Reading the Comics, November 10, 2015: Symbols And Meanings Edition
Eric the Circle for the 5th of November, by “andei”, is a mathematics-vocabulary pun. Ellipses are measured with a property called eccentricity. It measures, in a sense, how far any conic section is from being a circle. A circle has an eccentricity of zero. An ellipse, other than a circle, has an eccentricity between 0 and 1. The smaller the eccentricity the harder it is to tell the ellipse from a circle. The larger the eccentricity the longer one direction of the ellipse is compared to the other. For example, the Earth’s orbit around the sun, a very circular thing, has an eccentricity of about 0.0167 these days. Halley’s Comet, which gets closer to the Sun than Venus does, and farther from the sun than Neptune does, has an eccentricity of about 0.967. An eccentricity of exactly 1 means the shape is a parabola. An eccentricity of greater than 1 means the shape is a hyperbola.
Mark Pett’s Mr Lowe for the 5th of November (originally the 2nd of November, 2000) gives a lousy reason to learn long division. I admit I’m not sure I can give a good reason anyone needs to know long division now that calculators are a well-proven technology. Perhaps the best reason is that long division works like much of computational mathematics does. You make a best guess for an answer, and test it, and improve it as necessary. Needing to improve an answer does not mean one started out wrong. It just means that we can approximate and modify solutions.
Russell Myers’s Broom Hilda for the 6th of November is almost this entry’s anthropomorphic numerals joke. I’m not sure just how to categorize it. Perhaps “literal” is the best to be done.
Mark Anderson’s Andertoons for the 8th of November is a joke about turning a wrong answer into a “teach the controversy!” special plea. There are mathematical controversies. But I think the only ones thriving are in fields too abstract for the average person to know or care about. But we can look to controversies of the past. An example an elementary school kid might understand is “should 1 be considered a prime number?” It’s generally not regarded as a prime number. If it were, it would add special cases or extra words to many theorems about prime numbers. That would add boring parts to a lot of work. If we move the number 1 off to its own category (a “unit”), then we can talk about prime numbers and composite numbers more easily. Is that good enough reason? If it isn’t, then what would be a good enough reason?
Bill Amend’s FoxTrot for the 8th of November (a new strip, not a rerun) is a subverted word problem joke. It does contain a mention of curves (of happiness) going to infinity, and how they might do that. There’s some interesting linguistics at work here. A plot of a function — call it f(x), for convenience — is a graph that shows sets of values where the equation y = f(x) is true. We talk about functions “going to infinity”, although properly speaking they don’t “go” anywhere at all, any more than a photograph in a paper book moves.
But it’s hard to resist the image we get from imagining drawing the curve. The eye follows the pen that sweeps, usually left to right, fluttering up and down. And near some points the pen goes soaring off the top (or bottom) of the page. If we imagine zooming out, again and again, the pen still soars off the edge of the page. So we call that “going to infinity”. What we mean is there are some values in the domain which the function matches to numbers in the range that are greater than any finite number. (Or less than any finite but negative number, if we’re going off to negative infinity.)
We can even talk about how cuves “go to” infinity. If the function y = f(x) becomes infinitely large at some point, what does the function f(x)/x do? If that function stays finite we can say f(x) grows to infinity in the same way than x does. If f(x)/x grows infinitely large we can say that f(x) grows to infinity faster than x does. If f(x)/ex stays finite, we can say that f(x) grows to infinity in the same way that the exponential function ex does.
Rates of growth may seem like a dull thing to worry about. They become more obviously relevant if we’re interested in functions that measure, for example, how much of a resource is required to do something. Suppose we have different ways to find the best choice out of a set of things. How long finding that takes depends on how many things there are to look through. If we are looking at scalability — how well we’ll be able to find the best choice out of a much larger set of things — then the rate of growth of these functions can be quite important. If doubling the set of things to look through means searching takes ten thousand times longer, we know we’re probably searching wrong, and should find a better way to do it. If doubling the set of things to look through means we have to take one-and-a-half times as long to find what we want, we’re probably using a good approach.
Greg Evans and Karen Evans’s Luann for the 8th of November builds its joke on the idea that mathematical symbols are funny-looking things you have to interpret, just the same way emojis are. Gunther gives his best shot at explaining the various symbols. The grouping of them makes me wonder exactly what mathematics class he’s taking, though. I can’t think offhand of one that would have all of these in the same textbook.
There’s also an actual mistake right up front. He identifies “(f, g)” as the inner product. The “inner product” is a name we give to a collection of functions, all with different domains but all with the range of real numbers. It allows us to describe a “norm”, or size, of whatever kind of thing we have. It also allows us to describe something that works like an angle between two things, and from it, orthogonality. If we’re looking at vectors, then this inner product is also known as the dot product. The mistake, though, is that the inner product is normally written with angled braces, as <f, g> instead. Normal parentheses usually mean we are giving a set of coordinates or an n-tuple. They can also mean that we are taking a Cartesian product, which looks a lot like giving a set of coordinates or an n-tuple. Probably the writer or artist made an understandable mistake while transcribing notes.
The talk of an inner product suggests more than anything else that the subject is linear algebra. The reference to “Dim(U)” is consistent with this. If U is a matrix, we can talk about its dimension. This is a measure of how many of the rows of the matrix U cannot be made as the sum of scalar products of other rows. That’s useful because it tells us how many of the rows are “linearly independent”, or in a way, tell us something that we can’t get from other rows. So this is linear algebra work.
φ is indeed the Golden Ratio, the number approximately 1.618. It’s a famous number but it’s really got no mathematical significance. Its reciprocal, 1/φ, is about 0.618, and that’s pretty, but that’s all. Many have tried to imbue the Golden Ratio with biological or aesthetic significance, and have failed, because it has none. In mathematics, the Golden Ratio is one of those celebrities who’s famous for no discernable reason or accomplishment.
Δ is the Delta symbol, yes. It’s often used as a shorthand for “change in”. So “Δ x” means “the change in x”. We usually take this to mean a small but noticeable change. If we mean a much smaller change, or a perturbation from what we originally wanted, we might switch to a lowercase “δ x”. If we mean an incredibly tiny change we go to “dx”. This is important in calculus and analysis, as well as in many numerical methods classes.
∝ does mean proportional to. We use it to say one quantity varies as the other one does. For example, that the distance you go in an hour is proportional to how fast you go. Go twice as fast, you go twice as far. This turns up in analysis some, and in applied mathematics that tries to model real-world phenomena. We may be unsure of the precise relationship between two things, but we can say how we expect one thing to affect the other. ∝ is a symbol that lets us talk about qualitative relationships among things.
The equals sign with a triangle above it baffled me, and I had to search about for it. It seems to baffle a modest number of people. Apparently it’s used as a way of saying “is defined as”. That is, the term on the left side of this symbol is by definition equal to whatever appears on the right side. I don’t remember seeing it before, and I don’t get what role it serves that the three-line equals sign ≡ doesn’t already do. I’m not saying the Evanses are wrong to use it, just that it’s not one I’m familiar with.
But you see why I can’t figure what course Gunther is taking. Two of the symbols make sense for linear algebra. One fits in almost anywhere in calculus or applied mathematics. One is mostly an applied mathematics term. One is useless. The last is obscure, anyway. What do they have in common? And what could Tiffany’s message showing a heart-eyed smiley face, pizza, and two check marks mean? “I love to watch pizza voting”?
Dave Kellett’s science fiction/humor comic Drive for the 9th of November reveals the probability of a catastrophe has been mis-reported. The choice of numbers is amusing. It’s hard to have an instinctive feel for the difference between a chance of 1-in-600 and a chance of 1-in-400. The difference makes itself known after a few hundred attempts, at least.
Chris Giarrusso’s absurdist G-Man Webcomics for the 9th of November takes literally the problem of haunting, mysterious shapes.
Gary Wise and Lance Aldrich’s Real Life Adventures asks how to find the area of a trapezoid. I couldn’t dare say.