I had figured to do Reading the Comics posts weekly, and then last week went and gave me too big a flood of things to do. I have no idea what the rest of this week is going to look like. But given that I had four strips dated before last Sunday I’m going to err on the side of posting too much about comic strips.
Scott Metzger’s The Bent Pinky for the 24th uses mathematics as something that dogs can be adorable about not understanding. Thus all the heads tilted, as if it were me in a photograph. The graph here is from economics, which has long had a challenging relationship with mathematics. This particular graph is qualitative; it doesn’t exactly match anything in the real world. But it helps one visualize how we might expect changes in the price of something to affect its sales. A graph doesn’t need to be precise to be instructional.
Dave Whamond’s Reality Check for the 24th is this essay’s anthropomorphic-numerals joke. And it’s a reminder that something can be quite true without being reassuring. It plays on the difference between “real” numbers and things that really exist. It’s hard to think of a way that a number such as two could “really” exist that doesn’t also allow the square root of -1 to “really” exist.
And to be a bit curmudgeonly, it’s a bit sloppy to speak of “the square root of negative one”, even though everyone does. It’s all right to expand the idea of square roots to cover stuff it didn’t before. But there’s at least two numbers that would, squared, equal -1. We usually call them i and -i. Square roots naturally have this problem,. Both +2 and -2 squared give us 4. We pick out “the” square root by selecting the positive one of the two. But neither i nor -i is “positive”. (Don’t let the – sign fool you. It doesn’t count.) You can’t say either i or -i is greater than zero. It’s not possible to define a “greater than” or “less than” for complex-valued numbers. And that’s even before we get into quaternions, in which we summon two more “square roots” of -1 into existence. Octonions can be even stranger. I don’t blame 1 for being worried.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is a pleasant bit of pop-mathematics debunking. I’ve explained in the past how I’m a doubter of the golden ratio. The Fibonacci Sequence has a bit more legitimate interest to it. That’s sequences of numbers in which the next term is the sum of the previous two terms. The famous one of that is 1, 1, 2, 3, 5, 8, 13, 21, et cetera. It may not surprise you to know that the Fibonacci Sequence has a link to the golden ratio. As it goes on, the ratio between one term and the next one gets close to the golden ratio.
The Harmonic Series is much more deeply weird. A series is the number we get from adding together everything in a sequence. The Harmonic Series grows out of the first sequence you’d imagine ever adding up. It’s 1 plus 1/2 plus 1/3 plus 1/4 plus 1/5 plus 1/6 plus … et cetera. The first time you hear of this you get the surprise: this sum doesn’t ever stop piling up. We say it ‘diverges’. It won’t on your computer; the floating-point arithmetic it does won’t let you add enormous numbers like ‘1’ to tiny numbers like ‘1/531,325,263,953,066,893,142,231,356,120’ and get the right answer. But if you actually added this all up, it would.
The proof gets a little messy. But it amounts to this: 1/2 plus 1/3 plus 1/4? That’s more than 1. 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12? That’s also more than 1. 1/13 + 1/14 + 1/15 + et cetera up through + 1/32 + 1/33 + 1/34 is also more than 1. You need to pile up more and more terms each time, but a finite string of these numbers will add up to more than 1. So the whole series has to be more than 1 + 1 + 1 + 1 + 1 … and so more than any finite number.
That’s all amazing enough. And then the series goes on to defy all kinds of intuition. Obviously dropping a couple of terms from the series won’t change whether it converges or diverges. Multiplying alternating terms by -1, so you have (say) 1 – 1/2 + 1/3 – 1/4 + 1/5 et cetera produces something that looks like it converges. It equals the natural logarithm of 2. But if you take those terms and rearrange them, you can produce any real number, positive or negative, that you want.
And, as Weinersmith describes here, if you just skip the correct set of terms, you can make the sum converge. The ones with 9 in the denominator will be, then, 1/9, 1/19, 1/29, 1/90, 1/91, 1/92, 1/290, 1/999, those sorts of things. Amazing? Yes. Absurd? I suppose so. This is why mathematicians learn to be very careful when they do anything, even addition, infinitely many times.
John Deering’s Strange Brew for the 25th is a fear-of-mathematics joke. The sign the warrior’s carrying is legitimate algebra, at least so far as it goes. The right-hand side of the equation gets cut off. In time, it would get to the conclusion that x equals –19/2, or -9.5.