## Reading the Comics, December 11, 2017: Vamping For Andertoons Edition

So Mark Anderson’s Andertoons has been missing from the list of mathematically-themed the last couple weeks. Don’t think I haven’t been worried about that. But it’s finally given another on-topic-enough strip and I’m not going to include it here. I’ve had a terrible week and I’m going to use the comics we got in last week slowly.

Hector D Cantu and Carlos Castellanos’s Baldo for the 10th of December uses algebra as the type for homework you’d need help with. It reads plausibly enough to me, at least so far as I remember learning algebra.

Greg Evans’s Luann Againn for the 10th reprints the strip of the 10th of December, 1989. And as often happens, mathematics is put up as the stuff that’s too hard to really do. The expressions put up don’t quite parse; there’s nothing to solve. But that’s fair enough for a panicked brain. To not recognize what the problem even is makes it rather hard to solve.

Ruben Bolling’s Super-Fun-Pak Comix for the 10th is an installation of Quantum Mechanic, playing on the most fun example of non-commutative processes I know. That’s the uncertainty principle, which expresses itself as pairs of quantities that can’t be precisely measured simultaneously. There are less esoteric kinds of non-commutative processes. Like, rotating something 90 degrees along a horizontal and then along a vertical axis will turn stuff different from 90 degrees vertical and then horizontal. But that’s too easy to understand to capture the imagination, at least until you’re as smart as an adult and as thoughtful as a child.

Maria Scrivan’s Half Full for the 11th features Albert Einstein and one of the few equations that everybody knows. So that’s something.

Jeff Stahler’s Moderately Confused for the 11th features the classic blackboard full of equations, this time to explain why Christmas lights wouldn’t work. There is proper mathematics in lights not working. It’s that electrical-engineering work about the flow of electricity. The problem is, typically, a broken or loose bulb. Maybe a burnt-out fuse, although I have never fixed a Christmas lights problem by replacing the fuse. It’s just something to do so you can feel like you’ve taken action before screaming in rage and throwing the lights out onto the front porch. More interesting to me is the mathematics of strands getting tangled. The idea — a foldable thread, marked at regular intervals by points that can hook together — seems trivially simple. But it can give insight into how long molecules, particularly proteins, will fold together. It may help someone frustrated to ponder that their light strands are knotted for the same reasons life can exist. But I’m not sure it ever does.

## What You Need To Pass This Class. Also: It’s Algebra, Uncle Fletcher

The end of the (US) semester snuck up on me but, in my defense, I’m not teaching this semester. If you know someone who needs me to teach, please leave me a note. But as a service for people who are just trying to figure out exactly how much studying they need to do for their finals, knock it off. You’re not playing a video game. It’s not like you can figure out how much effort it takes to get an 83.5 on the final and then put the rest of your energy into your major’s classes.

But it’s a question people ask, and keep asking, so here’s my answers. This essay describes exactly how to figure out what you need, given whatever grade you have and whatever extra credit you have and whatever the weighting of the final exam is and all that. That might be more mechanism than you need. If you’re content with an approximate answer, here’s some tables for common finals weightings, and a selection of pre-final grades.

For those not interested in grade-grubbing, here’s some old-time radio. Vic and Sade was a longrunning 15-minute morning radio program written with exquisite care by Paul Rhymer. It’s not going to be to everyone’s taste. But if it is yours, it’s going to be really yours: a tiny cast of people talking not quite past one another while respecting the classic Greek unities. Part of the Overnightscape Underground is the Vic and Sadecast, which curates episodes of the show, particularly trying to explain the context of things gone by since 1940. This episode, from October 1941, is aptly titled “It’s Algebra, Uncle Fletcher”. Neither Vic nor Sade are in the episode, but their son Rush and Uncle Fletcher are. And they try to work through high school algebra problems. I’m tickled to hear Uncle Fletcher explaining mathematics homework. I hope you are too.

## Reading the Comics, December 9, 2017: Zach Weinersmith Wants My Attention Edition

If anything dominated the week in mathematically-themed comic strips it was Zach Weinersmith’s Saturday Morning Breakfast Cereal. I don’t know how GoComics selects the strips to (re?)print on their site. But there were at least four that seemed on-point enough for me to mention. So, okay. He’s got my attention. What’s he do with it?

On the 3rd of December is a strip I can say is about conditional probability. The mathematician might be right that the chance someone will be murdered by a serial killer are less than one in ten million. But that is the chance of someone drawn from the whole universe of human experiences. There are people who will never be near a serial killer, for example, or who never come to his attention or who evade his interest. But if we know someone is near a serial killer, or does attract his interest? The information changes the probability. And this is where you get all those counter-intuitive and somewhat annoying logic puzzles about, like, the chance someone’s other child is a girl if the one who just walked in was, and how that changes if you’re told whether the girl who just entered was the elder.

On the 5th is a strip about sequences. And built on the famous example of exponential growth from doubling a reward enough times. Well, you know these things never work out for the wise guy. The “Fibonacci Spiral” spoken of in the next-to-last panel is a spiral, like you figure. The dimensions of the spiral are based on those of golden-ratio rectangles. It looks a great deal like a logarithmic spiral to the untrained eye. Also to the trained eye, but you knew that. I think it’s supposed to be humiliating that someone would call such a spiral “random”. But I admit I don’t get that part.

The strip for the 6th has a more implicit mathematical content. It hypothesizes that mathematicians, given the chance, will be more interested in doing recreational puzzles than even in eating and drinking. It’s amusing, but I’ll admit I’ve found very few puzzles all that compelling. This isn’t to say there aren’t problems I keep coming back to because I’m curious about them, just that they don’t overwhelm my common sense. Don’t ask me when I last received actual pay for doing something mathematical.

And then on the 9th is one more strip, about logicians. And logic puzzles, such as you might get in a Martin Gardner collection. The problem is written out on the chalkboard with some shorthand logical symbols. And they’re symbols both philosophers and mathematicians use. The letter that looks like a V with a crossbar means “for all”. (The mnemonic I got was “it’s an A-for-all, upside-down”. This paired with the other common symbol, which looks like a backwards E and means there exists: “E-for-exists, backwards”. Later I noticed upside-down A and backwards E could both be just 180-degree-rotated A and E. But try saying “180-degree-rotated” in a quick way.) The curvy E between the letters ‘x’ and ‘S’ means “belongs to the set”. So that first line says “for all x that belong to the set S this follows”. Writing out “isLiar(x)” instead of, say, “L(x)”, is more a philosopher’s thing than a mathematician’s. But it wouldn’t throw anyway. And the T just means emphasizing that this is true.

And that is as much about Saturday Morning Breakfast Cereal as I have to say this week.

Sam Hurt’s Eyebeam for the 4th tells a cute story about twins trying to explain infinity to one another. I’m not sure I can agree with the older twin’s assertion that infinity means there’s no biggest number. But that’s just because I worry there’s something imprecise going on there. I’m looking forward to the kids learning about negative numbers, though, and getting to wonder what’s the biggest negative real number.

Percy Crosby’s Skippy for the 4th starts with Skippy explaining a story problem. One about buying potatoes, in this case. I’m tickled by how cranky Skippy is about boring old story problems. Motivation is always a challenge. The strip originally ran the 7th of October, 1930.

Dave Whamond’s Reality Check for the 6th uses a panel of (gibberish) mathematics as an example of an algorithm. Algorithms are mathematical, in origin at least. The word comes to us from the 9th century Persian mathematician Al-Khwarizmi’s text about how to calculate. The modern sense of the word comes from trying to describe the methods by which a problem can be solved. So, legitimate use of mathematics to show off the idea. The symbols still don’t mean anything.

Rick Detorie’s One Big Happy for the 7th has Joe trying to get his mathematics homework done at the last minute. … And it’s caused me to reflect on how twenty multiplication problems seems like a reasonable number to do. But there’s only fifty multiplications to even do, at least if you’re doing the times tables up to the 10s. No wonder students get so bored seeing the same problems over and over. It’s a little less dire if you’re learning times tables up to the 12s, but not that much better. Yow.

Olivia Walch’s Imogen Quest for the 8th looks pretty legitimate to me. It’s going to read as gibberish to people who haven’t done parametric functions, though. Start with the plane and the familiar old idea of ‘x’ and ‘y’ representing how far one is along a horizontal and a vertical direction. Here, we’re given a dummy variable ‘t’, and functions to describe a value for ‘x’ and ‘y’ matching each value of ‘t’. The plot then shows all the points that ever match a pair of ‘x’ and ‘y’ coordinates for some ‘t’. The top drawing is a shape known as the cardioid, because it kind of looks like a Valentine-heart. The lower figure is a much more complicated parametric equation. It looks more anatomically accurate,

Still no sign of Mark Anderson’s Andertoons and the drought is worrying me, yes.

But they’re still going on the cartoonist’s web site, so there’s that.

## As I Try To Make Wronski’s Formula For Pi Into Something I Like

Previously:

I remain fascinated with Józef Maria Hoëne-Wronski’s attempted definition of π. It had started out like this:

$\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}$

And I’d translated that into something that modern mathematicians would accept without flinching. That is to evaluate the limit of a function that looks like this:

$\displaystyle \lim_{x \to \infty} f(x)$

where

$f(x) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{x}} - \left(1 - \imath\right)^{\frac{1}{x}} \right\}$

So. I don’t want to deal with that f(x) as it’s written. I can make it better. One thing that bothers me is seeing the complex number $1 + \imath$ raised to a power. I’d like to work with something simpler than that. And I can’t see that number without also noticing that I’m subtracting from it $1 - \imath$ raised to the same power. $1 + \imath$ and $1 - \imath$ are a “conjugate pair”. It’s usually nice to see those. It often hints at ways to make your expression simpler. That’s one of those patterns you pick up from doing a lot of problems as a mathematics major, and that then look like magic to the lay audience.

Here’s the first way I figure to make my life simpler. It’s in rewriting that $1 + \imath$ and $1 - \imath$ stuff so it’s simpler. It’ll be simpler by using exponentials. Shut up, it will too. I get there through Gauss, Descartes, and Euler.

At least I think it was Gauss who pointed out how you can match complex-valued numbers with points on the two-dimensional plane. On a sheet of graph paper, if you like. The number $1 + \imath$ matches to the point with x-coordinate 1, y-coordinate 1. The number $1 - \imath$ matches to the point with x-coordinate 1, y-coordinate -1. Yes, yes, this doesn’t sound like much of an insight Gauss had, but his work goes on. I’m leaving it off here because that’s all that I need for right now.

So these two numbers that offended me I can think of as points. They have Cartesian coordinates (1, 1) and (1, -1). But there’s never only one coordinate system for something. There may be only one that’s good for the problem you’re doing. I mean that makes the problem easier to study. But there are always infinitely many choices. For points on a flat surface like a piece of paper, and where the points don’t represent any particular physics problem, there’s two good choices. One is the Cartesian coordinates. In it you refer to points by an origin, an x-axis, and a y-axis. How far is the point from the origin in a direction parallel to the x-axis? (And in which direction? This gives us a positive or a negative number) How far is the point from the origin in a direction parallel to the y-axis? (And in which direction? Same positive or negative thing.)

The other good choice is polar coordinates. For that we need an origin and a positive x-axis. We refer to points by how far they are from the origin, heedless of direction. And then to get direction, what angle the line segment connecting the point with the origin makes with the positive x-axis. The first of these numbers, the distance, we normally label ‘r’ unless there’s compelling reason otherwise. The other we label ‘θ’. ‘r’ is always going to be a positive number or, possibly, zero. ‘θ’ might be any number, positive or negative. By convention, we measure angles so that positive numbers are counterclockwise from the x-axis. I don’t know why. I guess it seemed less weird for, say, the point with Cartesian coordinates (0, 1) to have a positive angle rather than a negative angle. That angle would be $\frac{\pi}{2}$, because mathematicians like radians more than degrees. They make other work easier.

So. The point $1 + \imath$ corresponds to the polar coordinates $r = \sqrt{2}$ and $\theta = \frac{\pi}{4}$. The point $1 - \imath$ corresponds to the polar coordinates $r = \sqrt{2}$ and $\theta = -\frac{\pi}{4}$. Yes, the θ coordinates being negative one times each other is common in conjugate pairs. Also, if you have doubts about my use of the word “the” before “polar coordinates”, well-spotted. If you’re not sure about that thing where ‘r’ is not negative, again, well-spotted. I intend to come back to that.

With the polar coordinates ‘r’ and ‘θ’ to describe a point I can go back to complex numbers. I can match the point to the complex number with the value given by $r e^{\imath\theta}$, where ‘e’ is that old 2.71828something number. Superficially, this looks like a big dumb waste of time. I had some problem with imaginary numbers raised to powers, so now, I’m rewriting things with a number raised to imaginary powers. Here’s why it isn’t dumb.

It’s easy to raise a number written like this to a power. $r e^{\imath\theta}$ raised to the n-th power is going to be equal to $r^n e^{\imath\theta \cdot n}$. (Because $(a \cdot b)^n = a^n \cdot b^n$ and we’re going to go ahead and assume this stays true if ‘b’ is a complex-valued number. It does, but you’re right to ask how we know that.) And this turns into raising a real-valued number to a power, which we know how to do. And it involves dividing a number by that power, which is also easy.

And we can get back to something that looks like $1 + \imath$ too. That is, something that’s a real number plus $\imath$ times some real number. This is through one of the many Euler’s Formulas. The one that’s relevant here is that $e^{\imath \phi} = \cos(\phi) + \imath \sin(\phi)$ for any real number ‘φ’. So, that’s true also for ‘θ’ times ‘n’. Or, looking to where everybody knows we’re going, also true for ‘θ’ divided by ‘x’.

OK, on to the people so anxious about all this. I talked about the angle made between the line segment that connects a point and the origin and the positive x-axis. “The” angle. “The”. If that wasn’t enough explanation of the problem, mention how your thinking’s done a 360 degree turn and you see it different now. In an empty room, if you happen to be in one. Your pedantic know-it-all friend is explaining it now. There’s an infinite number of angles that correspond to any given direction. They’re all separated by 360 degrees or, to a mathematician, 2π.

And more. What’s the difference between going out five units of distance in the direction of angle 0 and going out minus-five units of distance in the direction of angle -π? That is, between walking forward five paces while facing east and walking backward five paces while facing west? Yeah. So if we let ‘r’ be negative we’ve got twice as many infinitely many sets of coordinates for each point.

This complicates raising numbers to powers. θ times n might match with some point that’s very different from θ-plus-2-π times n. There might be a whole ring of powers. This seems … hard to work with, at least. But it’s, at heart, the same problem you get thinking about the square root of 4 and concluding it’s both plus 2 and minus 2. If you want “the” square root, you’d like it to be a single number. At least if you want to calculate anything from it. You have to pick out a preferred θ from the family of possible candidates.

For me, that’s whatever set of coordinates has ‘r’ that’s positive (or zero), and that has ‘θ’ between -π and π. Or between 0 and 2π. It could be any strip of numbers that’s 2π wide. Pick what makes sense for the problem you’re doing. It’s going to be the strip from -π to π. Perhaps the strip from 0 to 2π.

What this all amounts to is that I can turn this:

$f(x) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{x}} - \left(1 - \imath\right)^{\frac{1}{x}} \right\}$

into this:

$f(x) = -4 \imath x \left\{ \left(\sqrt{2} e^{\imath \frac{\pi}{4}}\right)^{\frac{1}{x}} - \left(\sqrt{2} e^{-\imath \frac{\pi}{4}} \right)^{\frac{1}{x}} \right\}$

without changing its meaning any. Raising a number to the one-over-x power looks different from raising it to the n power. But the work isn’t different. The function I wrote out up there is the same as this function:

$f(x) = -4 \imath x \left\{ \sqrt{2}^{\frac{1}{x}} e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} - \sqrt{2}^{\frac{1}{x}} e^{-\imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\}$

I can’t look at that number, $\sqrt{2}^{\frac{1}{x}}$, sitting there, multiplied by two things added together, and leave that. (OK, subtracted, but same thing.) I want to something something distributive law something and that gets us here:

$f(x) = -4 \imath x \sqrt{2}^{\frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} - e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\}$

Also, yeah, that square root of two raised to a power looks weird. I can turn that square root of two into “two to the one-half power”. That gets to this rewrite:

$f(x) = -4 \imath x 2^{\frac{1}{2}\cdot \frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} - e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\}$

And then. Those parentheses. e raised to an imaginary number minus e raised to minus-one-times that same imaginary number. This is another one of those magic tricks that mathematicians know because they see it all the time. Part of what we know from Euler’s Formula, the one I waved at back when I was talking about coordinates, is this:

$\sin\left(\phi\right) = \frac{e^{\imath \phi} - e^{-\imath \phi}}{2\imath }$

That’s good for any real-valued φ. For example, it’s good for the number $\frac{\pi}{4}\cdot\frac{1}{x}$. And that means we can rewrite that function into something that, finally, actually looks a little bit simpler. It looks like this:

$f(x) = -2 x 2^{\frac{1}{2}\cdot \frac{1}{x}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{x}\right)$

And that’s the function whose limit I want to take at ∞. No, really.

## How November 2017 Treated My Mathematics Blog

I was barely done sulking about the drop in readership on my humor blog when I started preparing the mathematics-blog readership report. And readership did drop from October (and September). Not by much, though. There were 1,052 pages viewed here in November 2017, barely less than October’s 1,069. It’s a fair bit under September’s 1,232, but that’s to be expected when I don’t have a strong gimmick going on like an A To Z project.

The number of unique visitors dropped, down to 604 from October’s 614, again a trivial difference. September had seen 672 visitors and that’s a more noticeable drop. Still not much, considering. The number of likes rose a bit, up to 70 from October’s 64. Still down from September’s 98. And all that still way down from, like, a year ago. I don’t know if it’s my shuffling off into irrelevance or if there’s something making likes less of a thing lately. I’d be curious other bloggers’ experience.

I’d started December with 55,419 page views from an estimated 25,617 known unique visitors, although some of them I only know apart because of their nametags.

So what countries have sent me readers? 56 of them, up from October’s 51 but down from September’s 65. How many sent me multiple readers? All but 22 of those. That’s up from October — 13 single-reader countries — and September — 20 single-reader countries — but these things happen. Here’s the full roster:

United States 676
United Kingdom 87
India 40
Philippines 26
Australia 15
Singapore 11
Brazil 9
Spain 9
Hong Kong SAR China 7
Belgium 6
Germany 6
Israel 6
Slovenia 6
Uruguay 6
European Union 5
France 5
South Africa 5
South Korea 5
Switzerland 5
Denmark 4
Italy 4
Sweden 4
Egypt 3
Iceland 3
Indonesia 3
Netherlands 3
Austria 2
Ireland 2
Lithuania 2
Poland 2
Qatar 2
United Arab Emirates 2
Algeria 1
Argentina 1
Azerbaijan 1
Estonia 1
Ethiopia 1
Japan 1 (*)
Kazakhstan 1
Kuwait 1
Lebanon 1
Maldives 1
Mexico 1
Norway 1
Oman 1
Peru 1
Portugal 1
Romania 1
Saudi Arabia 1
Slovakia 1
St. Kitts and Nevis 1
Thailand 1
Tunisia 1
Zimbabwe 1

Japan’s the only country to have sent me a single reader last month too, and no countries have sent me single readers more than two months in a row currently.

So that’s general popularity. What articles were popular around here? One traditional piece. Reading the Comics pieces. And the lovably misguided attempt by Józef Maria Hoëne-Wronski to give us a culturally neutral definition of π broke out to … well, second and third place, anyway:

Also, clearly, I need to think of more simple mathematically-answerable questions that everybody wonders since that record-side question is always popular. And when that is less popular, the question about how many kinds of trapezoid there are turns up.

Anyway, should you have read this and decided you want to be among my hundreds of WordPress followers who somehow don’t show up on the readership statistics, please, do so. There should be a ‘Follow on WordPress’ button in the upper right corner of the page. There’s also a ‘Follow by e-mail’ if you want things sent to an already-overfull box of things you haven’t time to read. Thank you.

## Reading the Comics, December 2, 2017: Showing Intelligence Edition

November closed out with another of those weeks not quite busy enough to justify splitting into two. I blame Friday and Saturday. Nothing mathematically-themed was happening them. Suppose some days are just like that.

Johnny Hart’s Back To BC for the 26th is an example of using mathematical truths as profound statements. I’m not sure that I’d agree with just stating the Pythagorean Theorem as profound, though. It seems like a profound statement has to have some additional surprising, revelatory elements to it. Like, knowing the Pythagorean theorem is true means we can prove there’s exactly one line parallel to a given line and passing through some point. Who’d see that coming? I don’t blame Hart for not trying to fit all that into one panel, though. Too slow a joke. The strip originally ran the 4th of September, 1960.

Tom Toles’s Randolph Itch, 2 am rerun for the 26th is a cute little arithmetic-in-real-life panel. I suppose arithmetic-in-real-life. Well, I’m amused and stick around for the footer joke. The strip originally ran the 24th of February, 2002.

Zach Weinersmith’s Saturday Morning Breakfast Cereal makes its first appearance for the week on the 26th. It’s an anthropomorphic-numerals joke and some wordplay. Interesting trivia about the whole numbers that never actually impresses people: a whole number is either a perfect square, like 1 or 4 or 9 or 16 are, or else its square root is irrational. There’s no whole number with a square root that’s, like, 7.745 or something. Maybe I just discuss it with people who’re too old. It seems like the sort of thing to reveal to a budding mathematician when she’s eight.

Saturday Morning Breakfast Cereal makes another appearance the 29th. The joke’s about using the Greek ε, which has a long heritage of use for “a small, positive number”. We use this all the time in analysis. A lot of proofs in analysis are done by using ε in a sort of trick. We want to show something is this value, but it’s too hard to do. Fine. Pick any ε, a positive number of unknown size. So then we’ll find something we can calculate, and show that the difference between the thing we want and the thing we can do is smaller than ε. And that the value of the thing we can calculate is that. Therefore, the difference between what we want and what we can do is smaller than any positive number. And so the difference between them must be zero, and voila! We’ve proved what we wanted to prove. I have always assumed that we use ε for this for the association with “error”, ideally “a tiny error”. If we need another tiny quantity we usually go to δ, probably because it’s close to ε and ‘d’ is still a letter close to ‘e’. (The next letter after ε is ζ, which carries other connotations with it and is harder to write than δ is.) Anyway, Weinersmith is just doing a ha-ha, your penis is small joke.

Samson’s Dark Side of the Horse for the 28th is a counting-sheep joke. It maybe doesn’t belong here but I really, really like the art of the final panel and I want people to see it.

Bud Grace’s Piranha Club for the 29th is, as with Back to BC, an attempt at showing intelligence through mathematics. There are some flaws in the system. Fun fact: since one million is a perfect square, Arnold could have answered within a single panel. (Also fun fact: I am completely unqualified to judge whether something is a “fun” fact.)

Jason Chatfield’s Ginger Meggs for the 29th is Ginger subverting the teacher’s questions, like so many teacher-and-student jokes will do.

Dan Thompson’s Brevity for the 30th is the anthropomorphic geometric figures joke for the week.

There seems to be no Mark Anderson’s Andertoons for this week. There’ve been some great ones (like on the 26th or the 28th and the 29th) but they’re not at all mathematical. I apologize for the inconvenience and am launching an investigation into this problem.

## Deciphering Wronski, Non-Standardly

I ran out of time to do my next bit on Wronski’s attempted definition of π. Next week, all goes well. But I have something to share anyway. William Lane Craig, of the The author of Boxing Pythagoras blog was intrigued by the starting point. And as a fan of studying how people understand infinity and infinitesimals (and how they don’t), this two-century-old example of mixing the numerous and the tiny set his course.

So here’s his essay, trying to work out Wronski’s beautiful weird formula from a non-standard analysis perspective. Non-standard analysis is a field that’s grown in the last fifty years. It’s probably fairly close in spirit to what (I think) Wronski might have been getting at, too. Non-standard analysis works with ideas that seem to match many people’s intuitive feelings about infinitesimals and infinities.

For example, can we speak of a number that’s larger than zero, but smaller than the reciprocal of any positive integer? It’s hard to imagine such a thing. But what if we can show that if we suppose such a number exists, then we can do this logically sound work with it? If you want to say that isn’t enough to show a number exists, then I have to ask how you know imaginary numbers or negative numbers exist.

Standard analysis, you probably guessed, doesn’t do that. It developed over the 19th century when the logical problems of these kinds of numbers seemed unsolvable. Mostly that’s done by limits, showing that a thing must be true whenever some quantity is small enough, or large enough. It seems safe to trust that the infinitesimally small is small enough, and the infinitely large is large enough. And it’s not like mathematicians back then were bad at their job. Mathematicians learned a lot of things about how infinitesimals and infinities work over the late 19th and early 20th century. It makes modern work possible.

Anyway, Boxing Pythagoras goes over what a non-standard analysis treatment of the formula suggests. I think it’s accessible even if you haven’t had much non-standard analysis in your background. At least it worked for me and I haven’t had much of the stuff. I think it’s also accessible if you’re good at following logical argument and won’t be thrown by Greek letters as variables. Most of the hard work is really arithmetic with funny letters. I recommend going and seeing if he did get to π.

## How Drunk Can We Expect The Greatest Generation Podcast Hosts To Get?

Among my entertainments is listening to the Greatest Generation podcast, hosted by Benjamin Ahr Harrison and Adam Pranica. They recently finished reviewing all the Star Trek: The Next Generation episodes, and have started Deep Space Nine. To add some fun and risk to episode podcasts the hosts proposed to record some episodes while drinking heavily. I am not a fun of recreational over-drinking, but I understand their feelings. There’s an episode where Quark has a sex-change operation because he gave his mother a heart attack right before a politically charged meeting with a leading Ferengi soda executive. Nobody should face that mess sober.

At the end of the episode reviewing “Babel”, Harrison proposed: there’s 15 episodes left in the season. Use a random number generator to pick a number from 1 to 15; if it’s one, they do the next episode (“Captive Pursuit”) drunk. And it was; what are the odds? One in fifteen. I just said.

The question: how many episodes would they be doing drunk? As they discussed in the next episode, this would imply they’d always get smashed for the last episode of the season. This is a straightforward expectation-value problem. The expectation value of a thing is the sum of all the possible outcomes times the chance of each outcome. Here, the possible outcome is adding 1 to the number of drunk episodes. The chance of any particular episode being a drunk episode is 1 divided by ‘N’, if ‘N’ is the number of episodes remaining. So the next-to-the-last episode has 1 chance in 2 of being drunk. The second-from-the-last has 1 chance in 3 of being drunk. And so on.

This expectation value isn’t hard to calculate. If we start counting from the last episode of the season, then it’s easy. Add up $1 + \frac12 + \frac13 + \frac14 + \frac15 + \frac16 + \cdots$, ending when we get up to one divided by the number of episodes in the season. 25 or 26, for most seasons of Deep Space Nine. 15, from when they counted here. This is the start of the harmonic series.

The harmonic series gets taught in sequences and series in calculus because it does some neat stuff if you let it go on forever. For example, every term in this sequence gets smaller and smaller. (The “sequence” is the terms that go into the sum: $1, \frac12, \frac13, \frac14, \frac{1}{1054}, \frac{1}{2038}$, and so on. The “series” is the sum of a sequence, a single number. I agree it seems weird to call a “series” that sum, but it’s the word we’re stuck with. If it helps, consider: when we talk about “a TV series” we usually mean the whole body of work, not individual episodes.) You can pick any number, however tiny you like. I can then respond with the last term in the sequence bigger than your number. Infinitely many terms in the sequence will be smaller than your pick. And yet: you can pick any number you like, however big. And I can take a finite number of terms in this sequence to make a sum bigger than whatever number you liked. The sum will eventually be bigger than 10, bigger than 100, bigger than a googolplex. These two facts are easy to prove, but they seem like they ought to be contradictory. You can see why infinite series are fun and produce much screaming on the part of students.

No Star Trek show has a season has infinitely many episodes, though, however long the second season of Enterprise seemed to drag out. So we don’t have to worry about infinitely many drunk episodes.

Since there were 15 episodes up for drunkenness in the first season of Deep Space Nine the calculation’s easy. I still did it on the computer. For the first season we could expect $1 + \frac12 + \frac13 + \cdots + \frac{1}{15}$ drunk episodes. This is a number a little bigger than 3.318. So, more likely three drunk episodes, four being likely. For the 25-episode seasons (seasons four and seven, if I’m reading this right), we could expect $1 + \frac12 + \frac13 + \cdots + \frac{1}{25}$ or just over 3.816 drunk episodes. Likely four, maybe three. For the 26-episode seasons (seasons two, five, and six), we could expect $1 + \frac12 + \frac13 + \cdots + \frac{1}{26}$ drunk episodes. That’s just over 3.854.

The number of drunk episodes to expect keeps growing. The harmonic series grows without bounds. But it keeps growing slower, compared to the number of terms you add together. You need a 31-episode season to be able to expect at four drunk episodes. To expect five drunk episodes you’d need an 83-episode season. If the guys at Worst Episode Ever, reviewing The Simpsons, did all 625-so-far episodes by this rule we could only expect seven drunk episodes.

Still, three, maybe four, drunk episodes of the 15 remaining first season is a fair number. They shouldn’t likely be evenly spaced. The chance of a drunk episode rises the closer they get to the end of the season. Expected length between drunk episodes is interesting but I don’t want to deal with that. I’ll just say that it probably isn’t the five episodes the quickest, easiest suggested by taking 15 divided by 3.

And it’s moot anyway. The hosts discussed it just before starting “Captive Pursuit”. Pranica pointed out, for example, the smashed-last-episode problem. What they decided they meant was there would be a 1-in-15 chance of recording each episode this season drunk. For the 25- or 26-episode seasons, each episode would get its 1-in-25 or 1-in-26 chance.

That changes the calculations. Not in spirit: that’s still the same. Count the number of possible outcomes and the chance of each one being a drunk episode and add that all up. But the work gets simpler. Each episode has a 1-in-15 chance of adding 1 to the total of drunk episodes. So the expected number of drunk episodes is the number of episodes (15) times the chance each is a drunk episode (1 divided by 15). We should expect 1 drunk episode. The same reasoning holds for all the other seasons; we should expect 1 drunk episode per season.

Still, since each episode gets an independent draw, there might be two drunk episodes. Could be three. There’s no reason that all 15 couldn’t be drunk. (Except that at the end of reviewing “Captive Pursuit” they drew for the next episode and it’s not to be a drunk one.) What are the chances there’s no drunk episodes? What are the chances there’s two, or three, or eight drunk episodes?

There’s a rule for this. This kind of problem is a mathematically-famous one. We get our results from the “binomial distribution”. This applies whenever there’s a bunch of attempts at something. And each attempt can either clearly succeed or clearly fail. And the chance of success (or failure) each attempt is always the same. That’s what applies here. If there’s ‘N’ episodes, and the chance is ‘p’ that any one will be drunk, then we get the chance ‘y’ of turning up exactly ‘k’ drunk episodes by the formula:

$y = \frac{N!}{k! \cdot \left(n - k\right)!} p^k \left(1 - p\right)^{n - k}$

That looks a bit ugly, yeah. (I don’t like using ‘y’ as the name for a probability. I ran out of good letters and didn’t want to do subscripts.) It’s just tedious to calculate is all. Factorials and everything. Better to let the computer work it out. There is a formula that’s easy enough to work with, though. That’s because the chance of a drunk episode is the same each episode. I don’t know a formula to get the chance of exactly zero or one or four drunk episodes with the first, one-in-N chance. Probably the only thing to do is run a lot of simulations and trust that’s approximately right.

But for this rule it’s easy enough. There’s this formula, like I said. I figured out the chance of all the possible drunk episode combinations for the seasons. I mean I had the computer work it out. All I figured out was how to make it give me the results in a format I liked. Here’s what I got.

The chance of these many drunk episodes In a 15-episode season is
0 0.355
1 0.381
2 0.190
3 0.059
4 0.013
5 0.002
6 0.000
7 0.000
8 0.000
9 0.000
10 0.000
11 0.000
12 0.000
13 0.000
14 0.000
15 0.000

Sorry it’s so dull, but the chance of a one-in-fifteen event happening 15 times in a row? You’d expect that to be pretty small. It’s got a probability of something like 0.000 000 000 000 000 002 28 of happening. Not technically impossible, but yeah, impossible.

How about for the 25- and 26-episode seasons? Here’s the chance of all the outcomes:

The chance of these many drunk episodes In a 25-episode season is
0 0.360
1 0.375
2 0.188
3 0.060
4 0.014
5 0.002
6 0.000
7 0.000
8 or more 0.000

And things are a tiny bit different for a 26-episode season.

The chance of these many drunk episodes In a 26-episode season is
0 0.361
1 0.375
2 0.188
3 0.060
4 0.014
5 0.002
6 0.000
7 0.000
7 0.000
8 or more 0.000

Yes, there’s a greater chance of no drunk episodes. The difference is really slight. It only looks so big because of rounding. A no-drunk 25 episode season has a chance of about 0.3604, while a no-drunk 26 episodes season has a chance of about 0.3607. The difference comes from the chance of lots of drunk episodes all being even worse somehow.

And there’s some neat implications through this. There’s a slightly better than one in three chance that each of the second through seventh seasons won’t have any drunk episodes. We could expect two dry seasons, hopefully not the one with Quark’s sex-change episode. We can reasonably expect at least one season with two drunk episodes. There’s a slightly more than 40 percent chance that some season will have three drunk episodes. There’s just under a 10 percent chance some season will have four drunk episodes.

There’s no guarantees, though. Probability has a curious blend. There’s no predicting when any drunk episode will come. But we can make meaningful predictions about groups of episodes. These properties seem like they should be contradictions. And they’re not, and that’s wonderful.

## Reading the Comics, November 25, 2017: Shapes and Probability Edition

This week was another average-grade week of mathematically-themed comic strips. I wonder if I should track them and see what spurious correlations between events and strips turn up. That seems like too much work and there’s better things I could do with my time, so it’s probably just a few weeks before I start doing that.

Ruben Bolling’s Super-Fun-Pax Comics for the 19th is an installment of A Voice From Another Dimension. It’s in that long line of mathematics jokes that are riffs on Flatland, and how we might try to imagine spaces other than ours. They’re taxing things. We can understand some of the rules of them perfectly well. Does that mean we can visualize them? Understand them? I’m not sure, and I don’t know a way to prove whether someone does or does not. This wasn’t one of the strips I was thinking of when I tossed “shapes” into the edition title, but you know what? It’s close enough to matching.

Olivia Walch’s Imogen Quest for the 20th — and I haven’t looked, but it feels to me like I’m always featuring Imogen Quest lately — riffs on the Monty Hall Problem. The problem is based on a game never actually played on Monty Hall’s Let’s Make A Deal, but very like ones they do. There’s many kinds of games there, but most of them amount to the contestant making a choice, and then being asked to second-guess the choice. In this case, pick a door and then second-guess whether to switch to another door. The Monty Hall Problem is a great one for Internet commenters to argue about while the rest of us do something productive. The trouble — well, one trouble — is that whether switching improves your chance to win the car is that whether it does depends on the rules of the game. It’s not stated, for example, whether the host must open a door showing a goat behind it. It’s not stated that the host certainly knows which doors have goats and so chooses one of those. It’s not certain the contestant even wants a car when, hey, goats. What assumptions you make about these issues affects the outcome.

If you take the assumptions that I would, given the problem — the host knows which door the car’s behind, and always offers the choice to switch, and the contestant would rather have a car, and such — then Walch’s analysis is spot on.

Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 20th features a pretend virtual reality arithmetic game. The strip is of incredibly low mathematical value, but it’s one of those comics I like that I never hear anyone talking about, so, here.

Richard Thompson’s Cul de Sac rerun for the 20th talks about shapes. And the names for shapes. It does seem like mathematicians have a lot of names for slightly different quadrilaterals. In our defense, if you’re talking about these a lot, it helps to have more specific names than just “quadrilateral”. Rhomboids are those parallelograms which have all four sides the same length. A parallelogram has to have two pairs of equal-sized legs, but the two pairs’ sizes can be different. Not so a rhombus. Mathworld says a rhombus with a narrow angle that’s 45 degrees is sometimes called a lozenge, but I say they’re fibbing. They make even more preposterous claims on the “lozenge” page.

Todd Clark’s Lola for the 20th does the old “when do I need to know algebra” question and I admit getting grumpy like this when people ask. Do French teachers have to put up with this stuff?

Brian Fies’s Mom’s Cancer rerun for the 23rd is from one of the delicate moments in her story. Fies’s mother just learned the average survival rate for her cancer treatment is about five percent and, after months of things getting haltingly better, is shaken. But as with most real-world probability questions context matters. The five-percent chance is, as described, the chance someone who’d just been diagnosed in the state she’d been diagnosed in would survive. The information that she’s already survived months of radiation and chemical treatment and physical therapy means they’re now looking at a different question. What is the chance she will survive, given that she has survived this far with this care?

Mark Anderson’s Andertoons for the 24th is the Mark Anderson’s Andertoons for the week. It’s a protesting-student kind of joke. For the student’s question, I’m not sure how many sides a polygon has before we can stop memorizing them. I’d say probably eight. Maybe ten. Of the shapes whose names people actually care about, mm. Circle, triangle, a bunch of quadrilaterals, pentagons, hexagons, octagons, maybe decagon and dodecagon. No, I’ve never met anyone who cared about nonagons. I think we could drop heptagons without anyone noticing either. Among quadrilaterals, ugh, let’s see. Square, rectangle, rhombus, parallelogram, trapezoid (or trapezium), and I guess diamond although I’m not sure what that gets you that rhombus doesn’t already. Toss in circles, ellipses, and ovals, and I think that’s all the shapes whose names you use.

Stephan Pastis’s Pearls Before Swine for the 25th does the rounding-up joke that’s been going around this year. It’s got a new context, though.

## As I Try To Figure Out What Wronski Thought ‘Pi’ Was

A couple weeks ago I shared a fascinating formula for π. I got it from Carl B Boyer’s The History of Calculus and its Conceptual Development. He got it from Józef Maria Hoëne-Wronski, early 19th-century Polish mathematician. His idea was that an absolute, culturally-independent definition of π would come not from thinking about circles and diameters but rather this formula:

$\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}$

Now, this formula is beautiful, at least to my eyes. It’s also gibberish. At least it’s ungrammatical. Mathematicians don’t like to write stuff like “four times infinity”, at least not as more than a rough draft on the way to a real thought. What does it mean to multiply four by infinity? Is arithmetic even a thing that can be done on infinitely large quantities? Among Wronski’s problems is that they didn’t have a clear answer to this. We’re a little more advanced in our mathematics now. We’ve had a century and a half of rather sound treatment of infinitely large and infinitely small things. Can we save Wronski’s work?

Start with the easiest thing. I’m offended by those $\sqrt{-1}$ bits. Well, no, I’m more unsettled by them. I would rather have $\imath$ in there. The difference? … More taste than anything sound. I prefer, if I can get away with it, using the square root symbol to mean the positive square root of the thing inside. There is no positive square root of -1, so, pfaugh, away with it. Mere style? All right, well, how do you know whether those $\sqrt{-1}$ terms are meant to be $\imath$ or its additive inverse, $-\imath$? How do you know they’re all meant to be the same one? See? … As with all style preferences, it’s impossible to be perfectly consistent. I’m sure there are times I accept a big square root symbol over a negative or a complex-valued quantity. But I’m not forced to have it here so I’d rather not. First step:

$\pi = \frac{4\infty}{\imath}\left\{ \left(1 + \imath\right)^{\frac{1}{\infty}} - \left(1 - \imath\right)^{\frac{1}{\infty}} \right\}$

Also dividing by $\imath$ is the same as multiplying by $-\imath$ so the second easy step gives me:

$\pi = -4 \imath \infty \left\{ \left(1 + \imath\right)^{\frac{1}{\infty}} - \left(1 - \imath\right)^{\frac{1}{\infty}} \right\}$

Now the hard part. All those infinities. I don’t like multiplying by infinity. I don’t like dividing by infinity. I really, really don’t like raising a quantity to the one-over-infinity power. Most mathematicians don’t. We have a tool for dealing with this sort of thing. It’s called a “limit”.

Mathematicians developed the idea of limits over … well, since they started doing mathematics. In the 19th century limits got sound enough that we still trust the idea. Here’s the rough way it works. Suppose we have a function which I’m going to name ‘f’ because I have better things to do than give functions good names. Its domain is the real numbers. Its range is the real numbers. (We can define functions for other domains and ranges, too. Those definitions look like what they do here.)

I’m going to use ‘x’ for the independent variable. It’s any number in the domain. I’m going to use ‘a’ for some point. We want to know the limit of the function “at a”. ‘a’ might be in the domain. But — and this is genius — it doesn’t have to be. We can talk sensibly about the limit of a function at some point where the function doesn’t exist. We can say “the limit of f at a is the number L”. I hadn’t introduced ‘L’ into evidence before, but … it’s a number. It has some specific set value. Can’t say which one without knowing what ‘f’ is and what its domain is and what ‘a’ is. But I know this about it.

Pick any error margin that you like. Call it ε because mathematicians do. However small this (positive) number is, there’s at least one neighborhood in the domain of ‘f’ that surrounds ‘a’. Check every point in that neighborhood other than ‘a’. The value of ‘f’ at all those points in that neighborhood other than ‘a’ will be larger than L – ε and smaller than L + ε.

Yeah, pause a bit there. It’s a tricky definition. It’s a nice common place to crash hard in freshman calculus. Also again in Intro to Real Analysis. It’s not just you. Perhaps it’ll help to think of it as a kind of mutual challenge game. Try this.

1. You draw whatever error bar, as big or as little as you like, around ‘L’.
2. But I always respond by drawing some strip around ‘a’.
3. You then pick absolutely any ‘x’ inside my strip, other than ‘a’.
4. Is f(x) always within the error bar you drew?

Suppose f(x) is. Suppose that you can pick any error bar however tiny, and I can answer with a strip however tiny, and every single ‘x’ inside my strip has an f(x) within your error bar … then, L is the limit of f at a.

Again, yes, tricky. But mathematicians haven’t found a better definition that doesn’t break something mathematicians need.

To write “the limit of f at a is L” we use the notation:

$\displaystyle \lim_{x \to a} f(x) = L$

The ‘lim’ part probably makes perfect sense. And you can see where ‘f’ and ‘a’ have to enter into it. ‘x’ here is a “dummy variable”. It’s the falsework of the mathematical expression. We need some name for the independent variable. It’s clumsy to do without. But it doesn’t matter what the name is. It’ll never appear in the answer. If it does then the work went wrong somewhere.

What I want to do, then, is turn all those appearances of ‘∞’ in Wronski’s expression into limits of something at infinity. And having just said what a limit is I have to do a patch job. In that talk about the limit at ‘a’ I talked about a neighborhood containing ‘a’. What’s it mean to have a neighborhood “containing ∞”?

The answer is exactly what you’d think if you got this question and were eight years old. The “neighborhood of infinity” is “all the big enough numbers”. To make it rigorous, it’s “all the numbers bigger than some finite number that let’s just call N”. So you give me an error bar around ‘L’. I’ll give you back some number ‘N’. Every ‘x’ that’s bigger than ‘N’ has f(x) inside your error bars. And note that I don’t have to say what ‘f(∞)’ is or even commit to the idea that such a thing can be meaningful. I only ever have to think directly about values of ‘f(x)’ where ‘x’ is some real number.

So! First, let me rewrite Wronski’s formula as a function, defined on the real numbers. Then I can replace each ∞ with the limit of something at infinity and … oh, wait a minute. There’s three ∞ symbols there. Do I need three limits?

Ugh. Yeah. Probably. This can be all right. We can do multiple limits. This can be well-defined. It can also be a right pain. The challenge-and-response game needs a little modifying to work. You still draw error bars. But I have to draw multiple strips. One for each of the variables. And every combination of values inside all those strips has give an ‘f’ that’s inside your error bars. There’s room for great mischief. You can arrange combinations of variables that look likely to break ‘f’ outside the error bars.

So. Three independent variables, all taking a limit at ∞? That’s not guaranteed to be trouble, but I’d expect trouble. At least I’d expect something to keep the limit from existing. That is, we could find there’s no number ‘L’ so that this drawing-neighborhoods thing works for all three variables at once.

Let’s try. One of the ∞ will be a limit of a variable named ‘x’. One of them a variable named ‘y’. One of them a variable named ‘z’. Then:

$f(x, y, z) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{y}} - \left(1 - \imath\right)^{\frac{1}{z}} \right\}$

Without doing the work, my hunch is: this is utter madness. I expect it’s probably possible to make this function take on many wildly different values by the judicious choice of ‘x’, ‘y’, and ‘z’. Particularly ‘y’ and ‘z’. You maybe see it already. If you don’t, you maybe see it now that I’ve said you maybe see it. If you don’t, I’ll get there, but not in this essay. But let’s suppose that it’s possible to make f(x, y, z) take on wildly different values like I’m getting at. This implies that there’s not any limit ‘L’, and therefore Wronski’s work is just wrong.

Thing is, Wronski wouldn’t have thought that. Deep down, I am certain, he thought the three appearances of ∞ were the same “value”. And that to translate him fairly we’d use the same name for all three appearances. So I am going to do that. I shall use ‘x’ as my variable name, and replace all three appearances of ∞ with the same variable and a common limit. So this gives me the single function:

$f(x) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{x}} - \left(1 - \imath\right)^{\frac{1}{x}} \right\}$

And then I need to take the limit of this at ∞. If Wronski is right, and if I’ve translated him fairly, it’s going to be π. Or something easy to get π from.

I hope to get there next week.

## When Is Thanksgiving Most Likely To Happen?

I thought I had written this up. Which is good because I didn’t want to spend the energy redoing these calculations.

The date of Thanksgiving, as observed in the United States, is that it’s the fourth Thursday of November. So it might happen anytime from the 22nd through the 28th. But because of the quirks of the Gregorian calendar, it can happen that a particular date, like the 23rd of November, is more or less likely to be a Thursday than some other day of the week.

So here’s the results of what days are most and least likely to be Thanksgiving. It turns out the 23rd, this year’s candidate, is tied for the rarest of Thanksgiving days. It’s not that rare, in comparison. It happens only two fewer times every 400 years than do Thanksgivings on the 22nd of November, the (tied) most common day.

## Reading the Comics, November 18, 2017: Story Problems and Equation Blackboards Edition

It was a normal-paced week at Comic Strip Master Command. It was also one of those weeks that didn’t have anything from Comics Kingdom or Creators.Com. So I’m afraid you’ll all just have to click the links for strips you want to actually see. Sorry.

Bill Amend’s FoxTrot for the 12th has Jason and Marcus creating “mathic novels”. They, being a couple of mathematically-gifted smart people, credit mathematics knowledge with smartness. A “chiliagon” is a thousand-sided regular polygon that’s mostly of philosophical interest. A regular polygon with a thousand equal sides and a thousand equal angles looks like a circle. There’s really no way to draw one so that the human eye could see the whole figure and tell it apart from a circle. But if you can understand the idea of a regular polygon it seems like you can imagine a chilagon and see how that’s not a circle. So there’s some really easy geometry things that can’t be visualized, or at least not truly visualized, and just have to be reasoned with.

Rick Detorie’s One Big Happy for the 12th is a story-problem-subversion joke. The joke’s good enough as it is, but the supposition of the problem is that the driving does cover fifty miles in an hour. This may not be the speed the car travels at the whole time of the problem. Mister Green is maybe speeding to make up for all the time spent travelling slower.

Brandon Sheffield and Dami Lee’s Hot Comics for Cool People for the 13th uses a blackboard full of equations to represent the deep thinking being done on a silly subject.

Shannon Wheeler’s Too Much Coffee Man for the 15th also uses a blackboard full of equations to represent the deep thinking being done on a less silly subject. It’s a really good-looking blackboard full of equations, by the way. Beyond the appearance of our old friend E = mc2 there’s a lot of stuff that looks like legitimate quantum mechanics symbols there. They’re at least not obvious nonsense, as best I can tell without the ability to zoom the image in. I wonder if Wheeler didn’t find a textbook and use some problems from it for the feeling of authenticity.

Samson’s Dark Side of the Horse for the 16th is a story-problem subversion joke.

Jef Mallett’s Frazz for the 18th talks about making a bet on the World Series, which wrapped up a couple weeks ago. It raises the question: can you bet on an already known outcome? Well, sure, you can bet on anything you like, given a willing partner. But there does seem to be something fundamentally different between betting on something whose outcome isn’t in principle knowable, such as the winner of the next World Series, and betting on something that could be known but happens not to be, such as the winner of the last. We see this expressed in questions like “is it true the 13th of a month is more likely to be Friday than any other day of the week?” If you know which month and year is under discussion the chance the 13th is Friday is either 1 or 0. But we mean something more like, if we don’t know what month and year it is, what’s the chance this is a month with a Friday the 13th? Something like this is at work in this World Series bet. (The Astros won the recently completed World Series.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is also featured on some underemployed philosopher’s “Reading the Comics” WordPress blog and fair enough. Utilitarianism exists in an odd triple point, somewhere on the borders of ethics, economics, and mathematics. The idea that one could quantize the good or the utility or the happiness of society, and study how actions affect it, is a strong one. It fits very well the modern mindset that holds everything can be quantified even if we don’t know how to do it well just yet. And it appeals strongly to a mathematically-minded person since it sounds like pure reason. It’s not, of course, any more than any ethical scheme can be. But it sounds like the ethics a Vulcan would come up with and that appeals to a certain kind of person. (The comic is built on one of the implications of utilitarianism that makes it seem like the idea’s gone off the rails.)

There’s some mathematics symbols on The Utilitarian’s costume. The capital U on his face is probably too obvious to need explanation. The $\sum u$ on his chest relies on some mathematical convention. For maybe a half-millennium now mathematicians have been using the capital sigma to mean “take a sum of things”. The things are whatever the expression after that symbol is. Usually, the Sigma will have something below and above which carries meaning. It says what the index is for the thing after the symbol, and what the bounds of the index are. Here, it’s not set. This is common enough, though, if this is understood from context. Or if it’s obvious. The small ‘u’ to the right suggests the utility of whatever’s thought about. (“Utility” being the name for the thing measured and maximized; it might be happiness, it might be general well-being, it might be the number of people alive.) So the symbols would suggest “take the sum of all the relevant utilities”. Which is the calculation that would be done in this case.

## Reading the Comics, November 11, 2017: Pictured Comics Edition

And now the other half of last week’s comic strips. It was unusually rich in comics that come from Comics Kingdom or Creators.com, which have limited windows of access and therefore make me feel confident I should include the strips so my comments make any sense.

Rick Kirkman and Jerry Scott’s Baby Blues for the 9th mentions mathematics homework as a resolutely rage-inducing topic. It’s mathematics homework, obviously, or else it wouldn’t be mentioned around here. And even more specifically it’s Common Core mathematics homework. So it always is with attempts to teach subjects better. Especially mathematics, given how little confidence people have in their own mastery. I can’t blame parents for supposing any change to be just malice.

Bill Amend’s FoxTrot Classics for the 9th is about random numbers. As Jason says, it is hard to generate random numbers. Random numbers are a resource. Having a good source of them makes a lot of computation work. But they’re hard to make. It seems to be a contradiction to create random numbers by an algorithm. There’s reasons we accept pseudorandom numbers, or find quasirandom numbers. This strip originally ran the 16th of November, 2006.

Chris Browne’s Hagar the Horrible for the 10th is about the numerous. There’s different kinds of limits. There’s the greatest number of things we can count in an instant. There’s a limit to how long a string of digits or symbols we can remember. There’s the biggest number of things we can visualize. And “visualize” is a slippery concept. I think I have a pretty good idea what we mean when we say “a thousand” of something. I could calculate how long it took me to do something a thousand times, or to write a thousand of something. I know that it was at about a thousand words that, last A To Z sequence, I got to feeling I should wrap up any particular essay. But did I see any particular difference between word 999 and word 1,000? No; what I really knew was “about enough paragraphs” and maybe “fills just over two screens in my text editor”. So do I know what a thousand is? Anyway, we all have our limits, acknowledge them or not.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 17th is about Moose’s struggle with mathematics. Just writing “more or less” doesn’t fix an erroneous answer, true. But error margins, and estimates of where an answer should be, can be good mathematics. (Part of the Common Core that many parents struggle with is making the estimate of an answer the first step, and a refined answer later. Based on what I see crossing social media, this really offends former engineering majors who miss the value in having an expected approximate answer.) It’s part of how we define limits, and derivatives, and integrals, and all of calculus. But it’s in a more precise way than Moose tries to do.

Ted Shearer’s Quincy for the 18th of September, 1978 is a story-problem joke. Some of these aren’t complicated strips.

## Reading the Comics, November 8, 2017: Uses Of Mathematics Edition

Was there an uptick in mathematics-themed comic strips in the syndicated comics this past week? It depends how tight a definition of “theme” you use. I have enough to write about that I’m splitting the week’s load. And I’ve got a follow-up to that Wronski post the other day, so I’m feeling nice and full of content right now. So here goes.

Zach Weinersmith’s Saturday Morning Breakfast Cereal posted the 5th gets my week off to an annoying start. Science and mathematics and engineering people have a tendency to be smug about their subjects. And to see aptitude or interest in their subjects as virtue, or at least intelligence. (If they see a distinction between virtue and intelligence.) To presume that an interest in the field I like is a demonstration of intelligence is a pretty nasty and arrogant move.

And yes, I also dislike the attitude that school should be about training people. Teaching should be about letting people be literate with the great thoughts people have had. Mathematics has a privileged spot here. The field, as we’ve developed it, seems to build on human aptitudes for number and space. It’s easy to find useful sides to it. Doesn’t mean it’s vocational training.

Lincoln Peirce’s Big Nate on the 6th discovered mathematics puzzles. And this gave him the desire to create a new mathematical puzzle that he would use to get rich. Good luck with that. Coming up with interesting enough recreational mathematics puzzles is hard. Presenting it in a way that people will buy is another, possibly greater, challenge. It takes luck and timing and presentation, just as getting a hit song does. Sudoku, for example, spent five years in the Dell Magazine puzzle books before getting a foothold in Japanese newspapers. And then twenty years there before being noticed in the English-speaking puzzle world. Big Nate’s teacher tries to encourage him, although that doesn’t go as Mr Staples might have hoped. (The storyline continues to the 11th. Spoiler: Nate does not invent the next great recreational mathematics puzzle.)

Jef Mallett’s Frazz for the 7th start out in a mathematics class, at least. I suppose the mathematical content doesn’t matter, though. Mallett’s making a point about questions that, I confess, I’m not sure I get. I’ll leave it for wiser heads to understand.

Mike Thompson’s Grand Avenue for the 8th is a subverted word-problem joke. And I suppose a reminder about the need for word problems to parse as things people would do, or might be interested in. I can’t go along with characterizing buying twelve candy bars “gluttonous” though. Not if they’re in a pack of twelve or something like that. I may be unfair to Grand Avenue. Mind, until a few years ago I was large enough my main method of getting around was “being rolled by Oompa-Loompas”, so I could be a poor judge.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 8th does a rounding joke. It’s not much, but I’ve included appearances of this joke before and it seems unfair to skip it this time.

## What Only One Person Ever Has Thought ‘Pi’ Means, And Who That Was

I’ve been reading Carl B Boyer’s The History of Calculus and its Conceptual Development. It’s been slow going, because reading about how calculus’s ideas developed is hard. The ideas underlying it are subtle to start with. And the ideas have to be discussed using vague, unclear definitions. That’s not because dumb people were making arguments. It’s because these were smart people studying ideas at the limits of what we understood. When we got clear definitions we had the fundamentals of calculus understood. (By our modern standards. The future will likely see us as accepting strange ambiguities.) And I still think Boyer whiffs the discussion of Zeno’s Paradoxes in a way that mathematics and science-types usually do. (The trouble isn’t imagining that infinite series can converge. The trouble is that things are either infinitely divisible or they’re not. Either way implies things that seem false.)

Anyway. Boyer got to a part about the early 19th century. This was when mathematicians were discovering infinities and infinitesimals are amazing tools. Also that mathematicians should maybe learn if they follow any rules. Because you can just plug symbols in to formulas and grind out what looks like they might mean and get answers. Sometimes this works great. Grind through the formulas for solving cubic polynomials as though square roots of negative numbers make sense. You get good results. Later, we worked out a coherent scheme of “complex-valued numbers” that justified it all. We can get lucky with infinities and infinitesimals, sometimes.

And this brought Boyer to an argument made by Józef Maria Hoëne-Wronski. He was a Polish mathematician whose fantastic ambition in … everything … didn’t turn out many useful results. Algebra, the Longitude Problem, building a rival to the railroad, even the Kosciuszko Uprising, none quite panned out. (And that’s not quite his name. The ‘n’ in ‘Wronski’ should have an acute mark over it. But WordPress’s HTML engine doesn’t want to imagine such a thing exists. Nor do many typesetters writing calculus or differential equations books, Boyer’s included.)

But anyone who studies differential equations knows his name, for a concept called the Wronskian. It’s a matrix determinant that anyone who studies differential equations hopes they won’t ever have to do after learning it. And, says Boyer, Wronski had this notion for an “absolute meaning of the number π”. (By “absolute” Wronski means one that not drawn from cultural factors like the weird human interset in circle perimeters and diameters. Compare it to the way we speak of “absolute temperature”, where the zero means something not particular to western European weather.)

$\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} - \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}$

Well.

I will admit I’m not fond of “real” alternate definitions of π. They seem to me mostly to signal how clever the definition-originator is. The only one I like at all defines π as the smallest positive root of the simple-harmonic-motion differential equation. (With the right starting conditions and all that.) And I’m not sure that isn’t “circumference over diameter” in a hidden form.

And yes, that definition is a mess of early-19th-century wild, untamed casualness in the use of symbols. But I admire the crazypants beauty of it. If I ever get a couple free hours I should rework it into something grammatical. And then see if, turned into something tolerable, Wronski’s idea is something even true.

Boyer allows that “perhaps” because of the strange notation and “bizarre use of the symbol ∞” Wronski didn’t make much headway on this point. I can’t fault people for looking at that and refusing to go further. But isn’t it enchanting as it is?

## Reading the Comics, November 4, 2017: Slow, Small Week Edition

It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.

Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.

Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.

Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.

Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.

There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.

Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.

Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)

Guy Gilchrist’s Nancy for the 1st does an alphabet soup joke, so like I said, it’s been a slow week around here.

John Zakour and Scott Roberts’s Maria’s Day for the 2nd is really just mathematics being declared hated, so like I said, it’s been a slow week around here.

## How October 2017 Treated My Mathematics Blog

October paid less attention to my mathematics blog than did September. I expected that. I published rather fewer pieces in October as the A To Z project had finished. And there’s some extent to which publishing anything is valuable in getting readership. How important I don’t know. I’ve never tried testing the relationship between how many readers I get and how many articles I post. I imagine the number of confounding factors would make their relationship vague. But I could run it anyway, as an example of how to do that kind of calculation.

It also makes me wonder whether republishing older essays is worthwhile. Or at least posting links to older content. I worry about boring longtime readers, although I’m not sure how many of those I even have. And it happens two of my most popular essays this month were fairly old bits of writing. I like to list the top five around here, but there was a three-way tie for fifth place. Big in October were:

That “here’s a thing I read” also seems to be a reliably popular post suggests maybe I need to do a weekly post about just other mathematics stuff I’d read.

United States 632
United Kingdom 76
India 67
Philippines 60
Germany 16
Slovenia 16
Singapore 15
Australia 12
France 11
Austria 10
Romania 7
Spain 7
Malaysia 6
Brazil 5
Kuwait 5
Netherlands 5
Turkey 5
Belarus 4
Hong Kong SAR China 4
Italy 4
South Africa 4
European Union 3
Poland 3
Slovakia 3
South Korea 3
Argentina 2
Denmark 2
Indonesia 2
Iraq 2
Ireland 2
Mexico 2
Norway 2
St. Kitts and Nevis 2
Sweden 2
Thailand 2
Ukraine 2
United Arab Emirates 2
Albania 1
Belgium 1 (*)
Bulgaria 1 (*)
China 1
Hungary 1
Japan 1
Latvia 1
Macedonia 1
New Zealand 1 (*)
Russia 1
Switzerland 1
Taiwan 1

I make that out to be 51 countries sending me readers at all, down from September’s 65. There were 13 single-reader countries, down from September’s 20. Belgium, Bulgaria, and New Zealand were single-reader countries for two months in a row, and no country’s on a three-month single-reader streak. “European Union” is back after a month’s absence. I’m still surprised by the number of readers from the Philippines I’ve drawn two months in a row now.

All together there were 1,069 page views from 614 unique visitors in October. That’s down from 1,232 page views and 672 unique visitors in September, and an up-and-down split from the 1,030 page views from 680 unique visitors in August. In August there were 21 posts here, in September 20, and in October 13. I kind of get the feeling people like me, but only a certain amount of me, and then they drift off.

The number of ‘likes’ went back to cratering, down to 64 over the month of October. There’d been 98 in September and 147 in August. The number of comments fell too, to a meager 12 from September’s 42 and August’s 46. The A To Z format definitely looks more inviting and welcoming to commenters, I have to conclude.

October finished out with my page here having collected 54,336 total page views from some 25,288 admitted unique visitors. I believe there were a few more visitors but some of them were copying.

Insights says that the most popular day for page views was Monday, which drew 18 percent of page views, down just a bit from September’s 20 percent. In a major upset 6 pm was not the most popular hour for readers, though. 7 pm was, when 8 percent of page views came in. I’m not sure how that happened; 6 pm is when I set most stuff to post and readers seem to follow. Maybe it’s a Daylight Saving Time issue. Oh, come to think of it, this is one of the few weeks that Greenwich Time and Eastern Time aren’t in Daylight-Saving/Summer-Time synch, isn’t it? I started out with this as a joke but perhaps that’s really going on. (No, I guess not. 12:00 am is still my most popular hour on my humor blog.) Anyway, I’m figuring to skip future mentions of what Insights tells me about popular days or hours. I can’t figure how they’re indicating anything more than “I’m about equally popular-ish any hour of any day of the week”.

WordPress says I’m starting November with 709 WordPress.com followers, which is down from September’s 717. Well, I’m sure all 709 of them are live, active accounts from people who’ve used them more recently than three years ago when they posted twice. If you’d like to follow my mathematical chats here you can add it to your reader. Go to the upper right corner of this page and click the ‘Follow NebusResearch’ button. If you’d rather get things by e-mail, there should be a ‘Follow Blog Via E-Mail’ button there too. And if that’s all fine enough but you’d like to see me limited to about 22 words at a time, try out @Nebusj on Twitter. Thanks.

This attractive little tweet came across my feed yesterday:

This function — I guess it’s the “popcorn” function — is a challenge to our ideas about what a “continuous” function is. I’ve mentioned “continuous” functions before and said something like they’re functions you could draw without lifting your pen from the paper. That’s the colloquial, and the intuitive, idea of what they mean. And that’s all right for ordinary uses.

But the best definition mathematicians have thought of for a “continuous function” has some quirks. And here’s one of them. Define a function named ‘f’. Its domain is the real numbers. Its range is the real numbers. And the rule matching things in the domain to things in the range is, as pictured:

• If ‘x’ is zero then $f(x) = 1$
• If ‘x’ is an irrational number then $f(x) = 0$
• If ‘x’ is a rational number, then it’s equal in lowest terms to the whole number ‘p’ divided by the positive whole number ‘q’. And for this ‘x’, then $f(x) = \frac{1}{q}$

And as the tweet from Fermat’s Library says, this is a function that’s continuous on all the irrational numbers. It’s not continuous on any rational numbers. This seems like a prank. But it’s a common approach to finding intuition-testing ideas about continuity. Setting different rules for rational and irrational numbers works well for making these strange functions. And thinking of rational numbers as their representation in lowest terms is also common. (Writing it as ‘p divided by q’ suggests that ‘p’ and ‘q’ are going to be prime, but, no! Think of $\frac{3}{8}$ or of $\frac{4}{9}$.) If you stare at the plot you can maybe convince yourself that “continuous on the irrational numbers” makes sense here. That heavy line of dots at the bottom looks like it’s approaching a continuous blur, at least.

It can get weirder. It’s possible to create a function that’s continuous at only a single point of all the real numbers. This is why Real Analysis is such a good subject to crash hard against. But we accept weird conclusions like this because the alternative is to give up as “continuous” functions that we just know have to be continuous. Mathematical definitions are things we make for our use.

## Reading the Comics, October 2017: Mathematics Anxiety Edition

Comic Strip Master Command hasn’t had many comics exactly on mathematical points the past week. I’ll make do. There are some that are close enough for me, since I like the comics already. And enough of them circle around people being nervous about doing mathematics that I have a title for this edition.

Tony Cochrane’s Agnes for the 24th talks about math anxiety. It’s not a comic strip that will do anything to resolve anyone’s mathematics anxiety. But it’s funny about its business. Agnes usually is; it’s one of the less-appreciated deeply-bizarre comics out there.

John Atkinson’s Wrong Hands for the 24th might be the anthropomorphic numerals joke for this week. Or it might be the anthropomorphic letters joke. Or something else entirely.

Charles Schulz’s Peanuts for the 24th reruns the comic from the 2nd of November, 1970. It has Sally discovering that multiplication is much easier than she imagined. As it is, she’s not in good shape. But if you accept ‘tooty-two’ as another name for ‘four’ and ‘threety-three’ as another name for ‘nine’, why not? And she might do all right in group theory. In that you can select a bunch of things, called ‘elements’, and describe their multiplication to fit anything you like, provided there’s consistency. There could be a four-forty-four if that seems to answer some question.

Steve Kelley and Jeff Parker’s Dustin for the 25th might be tied in to mathematics anxiety. At least it expresses how the thought of mathematics will cause some people to shut down entirely. Shame for them, but I can’t deny it’s so.