## What I Learned Doing The End 2016 Mathematics A To Z

The slightest thing I learned in the most recent set of essays is that I somehow slid from the descriptive “End Of 2016” title to the prescriptive “End 2016” identifier for the series. My unscientific survey suggests that most people would agree that we had too much 2016 and would have been better off doing without it altogether. So it goes.

The most important thing I learned about this is I have to pace things better. The A To Z essays have been creeping up in length. I didn’t keep close track of their lengths but I don’t think any of them came in under a thousand words. 1500 words was more common. And that’s fine enough, but at three per week, plus the Reading the Comics posts, that’s 5500 or 6000 words of mathematics alone. And that before getting to my humor blog, which even on a brief week will be a couple thousand words. I understand in retrospect why November and December felt like I didn’t have any time outside the word mines.

I’m not bothered by writing longer essays, mind. I can apparently go on at any length on any subject. And I like the words I’ve been using. My suspicion is between these A To Zs and the Theorem Thursdays over the summer I’ve found a mode for writing pop mathematics that works for me. It’s just a matter of how to balance workloads. The humor blog has gotten consistently better readership, for the obvious reasons (lately I’ve been trying to explain what the story comics are doing), but the mathematics more satisfying. If I should have to cut back on either it’d be the humor blog that gets the cut first.

Another little discovery is that I can swap out equations and formulas and the like for historical discussion. That’s probably a useful tradeoff for most of my readers. And it plays to my natural tendencies. It is very easy to imagine me having gone into history than into mathematics or science. It makes me aware how mediocre my knowledge of mathematics history is, though. For example, several times in the End 2016 A To Z the Crisis of Foundations came up, directly or in passing. But I’ve never read a proper history, not even a basic essay, about the Crisis. I don’t even know of a good description of this important-to-the-field event. Most mathematics history focuses around biographies of a few figures, often cribbed from Eric Temple Bell’s great but unreliable book, or a couple of famous specific incidents. (Newton versus Leibniz, the bridges of Köningsburg, Cantor’s insanity, Gödel’s citizenship exam.) Plus Bourbaki.

That’s not enough for someone taking the subject seriously, and I do mean to. So if someone has a suggestion for good histories of, for example, how Fourier series affected mathematicians’ understanding of what functions are, I’d love to know it. Maybe I should set that as a standing open request.

In looking over the subjects I wrote about I find a pretty strong mix of group theory and real analysis. Maybe that shouldn’t surprise. Those are two of the maybe three legs that form a mathematics major’s education. So anyone wanting to understand mathematicians would see this stuff and have questions about it. (There are more things mathematics majors learn, but there are a handful of things almost any mathematics major is sure to spend a year being baffled by.)

The third leg, I’d say, is differential equations. That’s a fantastic field, but it’s hard to describe without equations. Also pictures of what the equations imply. I’ve tended towards essays with few equations and pictures. That’s my laziness. Equations are best written in LaTeX, a typesetting tool that might as well be the standard for mathematicians writing papers and books. While WordPress supports a bit of LaTeX it isn’t quite effortless. That comes back around to balancing my workload. I do that a little better and I can explain solving first-order differential equations by integrating factors. (This is a prank. Nobody has ever needed to solve a first-order differential equation by integrating factors except for mathematics majors being taught the method.) But maybe I could make a go of that.

I’m not setting any particular date for the next A-To-Z, or similar, project. I need some time to recuperate. And maybe some time to think of other running projects that would be fun or educational for me. There’ll be something, though.

• #### ivasallay 6:28 pm on Thursday, 5 January, 2017 Permalink | Reply

I hope you’ll enjoy a short recuperation. Happy 2017, too.

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• #### Joseph Nebus 4:09 am on Friday, 13 January, 2017 Permalink | Reply

Thank you. I am hoping to be back to normal soon.

Also I’m glad to see you back and posting Find The Factors puzzles again! I was so glad to see them turning back up in my Twitter timeline.

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• #### mathtuition88 12:11 pm on Saturday, 7 January, 2017 Permalink | Reply

If you ever need to convert normal LaTeX to WordPress LaTeX, you are welcome to try my converter: http://mathtuition88.blogspot.sg/2015/12/latex-to-wordpress-converter.html

Happy 2017!

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• #### Joseph Nebus 4:12 am on Friday, 13 January, 2017 Permalink | Reply

Oh, yes! You had shared that and I am embarrassed to admit I’d forgotten. Thank you for mentioning it again.

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## Reading the Comics, April 15, 2016: Remarkably, No Income Tax Comics Edition

I’m as startled as you are. While a couple comic strips mentioned United States Income Tax Day, they didn’t do so in a way that seemed on-point enough for this Reading The Comics post. Of course, United States Income Tax Day happens to be the 18th this year. I haven’t seen Sunday’s comics yet.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of April one again uses arithmetic puns for its business. Also, if some science fiction writer doesn’t take hold of “Gribth” as a name for something they’re missing a fine syllable. “Tahew” is no slouch in the made-up word leagues either.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of April, 2016. The link will probably expire sometime before the year 2112.

Ryan North’s Dinosaur Comics for the 12th of April obviously originally ran sometime in mid-March. I have similarly ambiguous feelings about the value of Pi Day. I suppose it’s nice for people to think of “fun” and “mathematics” close together. Utahraptor’s distinction between “Pi Day” of March 14 and “Approximate Pi Day” of the 22nd of July s a curious one, though. It’s not as though 3.14 is any more exactly π than 22/7 is. I suppose you can argue that at some moment on 3/14 between 1:59:26 and 1:59:27 there’s some moment, 1:59:26.5358979 et cetera going on forever. But that assumes that time is a continuous thing, and it’s not like you’ll ever know what that moment is. By the time you might recognize it, it’s passed. They are all Approximate Pi Days; we just have to decide what the approximation is.

Bill Schorr’s The Grizzwells for the 12th is a silly-homework problem question. I know the point is to joke about how Fauna misunderstands a word. But if we pretend the assignment is for real, what might its point be? To show that students know the parts of a right triangle? I guess that’s all right, but it doesn’t seem like much of an assignment. I don’t blame her for getting snarky in the face of that.

Rick Kirkman and Jerry Scott’s Baby Blues for the 13th is a gag about picking random numbers for arithmetic homework. The approach is doomed, surely, although it’s probably not completely doomed. I’m not sure Hammie’s age, but if his homework is about adding and subtracting numbers he probably mostly gets problems that give results between zero and twenty, and almost always less than a hundred. He might hit some by luck.

Rick Kirkman and Jerry Scott’s Baby Blues for the 13th of April, 2016. It’s only after Hammy walks away that Zoe wonders why he needs five random numbers?

I’ve mentioned some how people are awful at picking “random” numbers in their heads. Zoe shows off one of the ways people are bad at it. People asked to name numbers “randomly” pick odd numbers more than even numbers. Somehow they just feel random. I doubt Kirkman and Scott were thinking of that; among other things, five numbers is a very small sample. Four odds out of five isn’t peculiar, not yet. They were probably just trying to pick numbers that sounded funny while fitting the space available. I’m a bit surprised 37 didn’t make the list.

Mark Anderson’s Andertoons for the 13th is Mark Anderson’s Andertoons entry for this essay. I like the teacher’s answer, though.

Patrick Roberts’s Todd the Dinosaur for the 14th just uses arithmetic as the most economic way to fit several problems on-screen at once. They’ve got a compactness that sentence-diagramming just can’t match.

Patrick Roberts’s Todd the Dinosaur for the 14th of April, 2016. No fair wondering why his more distant eye is always the larger one.

Greg Cravens’s The Buckets for the 15th amuses me with its use of coin-tossing as a way of making choices. I’m also amused the coin might be wrong only about half the time.

John Deering’s Strange Brew for the 15th is a visual puzzle. It’s intending to make use of a board full of mathematical symbols to represent deep thought. But the symbols aren’t quite mathematics. They look much more like LaTeX, a typesetting code used to express mathematics in print. Some of the symbols are obscured, so I can’t say exactly what’s meant. But it should be something like this:

$F = \{F_{x} \in F_{c}: (is ... (1) ) \cap (minPixels < \|s\| < maxPixels ) \\ \partial{P} \\ (is_{connected}| > |s| - \epsilon) \}$

At the risk of disappointing, this appears to me gibberish. The appearance of words like ‘minPixels’ and ‘maxPixels’ suggest a bit of computer code. So does having a subscript that’s the full word “connected”. I wonder where Deering drew this example from.

• #### Jacob Kanev 9:46 pm on Tuesday, 19 April, 2016 Permalink | Reply

Nice. On a totally unrelated note, my favourite comic about random numbers is from Dilbert, when he visits the accounting department: http://dilbert.com/strip/2001-10-25

Regards from Jacob.

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• #### Joseph Nebus 2:08 am on Friday, 22 April, 2016 Permalink | Reply

Ha ha! Thank you. That’s a strip I had forgotten. It’s true, though; it’s so very hard to say what randomness is, or really pin down whether we’ve ever seen it.

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• #### elkement (Elke Stangl) 12:50 pm on Wednesday, 27 April, 2016 Permalink | Reply

I wonder why dinosaurs are so popular as characters? ;-)

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• #### Joseph Nebus 6:47 pm on Friday, 29 April, 2016 Permalink | Reply

Oh, dinosaurs have a lot going for them. They’ve got a great visual style and there’s at least one to fit any mood you might have. I’m a little surprised there are so few comic strips that have them. But modern comic strips have a strange aversion to funny-looking characters.

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## How To Build Infinite Numbers

I had missed it, as mentioned in the above tweet. The link is to a page on the Form And Formalism blog, reprinting a translation of one of Georg Cantor’s papers in which he founded the modern understanding of sets, of infinite sets, and of infinitely large numbers. Although it gets into pretty heady topics, it doesn’t actually require a mathematical background, at least as I look at it; it just requires a willingness to follow long chains of reasoning, which I admit is much harder than algebra.

Cantor — whom I’d talked a bit about in a recent Reading The Comics post — was deeply concerned and intrigued by infinity. His paper enters into that curious space where mathematics, philosophy, and even theology blend together, since it’s difficult to talk about the infinite without people thinking of God. I admit the philosophical side of the discussion is difficult for me to follow, and the theological side harder yet, but a philosopher or theologian would probably have symmetric complaints.

The translation is provided as scans of a typewritten document, so you can see what it was like trying to include mathematical symbols in non-typeset text in the days before LaTeX (which is great at it, but requires annoying amounts of setup) or HTML (which is mediocre at it, but requires less setup) or Word (I don’t use Word) were available. Somehow, folks managed to live through times like that, but it wasn’t pretty.

• #### elkement 11:03 am on Sunday, 8 March, 2015 Permalink | Reply

I remember that stuff – as one of the most intriguing things I learned in the Linear Algebra class in the first semester.

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• #### Joseph Nebus 11:51 pm on Monday, 9 March, 2015 Permalink | Reply

Linear Algebra? I’m intrigued it was put in that course. In my curriculum they were fit into real analysis and mathematical logic instead.

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• #### elkement 10:59 am on Tuesday, 10 March, 2015 Permalink | Reply

It was somewhere in the same chapter / lecture as different types of sets, infinite sets, and Russell’s paradox of the set of all sets and related proof of the inherent contradition …

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• #### Joseph Nebus 7:59 pm on Thursday, 12 March, 2015 Permalink | Reply

Ah, I see. I wouldn’t have thought to connect the topics quite that way, although it’s possible I’m just thinking too heavily of how it happened to be done the semesters I took linear algebra, which were pretty heavily biased towards the sorts of matrix and vector space stuff that would be helpful in physics. Maybe I failed to read the chapters the professor chose to skip.

(I didn’t have much choice: I lost my textbook after the first exam and couldn’t buy or borrow a second copy. Luckily homeworks were assigned by actually writing out the problems, rather than just ‘Chapter 2.3 3-9 odds, 12, 14’, so I could keep up, but it was tougher than it needed to be. I’m not positive the professor wasn’t kind to me with my final grade, or whether having to pay extremely close attention to definitions and proofs in class was better for me than trusting I could check the details in the textbook later on.)

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• #### elkement 8:21 pm on Thursday, 12 March, 2015 Permalink | Reply

Yes, the lecture was mainly matrices, vector spaces, and tensors. The set of sets and Cantor’s diagonal argument etc. were mentioned in one of the first chapters if I recall correctly. Russell’s proof (or some version of it) required mapping elements of a set onto their power set (or something ;-)) so this was introduced right after surjective and injective linear maps.

I am too lazy now, but I could check – I still do have this textbook, but it is literally falling apart!

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• #### Joseph Nebus 3:17 am on Saturday, 14 March, 2015 Permalink | Reply

Ah, OK. Now I see where it’d fit naturally in with the way the instructor was leading the course. It wasn’t something I had expected but I do see how that makes sense.

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• #### elkement 8:57 pm on Thursday, 12 March, 2015 Permalink | Reply

… and as you speak about Real Analysis this is maybe the time to ask a perfectly stupid question, but blame it on differences in our educational systems and my ignorance thereof: When I was a student of physics, I had two math classes in the first year, Linear Algebra and Real Analysis (two semesters each, first and second of my “undergrad” studies, though we had no bachelor degrees back then, only masters – this was just the first year of five).
So I always thought “Calculus” = “Real Analysis”. But it isn’t, right?
I know this may sound dumb but I have tacitly made this assumption so often, so I admit my blunder publicly now :-)

Real analysis was mainly theorems and proofs, “building math from scratch”, series and functions, their properties – continuous, differentiable etc.
Is “Calculus” more about learning rules how to integrate and differentiate, but without all those detailed proofs? I started thinking about it when I read a book by a science writer (an English major) who tought herself calculus later. It seems it had not been mandatory in her high school. Then I’d understand why colleges would have to teach calculus to make sure everybody has the same background. I can remember I had a few colleagues who came from a high school not at all specialized in science. We have e.g. something like “business highschools”, with accounting classes and the like…. but those students were unlikely to pick a science degree at the university so perhaps nobody cared that they had a really hard time with that rigorous, proof-based math right from day 1.

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• #### Joseph Nebus 3:24 am on Saturday, 14 March, 2015 Permalink | Reply

I had to think about this one a bit, but I believe there is a subtle difference between Calculus and Real Analysis. Real Analysis is the study of real-valued functions — how to define them, how to use them, how to manipulate them. But the most interesting stuff to do with real-valued functions that you can teach with the sorts of proofs that new students can follow or reconstruct are generally the things that we get in intro calculus: finding maximums and minimums, finding derivatives, integrating, that sort of thing. So Real Analysis tends to look like “Intro Calculus, only this time you have to do the proofs”.

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## Some Facts For The Day

I’d just wanted to note the creation of another fact-of-the-day Twitter feed from the indefatigable John D Cook. This one is dubbed Unit Facts, and it’s aiming at providing information about where various units of measure come from. The first few days have begun with, naturally enough, the base units of the Metric System (can you name all seven?), and has stretched out already to things like what a knot is, how picas and inches are related, and what are ems and fortnights besides useful to know for crossword puzzles, or how something might be measured, as in the marshmallow tweet above.

Cook offers a number of interesting fact-of-the-day style feeds, which I believe are all linked to one another through their “Following” pages. These include algebra, topology, probability, and analysis facts of the day, as well as Unix tool tips, RegExp and TeX/LaTeX trivia, symbols (including a lot of Unicode and HTML entities), and the like. If you’re of the sort to get interested in neatly delivered bits of science- and math- and computer-related trivia, well, good luck with your imminent archive-binge.

## ASCII-Art Math

I had forgotten the challenges of doing more than the most basic mathematics expressions in ASCII art, and had completely forgotten there were tools that tried to make it a bit easier. Enteropia here’s put forth a script which ought to make it a bit easier to go from LaTeX into an ASCII representation, and I have the feeling I’m going to want to find this again later on, so I’d best make some kind of link I can locate when I do.

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Many of computer algebra systems date back to the times when GUI machines were rare and expensive, if were present at all. Thus command line was a standard interface. Unfortunately text terminal doesn’t fit very well for displaying mathematical expressions which demand for rich typesetting. To represent math formulas CAS’s resorted to some kind of ASCII art:

 inf 1 ==== / n n \ [ x log (x) > I ---------- dx / ] gamma(n x) ==== / n = 0…

That’s the output of Maxima. Some of the systems went further and don’t restrict themselves to plain ASCII. Axiom can produce such a nice output:

                         2
x   - %A  ┌──┐
┌┐  %e     \│%A
│   ──────────── d%A
└┘    tan(%A) + 2


Even now most CAS’s retain command line interface, for example Mathematica 8’s terminal session:

In[14]:= Pi*(a+b^2/(Exp[12]+3/2 ">2)) 2 b Out[14]= (a + -------) Pi 3 12 - + E…

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