Reading the Comics, February 6, 2017: Another Pictureless Half-Week Edition


Got another little flood of mathematically-themed comic strips last week and so once again I’ll split them along something that looks kind of middle-ish. Also this is another bunch of GoComics.com-only posts. Since those seem to be accessible to anyone whether or not they’re subscribers indefinitely far into the future I don’t feel like I can put the comics directly up and will trust you all to click on the links that you find interesting. Which is fine; the new GoComics.com design makes it annoyingly hard to download a comic strip. I don’t think that was their intention. But that’s one of the two nagging problems I have with their new design. So you know.

Tony Cochran’s Agnes for the 5th sees a brand-new mathematics. Always dangerous stuff. But mathematicians do invent, or discover, new things in mathematics all the time. Part of the task is naming the things in it. That’s something which takes talent. Some people, such as Leonhard Euler, had the knack a great novelist has for putting names to things. The rest of us muddle along. Often if there’s any real-world inspiration, or resemblance to anything, we’ll rely on that. And we look for terminology that evokes similar ideas in other fields. … And, Agnes would like to know, there is mathematics that’s about approximate answers, being “right around” the desired answer. Unfortunately, that’s hard. (It’s all hard, if you’re going to take it seriously, much like everything else people do.)

Scott Hilburn’s The Argyle Sweater for the 5th is the anthropomorphic numerals joke for this essay.

Carol Lay’s Lay Lines for the 6th depicts the hazards of thinking deeply and hard about the infinitely large and the infinitesimally small. They’re hard. Our intuition seems well-suited to handing a modest bunch of household-sized things. Logic guides us when thinking about the infinitely large or small, but it takes a long time to get truly conversant and comfortable with it all.

Paul Gilligan’s Pooch Cafe for the 6th sees Poncho try to argue there’s thermodynamical reasons for not being kind. Reasoning about why one should be kind (or not) is the business of philosophers and I won’t overstep my expertise. Poncho’s mathematics, that’s something I can write about. He argues “if you give something of yourself, you inherently have less”. That seems to be arguing for a global conservation of self-ness, that the thing can’t be created or lost, merely transferred around. That’s fair enough as a description of what the first law of thermodynamics tells us about energy. The equation he reads off reads, “the change in the internal energy (Δ U) equals the heat added to the system (U) minus the work done by the system (W)”. Conservation laws aren’t unique to thermodynamics. But Poncho may be aware of just how universal and powerful thermodynamics is. I’m open to an argument that it’s the most important field of physics.

Jonathan Lemon’s Rabbits Against Magic for the 6th is another strip Intro to Calculus instructors can use for their presentation on instantaneous versus average velocities. There’s been a bunch of them recently. I wonder if someone at Comic Strip Master Command got a speeding ticket.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th is about numeric bases. They’re fun to learn about. There’s an arbitrariness in the way we represent concepts. I think we can understand better what kinds of problems seem easy and what kinds seem harder if we write them out different ways. But base eleven is only good for jokes.

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What Are Numbers Made Of?


To return to my second major theme: my Dearly Beloved told me that I must explain that trick where one adds up the digits of a number and finds out from that whether it’s divisible by 9. I wanted to anyway, but a request like that is irresistible. The answer can be given quickly — and several of my hopefully faithful readers did, in comments, last Friday — but I’d like to take the long way around because I do that and because it lets a lot of other interesting divisibility properties show themselves.

We use ten numerals and the place where we write them to express all the counting numbers out there. We put one of the numerals, such as `2′, in a place which denotes whether we mean to say two tens, or two hundreds, or two millions. That’s a clever tool, and not one inherent to the idea of numbers. We could as easily use different symbols for different magnitudes; the only familiar example of this (in the west) is Roman numerals, where we use I, X, C, and M for increasing powers of ten, and then notice we aren’t really quite sure what to do past M.

The Romans were not very sure either, and individual variations developed when someone found they needed to express an M of M very often. The system has fewer numerals, symbols representing numbers, than ours does, with V and L and D the only additional numerals reasonably common. By the Middle Ages some symbols were improvised to allow for extremely large numbers such as the hundred thousands, and some extra symbols were pulled in for numbers such as 7 or 40, but they have faded to the point of obscurity. This is a numbering system which runs out when the numbers get too large, which seems impossibly limited at first glance. But we haven’t changed much from these times: while we have a numbering system that can, in principle, work with arbitrarily big or tiny numbers, in practice we only use a small range of them. When we turn over arithmetic to computers, in fact, we accept numbering systems which have limits on how big (positive or negative) a number may be, or how close to zero one may work. We accept those limits because of their convenience and are only sometimes annoyed to find, for example, that the spreadsheet trying to calculate a bill has decided we want 0.9999999 of a penny.

Continue reading “What Are Numbers Made Of?”