newly accessible. It frames something in a way which makes ideas easier to see. So it is with simplifying algebraic expressions. Finding a version of a formula that makes clearer what you would like to do makes the formula more useful.
Simplifying like this, putting an expression into the fewest number of terms, is common. It typically makes it easier to calculate with a formula. We calculate with formulas all the time. It often makes it easier to compare one formula to another. We compare formulas some of the time. So we practice simplifying like this a lot. Occasionally we’ll have a problem where this simplification is counter-productive and we’d do better to write out something as, to make up an example, instead. Someone who’s gotten good at simplifications, to the point it doesn’t take very much work, is likely to spot cases where one wants to keep part of the expression un-simplified.
Chen Weng’s Messycow Comics for the 13th starts off with some tut-tutting of lottery players. Objectively, yes, money put on a lottery ticket is wasted; even, for example, pick-three or pick-four daily games are so unlikely to pay any award as to be worth it. But I cannot make myself believe that this is necessarily a more foolish thing to do with a couple dollars than, say, buying a candy bar or downloading a song you won’t put on any playlists.
And as the Cow points out, the chance of financial success in art — in any creative field — is similarly ridiculously slight. Even skilled people need a stroke of luck to make it, and even then, making it is a marginal matter. (There is a reason I haven’t quit my job to support myself by blog-writing.) People are terrible at estimating probabilities, especially in situations that are even slightly complicated.
When I collected last week’s mathematically-themed comic strips I thought this set an uninspiring one. That changed sometime while I wrote. That’s the sort of week I like to have.
Richard Thompson’s Richard’s Poor Almanac for the 28th is a repeat; all these strips are. And I’ve featured it here before too. But never before in color, so I’ll take this chance to show it one last time. One of the depicted plants is the “Non-Euclidean Creeper”, which “ignores the geometry of the space-time continuum”. Non-Euclidean is one of those few geometry-related words that people recognize — maybe even only learn — in their adulthood. It has connotations of the bizarre and the weird and the wrong.
And it is a bit weird. While we live in a non-Euclidean space, we never really notice. Euclidean space is the geometry we’re used to from drawing shapes on paper and putting boxes in the corners of basements. And from this we’ve given “non-Euclidean” this sinister reputation. We credit it with defying common sense and even logic itself, although it’s geometry. It can’t defy logic. It can defy intuition. Non-Euclidean geometries have the idea that there are no such things as parallel lines. Or the idea that there are too many parallel lines. And it can get to weird results, particularly if we look at more than three dimensions of space. Those also tax the imagination. It will get a weed a bad reputation.
Chen Weng’s Messycow Comics for the 30th is about a child’s delight in learning how to count. I don’t remember ever being so fascinated by counting that it would distract me permanently. I do remember thinking it was amazing that once a pattern was established it kept on, with no reason to ever stop, or even change. My recollection is I thought this somehow unfair to the alphabet, which had a very sudden sharp end.
The counting numbers — counting in general — seem to be things we’ve evolved to understand. Other animals know how to count. Here I recommend again Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics, which describes some of the things we know about how animals do mathematics. It also describes how children come to understand it.
Samson’s Dark Side of the Horse for the 31st is a bit of play with arithmetic. Horace simplifies his problem by catching all the numerals with loops in them — the zeroes and the eights — and working with what’s left. Evidently he’s already cast out all the nines. (This is me making a joke. Casting out nines is a simple checksum that you can do which can guard against some common arithmetic mistakes. It doesn’t catch everything. But it is simple enough to do that it can be worth using.)
The part that disappoints me is that to load the problem up with digits with loops, we get a problem that’s not actually hard: 100 times anything is easy. If the problem were, say, 189 times 80008005 then you’d have a problem someone might sensibly refuse to do. But without those zeroes at the start it’d be harder to understand what Horace was doing. Maybe if it were 10089 times 800805 instead.
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st is the anthropomorphic numerals joke for the week. Also the anthropomorphic letters joke. The capital B sees occasional use in mathematics. It can represent the ball, that is, the set of all points that represent the interior of a sphere of a set radius. Usually a radius of 1. It also sometimes appears in equations as a parameter, a number whose value is fixed for the length of the problem but whose value we don’t care about. I had thought there were a few other roles for B alone, such as a label to represent the Bessel functions. These are a family of complicated-looking polynomials with some nice properties it’s too great a diversion for me to discuss just now. But they seem to more often be labelled with a capital J for reasons that probably seemed compelling at the time. It’ll also get used in logic, where B might stand for the second statement of some argument. 4, meanwhile, is that old familiar thing.
This clears out last week’s comic strips. This present week’s strips should be at this link on Sunday. I haven’t yet read Friday or Saturday’s comics, so perhaps there’s been a flood, but this has been a slow week so far.
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