## The End 2016 Mathematics A To Z: Principal

Functions. They’re at the center of so much mathematics. They have three pieces: a domain, a range, and a rule. The one thing functions absolutely must do is match stuff in the domain to one and only one thing in the range. So this is where it gets tricky.

## Principal.

Thing with this one-and-only-one thing in the range is it’s not always practical. Sometimes it only makes sense to allow for something in the domain to match several things in the range. For example, suppose we have the domain of positive numbers. And we want a function that gives us the numbers which, squared, are whatever the original function was. For any positive real number there’s two numbers that do that. 4 should match to both +2 and -2.

You might ask why I want a function that tells me the numbers which, squared, equal something. I ask back, what business is that of yours? I want a function that does this and shouldn’t that be enough? We’re getting off to a bad start here. I’m sorry; I’ve been running ragged the last few days. I blame the flat tire on my car.

Anyway. I’d want something like that function because I’m looking for what state of things makes some other thing true. This turns up often in “inverse problems”, problems in which we know what some measurement is and want to know what caused the measurement. We do that sort of problem all the time.

We can handle these multi-valued functions. Of course we can. Mathematicians are as good at loopholes as anyone else is. Formally we declare that the range isn’t the real numbers but rather sets of real numbers. My what-number-squared function then matches ‘4’ in the domain to the set of numbers ‘+2 and -2’. The set has several things in it, but there’s just the one set. Clever, huh?

This sort of thing turns up a lot. There’s two numbers that, squared, give us any real number (except zero). There’s three numbers that, squared, give us any real number (again except zero). Polynomials might have a whole bunch of numbers that make some equation true. Trig functions are worse. The tangent of 45 degrees equals 1. So is the tangent of 225 degrees. Also 405 degrees. Also -45 degrees. Also -585 degrees. OK, a mathematician would use radians instead of degrees, but that just changes what the numbers are. Not that there’s infinitely many of them.

It’s nice to have options. We don’t always want options. Sometimes we just want one blasted simple answer to things. It’s coded into the language. We say “the square root of four”. We speak of “the arctangent of 1”, which is to say, “the angle with tangent of 1”. We only say “all square roots of four” if we’re making a point about overlooking options.

If we’ve got a set of things, then we can pick out one of them. This is obvious, which means it is so very hard to prove. We just have to assume we can. Go ahead; assume we can. Our pick of the one thing out of this set is the “principal”. It’s not any more inherently right than the other possibilities. It’s just the one we choose to grab first.

So. The principal square root of four is positive two. The principal arctangent of 1 is 45 degrees, or in the dialect of mathematicians π divided by four. We pick these values over other possibilities because they’re nice. What makes them nice? Well, they’re nice. Um. Most of their numbers aren’t that big. They use positive numbers if we have a choice in the matter. Deep down we still suspect negative numbers of being up to something.

If nobody says otherwise then the principal square root is the positive one, or the one with a positive number in front of the imaginary part. If nobody says otherwise the principal arcsine is between -90 and +90 degrees (-π/2 and π/2). The principal arccosine is between 0 and 180 degrees (0 and π), unless someone says otherwise. The principal arctangent is … between -90 and 90 degrees, unless it’s between 0 and 180 degrees. You can count on the 0 to 90 part. Use your best judgement and roll with whatever develops for the other half of the range there. There’s not one answer that’s right for every possible case. The point of a principal value is to pick out one answer that’s usually a good starting point.

When you stare at what it means to be a function you realize that there’s a difference between the original function and the one that returns the principal value. The original function has a range that’s “sets of values”. The principal-value version has a range that’s just one value. If you’re being kind to your audience you make some note of that. Usually we note this by capitalizing the start of the function: “arcsin z” gives way to “Arcsin z”. “Log z” would be the principal-value version of “log z”. When you start pondering logarithms for negative numbers or for complex-valued numbers you get multiple values. It’s the same way that the arcsine function does.

And it’s good to warn your audience which principal value you mean, especially for the arc-trigonometric-functions or logarithms. (I’ve never seen someone break the square root convention.) The principal value is about picking the most obvious and easy-to-work-with value out of a set of them. It’s just impossible to get everyone to agree on what the obvious is.

## Reading the Comics, November 5, 2016: Surprisingly Few Halloween Costumes Edition

Comic Strip Master Command gave me a light load this week, which suit me fine. I’ve been trying to get the End 2016 Mathematics A To Z comfortably under way instead. It does strike me that there were fewer Halloween-themed jokes than I’d have expected. For all the jokes there are to make about Halloween I’d imagine some with some mathematical relevance would come up. But they didn’t and, huh. So it goes. The one big exception is the one I’d have guessed would be the exception.

Bill Amend’s FoxTrot for the 30th — a new strip — plays with the scariness of mathematics. Trigonometry specifically. Trig is probably second only to algebra for the scariest mathematics normal people encounter. And that’s probably more because people get to algebra before they might get to trigonometry. Which is madness, in its way. Trigonometry is about how we can relate angles, arcs, and linear distances. It’s about stuff anyone would like to know, like how to go from an easy-to-make observation of the angle spanned by a thing to how big the thing must be. But the field does require a bunch of exotic new functions like sine and tangent and novelty acts like “arc-cosecant”. And the numbers involved can be terrible things. The sine of an angle, for example, is almost always going to be some irrational number. For common angles we use a lot it’ll be an irrational number with an easy-to-understand form. For example the sine of 45 degrees, mentioned here, is “one-half the square root of two”. Anyone not trying to be intimidating will use that instead. But the sine of, say, 50 degrees? I don’t know what that is either except that it’s some never-ending sequence of digits. People love to have digits, but when they’re asked to do something with them, they get afraid and I don’t blame them.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 30th uses sudoku as shorthand for “genius thinking”. I am aware some complain sudoku isn’t mathematics. It’s certainly logic, though, and if we’re going to rule out logic puzzles from mathematics we’re going to lose a lot of fun fields. One of the commenters provided what I suppose the solution to be. (I haven’t checked.) If wish to do the puzzle be careful about scrolling.

In Jef Mallet’s Frazz for the 2nd Caulfield notices something cute about 100. A perfect square is a familiar enough idea; it’s a whole number that’s the square of another whole number. The “roundest of round numbers” is a value judgement I’m not sure I can get behind. It’s a good round number, anyway, at least for stuff that’s sensibly between about 50 and 150. Or maybe between 50 and 500 if you’re just interested in about how big something might be. An irrational number, well, you know where that joke’s going.

Mrs Olsen doesn’t seem impressed by Caulfield’s discovery, although in fairness we don’t see the actual aftermath. Sometimes you notice stuff like that and it is only good for a “huh”. But sometimes you get into some good recreational mathematics. It’s the sort of thinking that leads to discovering magic squares and amicable numbers and palindromic prime numbers and the like. Do they lead to important mathematics? Some of them do. Or at least into interesting mathematics. Sometimes they’re just passingly amusing.

Greg Curfman’s Meg rerun for the 12th quotes Einstein’s famous equation as the sort of thing you could just expect would be asked in school. I’m not sure I ever had a class where knowing E = mc2 was the right answer to a question, though. Maybe as I got into physics since we did spend a bit of time on special relativity and E = mc2 turns up naturally there. Maybe I’ve been out of elementary school too long to remember.

Mark Tatulli’s Heart of the City for the 4th has Heart and Dean talking about postapocalyptic society. Heart doubts that postapocalyptic society would need people like him, “with long-division experience”. Ah, but, grant the loss of computing devices. People will still need to compute. Before the days of electrical, and practical mechanical, computing people who could compute accurately were in demand. The example mathematicians learn to remember is Zacharias Dase, a German mental calculator. He was able to do astounding work and in his head. But he didn’t earn so much money as pro-mental-arithmetic propaganda would like us to believe. And why work entirely in your head if you don’t need to?

Larry Wright’s Motley Classics rerun for the 5th is a word problem joke. And it’s mixed with labor relations humor for the sake of … I’m not quite sure, actually. Anyway I would have sworn I’d featured this strip in a long-ago Reading The Comics post, but I don’t see it on a casual search. So, go figure.

## Reading the Comics, October 19, 2016: An Extra Day Edition

I didn’t make noise about it, but last Sunday’s mathematics comic strip roundup was short one day. I was away from home and normal computer stuff Saturday. So I posted without that day’s strips under review. There was just the one, anyway.

Also I want to remind folks I’m doing another Mathematics A To Z, and taking requests for words to explain. There are many appealing letters still unclaimed, including ‘A’, ‘T’, and ‘O’. Please put requests in over on that page because. It’s easier for me to keep track of what’s been claimed that way.

Matt Janz’s Out of the Gene Pool rerun for the 15th missed last week’s cut. It does mention the Law of Cosines, which is what the Pythagorean Theorem looks like if you don’t have a right triangle. You still have to have a triangle. Bobby-Sue recites the formula correctly, if you know the notation. The formula’s $c^2 = a^2 + b^2 - 2 a b \cos\left(C\right)$. Here ‘a’ and ‘b’ and ‘c’ are the lengths of legs of the triangle. ‘C’, the capital letter, is the size of the angle opposite the leg with length ‘c’. That’s a common notation. ‘A’ would be the size of the angle opposite the leg with length ‘a’. ‘B’ is the size of the angle opposite the leg with length ‘b’. The Law of Cosines is a generalization of the Pythagorean Theorem. It’s a result that tells us something like the original theorem but for cases the original theorem can’t cover. And if it happens to be a right triangle the Law of Cosines gives us back the original Pythagorean Theorem. In a right triangle C is the size of a right angle, and the cosine of that is 0.

That said Bobby-Sue is being fussy about the drawings. No geometrical drawing is ever perfectly right. The universe isn’t precise enough to let us draw a right triangle. Come to it we can’t even draw a triangle, not really. We’re meant to use these drawings to help us imagine the true, Platonic ideal, figure. We don’t always get there. Mock proofs, the kind of geometric puzzle showing something we know to be nonsense, rely on that. Give chalkboard art a break.

Samson’s Dark Side of the Horse for the 17th is the return of Horace-counting-sheep jokes. So we get a π joke. I’m amused, although I couldn’t sleep trying to remember digits of π out quite that far. I do better working out Collatz sequences.

Hilary Price’s Rhymes With Orange for the 19th at least shows the attempt to relieve mathematics anxiety. I’m sympathetic. It does seem like there should be ways to relieve this (or any other) anxiety, but finding which ones work, and which ones work best, is partly a mathematical problem. As often happens with Price’s comics I’m particularly tickled by the gag in the title panel.

Hilary Price’s Rhymes With Orange for the 19th of October, 2016. I don’t think there’s enough data given to solve the problem. But it’s a start at least. Start by making a note of it on your suspiciously large sheet of paper.

Norm Feuti’s Gil rerun for the 19th builds on the idea calculators are inherently cheating on arithmetic homework. I’m sympathetic to both sides here. If Gil just wants to know that his answers are right there’s not much reason not to use a calculator. But if Gil wants to know that he followed the right process then the calculator’s useless. By the right process I mean, well, the work to be done. Did he start out trying to calculate the right thing? Did he pick an appropriate process? Did he carry out all the steps in that process correctly? If he made mistakes on any of those he probably didn’t get to the right answer, but it’s not impossible that he would. Sometimes multiple errors conspire and cancel one another out. That may not hurt you with any one answer, but it does mean you aren’t doing the problem right and a future problem might not be so lucky.

Zach Weinersmith’s Saturday Morning Breakfast Cereal rerun for the 19th has God crashing a mathematics course to proclaim there’s a largest number. We can suppose there is such a thing. That’s how arithmetic modulo a number is done, for one. It can produce weird results in which stuff we just naturally rely on doesn’t work anymore. For example, in ordinary arithmetic we know that if one number times another equals zero, then either the first number or the second, or both, were zero. We use this in solving polynomials all the time. But in arithmetic modulo 8 (say), 4 times 2 is equal to 0.

And if we recklessly talk about “infinity” as a number then we get outright crazy results, some of them teased in Weinersmith’s comic. “Infinity plus one”, for example, is “infinity”. So is “infinity minus one”. If we do it right, “infinity minus infinity” is “infinity”, or maybe zero, or really any number you want. We can avoid these logical disasters — so far, anyway — by being careful. We have to understand that “infinity” is not a number, though we can use numbers growing infinitely large.

Induction, meanwhile, is a great, powerful, yet baffling form of proof. When it solves a problem it solves it beautifully. And easily, too, usually by doing something like testing two special cases. Maybe three. At least a couple special cases of whatever you want to know. But picking the cases, and setting them up so that the proof is valid, is not easy. There’s logical pitfalls and it is so hard to learn how to avoid them.

Jon Rosenberg’s Scenes from a Multiverse for the 19th plays on a wonderful paradox of randomness. Randomness is … well, unpredictable. If I tried to sell you a sequence of random numbers and they were ‘1, 2, 3, 4, 5, 6, 7’ you’d be suspicious at least. And yet, perfect randomness will sometimes produce patterns. If there were no little patches of order we’d have reason to suspect the randomness was faked. There is no reason that a message like “this monkey evolved naturally” couldn’t be encoded into a genome by chance. It may just be so unlikely we don’t buy it. The longer the patch of order the less likely it is. And yet, incredibly unlikely things do happen. The study of impossibly unlikely events is a good way to quickly break your brain, in case you need one.

## Reading the Comics, May 12, 2016: No Pictures Again Edition

I’ve hardly stopped reading the comics. I doubt I could even if I wanted at this point. But all the comics this bunch are from GoComics, which as far as I’m aware doesn’t turn off access to comic strips after a couple of weeks. So I don’t quite feel justified including the images of the comics when you can just click links to them instead.

It feels a bit barren, I admit. I wonder if I shouldn’t commission some pictures so I have something for visual appeal. There’s people I know who do comics online. They might be able to think of something to go alongside every “Student has snarky answer for a word problem” strip.

Brian and Ron Boychuk’s The Chuckle Brothers for the 8th of May drops in an absolute zero joke. Absolute zero’s a neat concept. People became aware of it partly by simple extrapolation. Given that the volume of a gas drops as the temperature drops, is there a temperature at which the volume drops to zero? (It’s complicated. But that’s the thread I use to justify pointing out this strip here.) And people also expected there should be an absolute temperature scale because it seemed like we should be able to describe temperature without tying it to a particular method of measuring it. That is, it would be a temperature “absolute” in that it’s not explicitly tied to what’s convenient for Western Europeans in the 19th century to measure. That zero and that instrument-independent temperature idea get conflated, and reasonably so. Hasok Chang’s Inventing Temperature: Measurement and Scientific Progress is well-worth the read for people who want to understand absolute temperature better.

Gene Weingarten, Dan Weingarten & David Clark’s Barney and Clyde for the 9th is another strip that seems like it might not belong here. While it’s true that accidents sometimes lead to great scientific discoveries, what has that to do with mathematics? And the first thread is that there are mathematical accidents and empirical discoveries. Many of them are computer-assisted. There is something that feels experimental about doing a simulation. Modern chaos theory, the study of deterministic yet unpredictable systems, has at its founding myth Edward Lorentz discovering that tiny changes in a crude weather simulation program mattered almost right away. (By founding myth I don’t mean that it didn’t happen. I just mean it’s become the stuff of mathematics legend.)

But there are other ways that “accidents” can be useful. Monte Carlo methods are often used to find extreme — maximum or minimum — solutions to complicated systems. These are good if it’s hard to find a best possible answer, but it’s easy to compare whether one solution is better or worse than another. We can get close to the best possible answer by picking an answer at random, and fiddling with it at random. If we improve things, good: keep the change. You can see why this should get us pretty close to a best-possible-answer soon enough. And if we make things worse then … usually but not always do we reject the change. Sometimes we take this “accident”. And that’s because if we only take improvements we might get caught at a local extreme. An even better extreme might be available but only by going down an initially unpromising direction. So it’s worth allowing for some “mistakes”.

Mark Anderson’s Andertoons for the 10th of Anderson is some wordplay on volume. The volume of boxes is an easy formula to remember and maybe it’s a boring one. It’s enough, though. You can work out the volume of any shape using just the volume of boxes. But you do need integral calculus to tell how to do it. So maybe it’s easier to memorize the formula for volumes of a pyramid and a sphere.

Berkeley Breathed’s Bloom County for the 10th of May is a rerun from 1981. And it uses a legitimate bit of mathematics for Milo to insult Freida. He calls her a “log 10 times 10 to the derivative of 10,000”. The “log 10” is going to be 1. A reference to logarithm, without a base attached, means either base ten or base e. “log” by itself used to invariably mean base ten, back when logarithms were needed to do ordinary multiplication and division and exponentiation. Now that we have calculators for this mathematicians have started reclaiming “log” to mean the natural logarithm, base e, which is normally written “ln”, but that’s still an eccentric use. Anyway, the logarithm base ten of ten is 1: 10 is equal to 10 to the first power.

10 to the derivative of 10,000 … well, that’s 10 raised to whatever number “the derivative of 10,000” is. Derivatives take us into calculus. They describe how much a quantity changes as one or more variables change. 10,000 is just a number; it doesn’t change. It’s called a “constant”, in another bit of mathematics lingo that reminds us not all mathematics lingo is hard to understand. Since it doesn’t change, its derivative is zero. As anything else changes, the constant 10,000 does not. So the derivative of 10,000 is zero. 10 to the zeroth power is 1.

So, one times one is … one. And it’s rather neat that kids Milo’s age understand derivatives well enough to calculate that.

Ruben Bolling’s Super-Fun-Pak Comix rerun for the 10th happens to have a bit of graph theory in it. One of Uncle Cap’n’s Puzzle Pontoons is a challenge to trace out a figure without retracting a line or lifting your pencil. You can’t, not this figure. One of the first things you learn in graph theory teaches how to tell, and why. And thanks to a Twitter request I’m figuring to describe some of that for the upcoming Theorem Thursdays project. Watch this space!

Charles Schulz’s Peanuts Begins for the 11th, a rerun from the 6th of February, 1952, is cute enough. It’s one of those jokes about how a problem seems intractable until you’ve found the right way to describe it. I can’t fault Charlie Brown’s thinking here. Figuring out a way the problems are familiar and easy is great.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 12th is a “see, we use mathematics in the real world” joke. In this case it’s triangles and triangulation. That’s probably the part of geometry it’s easiest to demonstrate a real-world use for, and that makes me realize I don’t remember mathematics class making use of that. I remember it coming up some, particularly in what must have been science class when we built and launched model rockets. We used a measure of how high an angle the rocket reached, and knowledge of how far the observing station was from the launchpad. But that wasn’t mathematics class for some reason, which is peculiar.

## Lewis Carroll Tries Changing The Way You See Trigonometry

Today’s On This Day In Math tweet was well-timed. I’d recently read Robin Wilson’s Lewis Carroll In Numberland: His Fantastical Mathematical Logical Life. It’s a biography centered around Charles Dodgson’s mathematical work. It shouldn’t surprise you that he was fascinated with logic, and wrote texts — and logic games — that crackle with humor. People who write logic texts have a great advantage on other mathematicians (or philosophers). Almost any of their examples can be presented as a classically structured joke. Vector calculus isn’t so welcoming. But Carroll was good at logic-joke writing.

Developing good notation was one of Dodgson/Carroll’s ongoing efforts, though. I’m not aware of any of his symbols that have got general adoption. But he put forth some interesting symbols to denote the sine and cosine and other trigonometric functions. In 1861, the magazine The Athanaeum reviewed one of his books, with its new symbols for the basic trigonometric functions. (The link shows off all these symbols.) The reviewer was unconvinced, apparently.

I confess that I am, too, but mostly on typographical grounds. It is very easy to write or type out “sin θ” and get something that makes one think of the sine of angle θ. And I’m biased by familiarity, after all. But Carroll’s symbols have a certain appeal. I wonder if they would help people learning the functions keep straight what each one means.

The basic element of the symbols is a half-circle. The sine is denoted by the half-circle above the center, with a vertical line in the middle of that. So it looks a bit like an Art Deco ‘E’ fell over. The cosine is denoted by the half circle above the center, but with a horizontal line underneath. It’s as if someone started drawing Chad and got bored and wandered off. The tangent gets the same half-circle again, with a horizontal line on top of the arc, literally tangent to the circle.

There’s a subtle brilliance to this. One of the ordinary ways to think of trigonometric functions is to imagine a circle with radius 1 that’s centered on the origin. That is, its center has x-coordinate 0 and y-coordinate 0. And we imagine drawing the line that starts at the origin, and that is off at an angle θ from the positive x-axis. (That is, the line that starts at the origin and goes off to the right. That’s the direction where the x-coordinate of points is increasing and the y-coordinate is always zero.) (Yes, yes, these are line segments, or rays, rather than lines. Let it pass.)

The sine of the angle θ is also going to be the y-coordinate of the point where the line crosses the unit circle. That is, it’s the vertical coordinate of that point. So using a vertical line touching a semicircle to suggest the sine represents visually one thing that the sine means. And the cosine of the angle θ is going to be the x-coordinate of the point where the line crosses the unit circle. So representing the cosine with a horizontal line and a semicircle again underlines one of its meanings. And, for that matter, the line might serve as a reminder to someone that the sine of a right angle will be 1, while the cosine of an angle of zero is 1.

The tangent has a more abstract interpretation. But a line that comes up to and just touches a curve at a single point is, literally, a tangent line. This might not help one remember any useful values for the tangent. (That the tangent of zero is zero, the tangent of half a right angle is 1, the tangent of a right angle is undefined). But it’s still a guide to what things mean.

The cotangent is just the tangent upside-down. Literally; it’s the lower half of a circle, with a horizontal line touching it at its lowest point. That’s not too bad a symbol, actually. The cotangent of an angle is the reciprocal of the tangent of an angle. So making its symbol be the tangent flipped over is mnemonic.

The secant and cosecant are worse symbols, it must be admitted. The secant of an angle is the reciprocal of the cosine of the angle, and the cosecant is the reciprocal of the sine. As far as I can tell they’re mostly used because it’s hard to typeset $\frac{1}{\sin\left(\theta\right)}$. And to write instead $\sin^{-1}\left(\theta\right)$ would be confusing as that’s often used for the inverse sine, or arcsine, function. I don’t think these symbols help matters any. I’m surprised Carroll didn’t just flip over the cosine and sine symbols, the way he did with the cotangent.

The versed sine function is one that I got through high school without hearing about. I imagine you have too. The versed sine, or the versine, of an angle is equal to one minus the cosine of the angle. Why do we need such a thing? … Computational convenience is the best answer I can find. It turns up naturally if you’re trying to work out the distance between points on the surface of a sphere, so navigators needed to know it.

And if we need to work with small angles, then this can be more computationally stable than the cosine is. The cosine of a small angle is close to 1, and the difference between 1 and the cosine, if you need such a thing, may be lost to roundoff error. But the versed sine … well, it will be the same small number. But the table of versed sines you have to refer to will list more digits. There’s a difference between working out “1 – 0.9999” and working with “0.0001473”, if you need three digits of accuracy.

But now we don’t need printed tables of trigonometric functions to get three (or many more) digits of accuracy. So we can afford to forget the versed sine ever existed. I learn (through Wikipedia) that there are also functions called versed cosines, coversed sines, hacoversed cosines, and excosecants, among others. These names have a wonderful melody and are almost poems by themselves. Just the same I’m glad I don’t have to remember what they all are.

Carroll’s notation just replaces the “sin” or “cos” or “tan” with these symbols, so you would have the half-circle and the line followed by θ or whatever variable you used for the angle. So the symbols don’t save any space on the line. They take fewer pen strokes to write, just two for each symbol. Writing the symbols out by hand takes three or four (or for cosecant, as many as five), unless you’re writing in cursive. They’re still probably faster than the truncated words, though. So I don’t know why precisely the symbols didn’t take hold. I suppose part is that people were probably used to writing “sin θ”. And typesetters already got enough hazard pay dealing with mathematicians and their need for specialized symbols. Why add in another half-dozen or more specialized bits of type for something everyone’s already got along without?

Still, I think there might be some use in these as symbols for mathematicians in training. I’d be interested to know how they serve people just learning trigonometry.

• #### sheldonk2014 10:46 pm on Monday, 27 July, 2015 Permalink | Reply

It’s really fascinating that the writer had a thing for numbers
What you won’t find……
Sheldon

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• #### Joseph Nebus 4:37 am on Tuesday, 28 July, 2015 Permalink | Reply

It really stands out how much Lewis Carroll liked the playful side of mathematics. If I have a slow stretch I might just pull out various puzzles and games he developed — there were a lot of them — to show how much recreational mathematics he had to share.

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• #### howardat58 12:14 am on Tuesday, 28 July, 2015 Permalink | Reply

There is a crying need to get rid of sin x, along with sin^2 x (write it with superscript).
Why don’t we write y = fx for any old function, or log^2 x for the square of log x ?
Consistency is a non-feature of elementary math, and much confusion is a result.

PUT THE BRACKETS IN ! (parentheses)
sin(x), cos(x), sin(x+a), (sin(x))^2, or maybe ok with sin(x)^2
and as for \sin^{-1}\left(\theta\right) then 1/sin(x) and arcsin(x) are far far better.

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• #### Joseph Nebus 4:48 am on Tuesday, 28 July, 2015 Permalink | Reply

There are some fields of analysis in which writing y = fx is the thing to do. But that does require working out how you want the composition of functions to look, and what f2 ought to mean in that context.

There’s probably no way to be perfectly consistent in notation throughout mathematics. There’s just too much and some stuff is useful in some contexts that isn’t in others. That parentheses mark a group of symbols as a common unit is pretty nearly universal. We always need some aggregation term, after all, and just drawing in a vinculum won’t always cut it.

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• #### Thumbup 11:44 am on Tuesday, 28 July, 2015 Permalink | Reply

He is a child molester.

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• #### J Brian. 8:57 pm on Monday, 3 August, 2015 Permalink | Reply

There are several notations that irritate students, often, to be handled while writing or speaking and they often ask why we can’t write a certain notation this way or a that…! I feel there’s a particular reason. Which is, if you’ve done some extraordinary work in your discipline, and you’ve conquered some big problem of your world. Only after that you can challenge the previously defined notations. I think, except doing that, nobody is going to give value to your newly defined notation/symbols, doesn’t matter if the new notations are better.

I hope you people know the Kumar Jan’s story.

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• #### Joseph Nebus 5:00 am on Tuesday, 4 August, 2015 Permalink | Reply

Most notation does have a history behind it. For the prominent symbols everyone uses there’s typically great names attached, too, and a string of attempted or possible alternate symbols. Sometimes they give hints about how the understanding of an idea developed. The decimal point, for example, is the result of a lot of uncertainty about how to express this radical idea, and how to make it clear, versus how to make it compact. We’ve got the idea down now to where it’s hard to imagine it being made much better. I think the only thing we really need to fix yet is how to aggregate digits after the decimal: 2,038.12834200 ends in a bit of a mess, but how to improve it?

I think the problem in taking on well-established notation is mostly one of inertia. If we’ve settled on “sin(x)” as a tolerable way to express “the sine of the number called x” then to overthrow it we have to put forth something that’s somehow clearer enough to be worth changing. “sin(x)” has a lot going for it, though. In shape it parallels the “f(x)” notation that’s the most familiar way to call functions. By being several letters long you make clear you mean one of a common set of special functions. Of the common special functions — the trig functions, logarithms, exponentials, the Gaussian error function and so on — the letters “sin” pick out “sine” with very little room for alternate interpretations. The notation has a lot going for it. It’s difficult to picture something that would do better.

I am not familiar with the Kumar Jan’s story; could you share it?

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• #### J Brian. 12:16 pm on Thursday, 6 August, 2015 Permalink | Reply

Yeah…! I don’t know where you can find his story. But I heard it from my lecturer in my university. Kumar jan was an Indian mathematician, also he was very young, perhaps below age of 19 years. His could only complete his high school education and couldn’t afford expenses for graduation. But he did great work on a topic in mathematics (pardon me, I forgot what it was, but it was some new development in the form of a theorem). He sent his work to a lot of P.HD doctors across the world but no one gave him any importance. A professor named Howdy also got his research paper. But after reading, he just put it somewhere between files and folders. After one year, he again opened that paper and after reading it again, he was amazed a lot and he thought, the author of this paper is really a great mathematician. He replied to Kumar jan, and offered him a free P.HD education under his supervision. But till that, Kumar Jan was died by a natural death. Also, he couldn’t complete his p.hd because he didn’t had a graduation degree. So much big unfortunate he was. I told this story to explain my last message, that if you have to change something, you’ve to do something like Kumar Jan did.

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• #### Joseph Nebus 4:36 am on Saturday, 8 August, 2015 Permalink | Reply

I am not certain, but is it possible that you’re thinking of Srinivasa Ramanujan? The mention of a genius Indian mathematician discovered by a contact with “Howdy” makes me think of Ramanujan, brought out of obscurity by his contact with G H Hardy. Ramanujan did have a sadly short life, but he did receive his doctorate and world renown for his work. His birthday was declared India’s National Mathematics Day (although it’s not clear to me if that was a one-time event, for the 125th anniversary of his birth, or an annual occurrence).

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• #### Ghufran 6:58 am on Saturday, 8 August, 2015 Permalink | Reply

Hi joseph, as you’re telling, I assume that you’re quite right. As I already mentioned, I heard this story in my classroom by my lecturer, that’s why I was pronouncing Dr. Hardy as Howdy haha, Pardon me for that. And that’s why I already stated you people knew this story :) . Thank you for sharing this info.

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• #### Joseph Nebus 4:27 am on Monday, 10 August, 2015 Permalink | Reply

Oh, good. I’m glad I could help match your recollection of an amazing person with the actual person.

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• #### elkement 10:53 am on Tuesday, 4 August, 2015 Permalink | Reply

Trying to answer your question: Learning theoretical physics I was conditioned to use the exponential function instead of cosine and sine, so I hardly used even those, let alone secant and cosecant (or sinh and cosh). I am in favor of using the lowest number of special functions as possible; and I think it also helps to to recognize that grand unification behind all that stuff.

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• #### Joseph Nebus 4:18 am on Saturday, 8 August, 2015 Permalink | Reply

Generically, I like using as few tools as possible. But that has to yield sometimes. It can be that you spend so much time getting from the primordial tool to the thing you actually mean to use that it’s counterproductive. The prime example there is how yes, you can do every common binary logical operation as a string of NAND operations. But that turns into obscurantism too easily.

So … sines and cosines. In analytical work it usually feels easier to me to use exponential functions. That’s surely because they make integration and differentiation such effortless things. But I don’t think I could do away with them, especially when studying the relationships between angles and arcs and lengths of things. I grant that I haven’t tried, though.

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## Reading the Comics, June 11, 2015: Bonus Education Edition

The coming US summer vacation suggests Comic Strip Master Command will slow down production of mathematics-themed comic strips. But they haven’t quite yet. And this week I also found a couple comics that, while not about mathematics, amused me enough that I want to include them anyway. So those bonus strips I’ll run at the end of my regular business here.

Bill Hinds’s Tank McNamara (June 6) does a pi pun. The pithon mathematical-snake idea is fun enough and I’d be interested in a character design. I think the strip’s unjustifiably snotty about tattoos. But comic strips have a strange tendency to get snotty about other forms of art.

A friend happened to mention one problem with tattoos that require straight lines or regular shapes is that human skin has a non-flat Gaussian curvature. Yes, that’s how the friend talks. Gaussian curvature is, well, a measure of how curved a surface is. That sounds obvious enough, but there are surprises: a circular cylinder, such as the label of a can, has the same curvature as a flat sheet of paper. You can see that by how easy it is to wrap a sheet of paper around a can. But a ball hasn’t, and you see that by how you can’t neatly wrap a sheet of paper around a ball without crumpling or tearing the paper. Human skin is kind of cylindrical in many places, but not perfectly so, and it changes as the body moves. So any design that looks good on paper requires some artistic imagination to adapt to the skin.

Bill Amend’s FoxTrot (June 7) sets Jason and Marcus working on their summer tans. It’s a good strip for adding to the cover of a trigonometry test as part of the cheat-sheet.

Dana Simpson’s Phoebe and her Unicorn (June 8) makes what I think is its first appearance in my Reading the Comics series. The strip, as a web comic, had been named Heavenly Nostrils. Then it got the vanishingly rare chance to run as a syndicated newspaper comic strip. And newspaper comics page editors don’t find the word “nostril” too inherently funny to pass up. Thus the more marketable name. After that interesting background I’m sad to say Simpson delivers a bog-standard “kids not understanding fractions” joke. I can’t say much about that.

Ruben Bolling’s Super Fun-Pak Comix (June 10, rerun) is an installment of everyone’s favorite literary device model of infinite probabilities. A Million Monkeys At A Million Typewriters subverts the model. A monkey thinking about the text destroys the randomness that it depends upon. This one’s my favorite of the mathematics strips this time around.

And Dan Thompson’s traditional Brevity appearance is the June 11th strip, an Anthropomorphic Numerals joke combining a traditional schoolyard gag with a pun I didn’t notice the first time I read the panel.

And now here’s a couple strips that aren’t mathematical but that I just liked too much to ignore. Also this lets Mark Anderson’s Andertoons get back on my page. The June 10th strip is a funny bit of grammar play.

Percy Crosby’s Skippy (June 6, rerun from sometime in 1928) tickles me for its point about what you get at the top and the bottom of the class. Although tutorials and office hours and extracurricular help, and automated teaching tools, do customize things a bit, teaching is ultimately a performance given to an audience. Some will be perfectly in tune with the performance, and some won’t. Audiences are like that.

• #### poetry and chocolate and books 9:29 pm on Thursday, 11 June, 2015 Permalink | Reply

Hey! I have nominated you for the quote challenge :) Check it here, please:
https://poetryandchocolateandbooks.wordpress.com/2015/06/11/if-you-are-a-quote-lover-day-1/

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• #### Joseph Nebus 4:14 am on Saturday, 13 June, 2015 Permalink | Reply

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• #### poetry and chocolate and books 6:35 am on Saturday, 13 June, 2015 Permalink | Reply

Okay! No problem :)

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• #### l1lyl4u 4:18 pm on Friday, 12 June, 2015 Permalink | Reply

I especially love Skippy! Thanks for these :)

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• #### ivasallay 2:24 pm on Saturday, 13 June, 2015 Permalink | Reply

I always enjoy Fox Trot especially when it has something to do with mathematics. That Andertoons strip was definitely worth sharing, too.

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• #### Joseph Nebus 2:19 am on Tuesday, 16 June, 2015 Permalink | Reply

FoxTrot is almost the leader in comic strips that are made for successful newspaper production while being as nerdy and mathematical as fans would like. It’s easy to suspect that Amend does that partly out of training — he was a physics major — although it’s just as easy to suppose he simply likes getting the mathematical stuff right because it makes the joke stronger.

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## My Math Blog Statistics, November 2014

October 2014 was my fourth-best month in the mathematics blog here, if by “best” we mean “has a number of page views”, and second-best if by “best” we mean “has a number of unique visitors”. And now November 2014 has taken October’s place on both counts, by having bigger numbers for both page views and visitors, as WordPress reveals such things to me. Don’t tell October; that’d just hurt its feelings. Plus, I got to the 19,000th page view, and as of right now I’m sitting at 19,181; it’s conceivable I might reach my 20,000th viewer this month, though that would be a slight stretch.

But the total number of page views grew from 625 up to 674, and the total number of visitors from 323 to 366. The number of page views is the highest since May 2014 (751), although this is the greatest number of visitors since January 2014 (473), the second month when WordPress started revealing those numbers to us mere bloggers. I like the trends, though; since June the number of visitors has been growing at a pretty steady rate, although steadily enough I can’t say whether it’s an arithmetic or geometric progression. (In an arithmetic progression, the difference between two successive numbers is about constant, for example: 10, 15, 20, 25, 30, 35, 40. In a geometric progression, the ratio between two successive numbers is about constant, for example: 10, 15, 23, 35, 53, 80, 120.) Views per visitor dropped from 1.93 to 1.84, although I’m not sure even that is a really significant difference.

The countries sending me the most readers were just about the same set as last month: the United States at 458; Canada recovering from a weak October with 27 viewers; Argentina at 20; Austria and the United Kingdom tied at 19; Australia at 17; Germany at 16 and Puerto Rico at 14.

Sending only one reader this month were: Belgium, Bermuda, Croatia, Estonia, Guatemala, Hong Kong, Italy, Lebanon, Malaysia, the Netherlands, Norway, Oman, the Philippines, Romania, Singapore, South Korea, and Sweden. (Really, Singapore? I’m a little hurt. I used to live there.) The countries repeating that from October were Estonia, the Netherlands, Norway, and Sweden; Sweden’s going on three months with just a single reader each. I don’t know what’s got me only slightly read in Scandinavia and the Balkans.

My most-read articles for November were pretty heavily biased towards the comics, with a side interest in that Pythagorean triangle problem with an inscribed circle. Elke Stangl had wondered about the longevity of my most popular posts, and I was curious too, so I’m including in brackets a note about the number of days between the first and the last view which WordPress has on record. This isn’t a perfect measure of longevity, especially for the most recent posts, but it’s a start.

As ever there’s no good search term poetry, but among the things that brought people here were:

• trapezoid
• how many grooves are on one side of an lp record?
• origin is the gateway to your entire gaming universe.
• cauchy funny things done
• trapezoid funny
• yet another day with no plans to use algebra

Won’t lie; that last one feels a little personal. But the “origin is the gateway” thing keeps turning up and I don’t know why. I’d try to search for it but that’d just bring me back here, leaving me no more knowledgeable, wouldn’t it?

## Reading the Comics, October 8, 2013

As promised, I’ve got a fresh round of mathematics-themed comic strips to discuss, something that’s rather fun to do because it offers such an easy answer to the question of what to write about today. Once you have the subject and a deadline the rest of the writing isn’t so very hard. So here’s some comics with all the humor safely buried in exposition:

Allison Barrows’s PreTeena (September 24, Rerun) brings the characters to “Performance Camp” and a pun on one of the basic tools of trigonometry. The pun’s routine enough, but I’m delighted to see that Barrows threw in a (correct) polynomial expression for the sine of an angle, since that’s the sort of detail that doesn’t really have to be included for the joke to read cleanly but which shows that Barrows made the effort to get it right.

Polynomial expansions — here, a Taylor series — are great tools to have, because, generally, polynomials are nice and well-behaved things. They’re easy to compute, they’re easy to analyze, they’re pretty much easy to do whatever you might want to do. Being able to shift a complicated or realistic function into a polynomial, even a polynomial with infinitely many terms, is often a big step towards making a complicated problem easy.

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