How All Of 2019 Treated My Mathematics Blog


I’d promised during my review of the past month that I’d also look at my readership for the whole of 2019. It took a bit longer than I figured, but I’ve gotten there. 2019 was the eighth full year that I’ve been mathematics-blogging. I started in September of 2011 and needed a while to figure out what the heck I was doing. I think I knew what I was doing for roughly half of last year’s A-to-Z sequence. I’ve since forgotten it.

2019 was my most-read year to date: 24,662 page views from 16,718 unique visitors. It’s a heck of growth from even my 2018 figures, of 16,597 page views and 9,769 unique visitors. This 49 percent growth in year-to-year page views is the second greatest I’ve had. 2014-to-2015 saw a 60 percent growth. 2015 is also the first year I did an A-to-Z and I’m certain that made a difference. The 71 percent growth in unique visitors was the greatest growth in that statistic.

Bar chart showing annual readership, which rose significantly in 2012, 2015, 2018 and 2019, and held steady or drifted downward in the other years.
WordPress only started tracking, or at least reporting, unique visitors partway through 2012 which is why the darker unique-visitors bar in these makes it look like I had two visitors who read three thousand posts each that year.

A good part of that is a fluke event, though. One post in my A-to-Z sequence got linked from somewhere and that brought a flood of readers in. Easily something like five thousand people came in, read one or two posts, and left again. I’d still have a record year without that influx. But I don’t see anything else getting a reference like that, so I have to suppose that 2020 is going to be a more challenging year.

Bar chart showing the number of likes over the years; 2014, 2017, 2018, and 2019 were at roughly a thousand likes, with 2015 and 2016 four times and two times that figure.
Again, those 2012 and 2013 figures look suspiciously low, but I can’t think of any changes in how WordPress reported statistics which would make that make sense.

I always talk about how I’m getting fewer likes and even fewer comments than I used to. The yearly statistics show just how big the drop off is. There were 798 things liked in 2019, the lowest number since 2013. I’m not sure that the statistics for 2011 through 2013 are quite right. The jump between 2013’s 262 and 2014’s 1,045 seems suspicious. I’ve had a steady decline since 2015, though.

Bar chart showing the number of comments over the years. 2015 had the unquestioned peak, with 2016 and 2018 next-best, but the progression is a decline from that 2015 high point.
I certainly am not looking at this chart and the Likes chart and thinking that my chance to be a successful blogger came and went in 2015, nope!

And there were 181 comments in all of 2019. That’s half of 2018’s comment count. It’s my lowest number since 2013. I suspect part of the trouble is Reading the Comics posts. They’re good content, yes, but as initial posts they’re fairly closed things. Even the A-to-Z posts, apart from the appeals for subject matter, are pretty closed topics. I’ve clearly forgotten how to write open essays.

Besides my home page there were 797 pages that got at least one page view over 2019. There were 635 that got at least two page views, 304 getting at least ten views, 16 getting at least a hundred, and two that got over a thousand page views. Also, 109 of the pages viewed were Reading the Comics posts. The most popular of these were:

The first and third of these were posted in 2019. The top five essays posted in 2019 would be the linear programming and the Hamiltonian essays, plus:

Apart from the linear programming essay, I understand why these A-to-Z topics should be so popular. They’re big topics, ones that support wide swaths of mathematics.

Over the whole of 2019, people from 148 countries or country-like entities read something here. I feel pretty good about the spread of people, really. The only anomaly is that it’s been yet another year with no Greenland readers. I know there’s 14 people in Greenland but it does seem like someone would have read a page of mine by accident. Madagascar is a similar curious anomaly. 31 countries had only a single page view, which is really not that different to how many single-view countries I’ll have in any one month. Here’s the full roster of reading countries:

Mercator-style map of the world. The United States is in the darkest red, but most of the world has at least some pink in it. The exceptions are some of Saharan Africa, Greenland, Cuba, Madagascar, Mozambique, Afghanistan, Turkmenistan, and Tajikistan.
I normally don’t mind Mercator but it does create this enormous visual over-estimation of how many people in Greenland refuse to read me.
Country Readers
United States 13,872
India 1,161
United Kingdom 1,153
Canada 997
Philippines 907
Germany 562
Australia 466
France 347
Sweden 294
Singapore 250
Italy 245
Brazil 244
Netherlands 232
South Africa 180
Finland 176
Denmark 175
Spain 166
Russia 148
Poland 146
Switzerland 129
Ireland 121
Hong Kong SAR China 120
Norway 111
Japan 110
Belgium 106
Mexico 106
Pakistan 89
Slovenia 86
Turkey 85
Malaysia 77
New Zealand 74
Austria 66
Thailand 65
Indonesia 63
Portugal 62
Israel 59
Czech Republic 58
China 54
Greece 54
South Korea 54
Romania 52
Taiwan 52
United Arab Emirates 52
Colombia 51
European Union 47
Argentina 42
Ukraine 40
Hungary 39
Vietnam 39
Nepal 36
American Samoa 35
Latvia 32
Macedonia 31
Serbia 31
Slovakia 31
Bangladesh 30
Croatia 28
Chile 25
Kenya 24
Saudi Arabia 24
Nigeria 23
Egypt 18
Lithuania 18
Peru 18
Puerto Rico 18
Sri Lanka 17
Ecuador 16
Bulgaria 15
Jordan 15
Jamaica 14
Morocco 12
Lebanon 11
Belarus 10
Algeria 9
Belize 9
Uruguay 9
Bosnia & Herzegovina 8
Guatemala 8
Iceland 8
Malta 8
Myanmar (Burma) 8
Panama 8
Uganda 8
Costa Rica 7
Estonia 7
Tanzania 7
Cyprus 6
Ghana 6
Guam 6
Iraq 6
Trinidad & Tobago 6
Tunisia 6
Bolivia 5
Cape Verde 5
Georgia 5
Luxembourg 5
Venezuela 5
Zimbabwe 5
Armenia 4
Bahrain 4
Ethiopia 3
Kuwait 3
Mongolia 3
Albania 2
Azerbaijan 2
Botswana 2
Cambodia 2
Dominican Republic 2
El Salvador 2
Fiji 2
Martinique 2
Mauritius 2
Namibia 2
Papua New Guinea 2
Paraguay 2
Rwanda 2
Uzbekistan 2
Angola 1
Bermuda 1
Brunei 1
Burundi 1
Cameroon 1
Congo – Kinshasa 1
Côte d’Ivoire 1
Curaçao 1
Djibouti 1
Faroe Islands 1
Guyana 1
Honduras 1
Iran 1
Kazakhstan 1
Laos 1
Maldives 1
Marshall Islands 1
Moldova 1
Montenegro 1
Nicaragua 1
Oman 1
Palestinian Territories 1
Qatar 1
Réunion 1
Senegal 1
Sint Maarten 1
Somalia 1
Sudan 1
Turks & Caicos Islands 1
U.S. Virgin Islands 1
Zambia 1

I’m delighted there were three countries that had at least a thousand page views. I’ll try not to think how there could have been a fourth thousand-view country if only I’d hit refresh a couple times more when I was in Canada back in June.

So for the whole of 2019 I posted 173,087 words, according to WordPress’s figures. This was the third-greatest number of words I’ve written in a year, after 2016’s 199,465 words and 2018’s 186,639 words. These were spread over 201 posts. That’s my second-greatest number of posts in a year, after 2016’s 213 posts. This implies my average posting was 861 words. This I’m glad to see. It’s the first time in four years that I’ve averaged under 900 words per posting.

For the year, I averaged 1.5 comments per posting. That’s the lowest figure I’ve had for any completed year. It’s under half the average for each year from 2013 through 2018. The average likes per post is a less dire dropoff. For 2019 I had an average 3.8 likes per posting; that’s the first time since 2013 that it’s been fewer than five likes per posting.

Twice over 2019 I set a new record for daily views. My record now was set the 16th of October, when 5,003 page views came in. 720 came in the next day. It was a bit much. That 16th of October, I believe, upset the previous record that was set the 2nd of October. Before that, my greatest number of page views had been some weird day back in … I want to say March 2014. Sometime around then, anyway.

And that’s last year, in reading around here. I remain quite happy to have you as reader here this year. You can do that by using the “Follow Nebusresearch” button that’s currently on the upper-right corner of the page. (I am doing my annual thinking about changing the theme around here, if I can find a new theme that I like at all. If I do change, that might relocate the button.) Or you can use an RSS reader with the feed https://nebusresearch.wordpress.com/feed to view posts as they come in without my being able to track anything. And again, a free account in Dreamdwidth or Livejournal, which both still exist, lets you use their Friends page as RSS reader.

Reading the Comics, January 18, 2020: Decimals In Fractions Edition


Let me first share the other comic strips from last week which mentioned mathematics, but in a casual way.

Jerry Scott and Jim Borgman’s Zits for the 14th used the phrase “do the math”, and snarked on the younger generation doing mathematics. This was as part of the longrunning comic’s attempt to retcon the parents from being Baby Boomers to being Generation X. Scott and Borgman can do as they like but, I mean, their kids are named Chad and Jeremy. That’s only tenable if they’re Boomers. (I’m not sure Chad has returned from college in the past ten years.) And even then it was marginal.

John Kovaleski’s Bo Nanas rerun for the 14th is a joke about the probability of birthdays.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th features “the Bertrand Russell Drinking Game”, playing on the famous paradox about self-referential statements of logic.

Stephan Pastis’s Pearls Before Swine for the 17th has Rat use a bunch of mathematical jargon to give his declarations authority.

Cy Olson’s Office Hours for the 18th, rerunning a strip from the 9th of November, 1971, is in the line of jokes about parents not understanding their children’s arithmetic. It doesn’t seem to depend on mocking the New Math, which is a slight surprise for a 1971 comic.


Classroom. The blackboard problem is 0.25 / 0.05 = ? Wavehead, to teacher: 'Decimals *in* fractions?! Have you no shame?!'
Mark Anderson’s Andertoons for the 12th of January, 2020. This and other essays with some topic raised by Andertoons should appear at this link.

So Mark Anderson’s Andertoons for the 12th is the only comic strip of some substance that I noticed last week. You see what a slender month it’s been. It does showcase the unsettling nature of seeing notations for similar things mixed. It’s not that there’s anything which doesn’t parse about having decimals in the numerator or denominator. It just looks weird. And that can be enough to throw someone out of a problem. They might mistake the problem for one that doesn’t have a coherent meaning. Or they might mistake it for one too complicated to do. Learning to not be afraid of a problem that looks complicated is worth doing. As is learning how to tell whether a problem parses at all, even if it looks weird.


And that’s an end to last week in comics. I plan to have a fresh Reading the Comics post on Sunday. Thank you for reading in the meanwhile.

Reading the Comics, January 13, 2020: The State Pinball Championships Were Yesterday Edition


I am not my state’s pinball champion, although for the first time I did make it through the first round of play. What is important about this is that between that and a work trip I needed time for things which were not mathematics this past week. So my first piece this week will be a partial listing of comic strips that, last week, mentioned mathematics but not in a way I could build an essay around. … It’s not going to be a week with long essays, either, though. Here’s a start, though.

Henry Scarpelli’s Archie rerun for the 12th of January was about Moose’s sudden understanding of algebra, and wish for it to be handy. Well, every mathematician knows the moment when suddenly something makes sense, maybe even feels inevitably true. And then we do go looking for excuses to show it off.

Art Sansom and Chip Sansom’s The Born Loser for the 12th has the Loser helping his kid with mathematics homework. And the kid asking about when they’ll use it outside school.

Jason Chatfield’s Ginger Meggs for the 13th has Meggs fail a probability quiz, an outcome his teacher claims is almost impossible. If the test were multiple-choice (including true-or-false) it is possible to calculate the probability of a person making wild guesses getting every answer wrong (or right) and it usually is quite the feat, at least if the test is of appreciable length. For more open answers it’s harder to say what the chance of someone getting the question right, or wrong, is. And then there’s the strange middle world of partial credit.

My love does give multiple-choice quizzes occasionally and it is always a source of wonder when a student does worse than blind chance would. Everyone who teaches has seen that, though.

Jan Eliot’s Stone Soup Classics for the 13th just mentions the existence of mathematics homework, as part of the morning rush of events.

Ed Allison’s Unstrange Phenomenon for the 13th plays with optical illusions, which include several based on geometric tricks. Humans have some abilities at estimating relative areas and distances and lengths. But they’re not, like, smart abilities. They can be fooled, basically because their settings are circumstances where there’s no evolutionary penalty for being fooled this way. So we can go on letting the presence of arrow pointers mislead us about the precise lengths of lines, and that’s all right. There are, like, eight billion cognitive tricks going on all around us and most of them are much more disturbing.


That’s a fair start for the week. I hope to have a second part to this Tuesday. Thanks for reading.

Reading the Comics, January 11, 2020: Saturday was Quiet Too Edition


So I did get, as I hoped, to Saturday’s comics and they didn’t have much of deep mathematical content. There was an exception, though.

Morrie Turner’s Wee Pals for the 8th has Rocky failing a mathematics test.

Lorie Ransom’s The Daily Drawing for the 10th is the anthropomorphic geometric-figures joke for the week.

Mark Pett’s Mr Lowe rerun for the 11th has Quentin sitting through a dull mathematics class. And then, ah, the exceptional case …

Ryan North’s Dinosaur Comics for the 10th sees T-Rex pondering the point of solitaire. As he notes, there’s the weird aspect of solitaires that many of them can’t be won, even if you play perfectly. This comes close, without mentioning, an important event in numerical mathematics. So let me mention it.

T-Rex: 'Let's say you're alone in the universe with a deck of cards, and you're like, 'Welp, guess I'll make it possible to lose at sorting this deck of cards!' You put them in piles and moves them around by rules someone else invented. Eventually you think about cheating. But you're playing by yourself; who are you cheating? Yourself? THe game? Would it help if I told you almost 20% of solitaire games are provably unwinnable?' Utahraptor: 'No way!' T-Rex: 'Science confirms it! You'll lose a non-trivial amount of the time, and not in some novel way. The only novel way to lose is by dying in real life, but you do that once, and if you do, your last words are 'Oh look, a four of hearts! I can put that on the tree of hearts.' As last words go: a solid eight on ten?
Ryan North’s Dinosaur Comics for the 10th of January, 2020. Essays which discuss some aspect of Dinosaur Comics appear at this link.

There have always been things we could compute by random experiments. The digits of π, for example, if we’re willing to work at it. The catch is that this takes a lot of work. So we did not do much of this before we had computers, which are able to do a lot of work for the cost of electricity. There is a deep irony in this, since computers are — despite appearances — deterministic. They cannot do anything unpredictable. We have to provide random numbers, somehow. Or numbers that look enough like random numbers that we won’t make a grave error by using them.

Many of these techniques are known as Monte Carlo methods. These were developed in the 1940s. Stanislaw Ulam described convalescing from an illness, and playing a lot of solitaire. He pondered particularly the chance of winning a Canfield solitaire, a kind of game I have never heard of outside this anecdote. There seemed no way to work out this problem by reason alone. But he could imagine doing it in simulation, and with John von Neumann began calculating. Nicholas Metropolis gave it the gambling name, although something like that would be hard to resist. This is far from the only game that’s inspired useful mathematics. It is a good one, though.


That’s the mathematical comics for the week. Sunday, at this link, should see my next posting, with whatever comics up this week. Thanks for reading me reading the comics.

Reading the Comics, January 7, 2020: I Could Have Slept In Edition


It’s been another of those weeks where the comic strips mentioned mathematics but not in any substantive way. I haven’t read Saturday’s comics yet, as I write this, so perhaps the last day of the week will revolutionize things. In the meanwhile, here’s the strips you can look at and agree say ‘mathematics’ in them somewhere.

Dave Whamond’s Reality Check for the 5th of January, 2020, uses a blackboard full of arithmetic as signifier of teaching. The strip is an extended riff on Extruded Inspirational Teacher Movie product. I like the strip, but I don’t fault you if you think it’s a lot of words deployed against a really ignorable target.

Henry Scarpelli’s Archie rerun for the 7th has Archie not doing his algebra homework.

Bill Bettwy’s Take It From The Tinkersons on the 6th started a short series about Tillman Tinkerson and his mathematics homework. The storyline went on Tuesday, and Wednesday, and finished on Thursday.

Nate Fakes’s Break of Day for the 7th uses arithmetic as signifier of a child’s intelligence.

Mark Pett’s Mr Lowe rerun for the 7th has Lowe trying to teach arithmetic. Also, the strip is rerunning again, which is nice to see.


And that’s enough for now. I’ll read Saturday’s comics next and maybe have another essay at this link, soon. Thanks for reading.

How December 2019 Treated My Mathematics Blog


I have not been putting off the regular monthly review of my readership statistics because I didn’t like how they looked. I’ve had things occupying my day and haven’t had time to tend the blog is all. That’s come to a stop now, though, and I can look seriously at how things went around here last month. Later, I hope to do a review of the last year.

So, I don’t like how the readership around here looked. I knew there’d be some falling off, as last year’s A to Z project wrapped up. Fewer posts correspond very well with fewer page views, and fewer unique visitors. I wasn’t expecting the fall-off to be this severe, though. Here’s how it looked.

Bar chart of views per month for the last several years. The bars grow quite large in fall of 2019, with a spike about three times anything else's size for October 2019. They've dropped considerably two months in a row.
Still, the views per visitor of about 1.5 is holding around my average. That’s mediocre enough, isn’t it?

There were only 1,386 page views around here in December. That’s the lowest page views count since July. It’s well below the twelve-month running average of 2,057.1 visitors per month, too. The consoling thing: there were “only” eighteen posts in December. This is 77.0 views per posting, which is well below the average of 114.7 views per posting of the past year. But it’s basically identical to November’s record of 77.8 views per posting, and to September’s 81.5 views per posting. October 2019 is and will long remain an anomaly, unless someone else discovers me in some forum.

There were 909 unique visitors in December, again the smallest number since July. And well below the running average of 1,390.3 unique visitors. Per post, it’s 50.5 visitors, which is again way below the running average of 76.7 visitors per post. But it’s right in line with November’s 52.3 and September’s 46.2 visitors per post.

Still there’s things to be discontent about. There were 44 things liked in December, a mere 2.4 likes given per posting. That’s below the running averages of 69.7 likes per month, and of 4.4 likes per post. And then the most shocking statistics of all: zero comments in the whole month. I can’t find that that’s ever happened before, even in the earliest days of the blog when I would hit refresh to make the place seem busier than it is. The running averages are 16.4 comments per month, and 1.1 per post, and it’s hard to believe how far short of that I fell.

Well, there’s not much for me to do but lick my wounds. And to think about what I want in the blog: do I want a chatty comments section? If so, why? I like writing. But I do seem to not be good at blog conversations. I can either work to be better at that, or I can focus on what I am already enjoying. There are good things to say about both approaches.

There were popular posts in December, no matter how much I wasn’t particularly liked. The five posts most often read in December 2019 were:

All told there were 224 pages, including the home page, that got at least a single view in December. That’s down from November’s 300 and October’s 311. 102 of them got more than a single view, down from 160 and 187 the previous few months. 27 got at least ten views, down from 42 and 52 in recent months. Mm.

There were only 60 countries that sent me any page views in December. That’s down from 94 in November and 116 in October, although it’s getting close to September’s 69. There were 18 single-view countries, down from November’s and October’s 24. Here’s the readership figures for them all:

Mercator-style map of the world, with the United States in darkest red. Canada, Mexico, Brazil, most of Europe, South Asia, and Australia and New Zealand are in a fairly uniform pink too. There are a smattering of African and South American countries also offering readers.
I like to think I did Australia any bit of good by letting a couple dozen of its residents read about comic strips or other such nonsense given everything they’ve been going through, but I’m not positive this doesn’t reflect some people having left ‘curl’ processes running while they were out dealing with the fire crises.
Country Readers
United States 875
India 62
Philippines 58
Australia 55
Canada 47
United Kingdom 35
Germany 28
Brazil 23
Finland 13
Italy 13
Singapore 13
France 10
Denmark 9
Sweden 9
Israel 8
Belgium 7
Mexico 7
Bangladesh 6
Indonesia 6
Netherlands 6
Pakistan 6
Romania 6
Turkey 6
Russia 5
Spain 5
Switzerland 5
Thailand 5
Poland 4
South Africa 4
South Korea 4
Belize 3
Japan 3
New Zealand 3
Portugal 3
Chile 2
Colombia 2
Egypt 2
Nigeria 2
Serbia 2
Taiwan 2
Ukraine 2
Vietnam 2
American Samoa 1
Belarus 1
Bolivia 1
Bulgaria 1
Czech Republic 1
Georgia 1
Greece 1
Ireland 1
Jordan 1 (*)
Kenya 1
Kuwait 1
Latvia 1
Myanmar (Burma) 1
Norway 1
Oman 1
Saudi Arabia 1
Slovakia 1
Sri Lanka 1 (*)

Jordan and Sri Lanka were the only single-view countries in November, and neither of them was a single-view country in October too.

In December I had 18 posts. These had a total of 8,842 words, for an average of 491.2 words per posting. I’m surprised there’s so few of them too. This is obviously quite below the year’s average of 861 words per posting. December did a fair bit at bringing my words-per-post count down, too. December ended with my having written 201 posts over the whole year, my second-greatest number ever. And 173,087 words in total, my third-most-verbose year. I’ll get into the statistics for the full year in the look back at all 2019 that I mean to write soon. But 861 words per posting is the median of my words-per-posting average, so far.

From the dawn of time to the start of 2020 I’d had 1,403 posts around here. They drew a total of 97,577 views, from 52,978 logged unique visitors.

If you’d like to read these posts regularly, please do. You’ve already gotten much of the way there. If you have a WordPress account you can use the “Follow Nebusresearch” button on the upper right corner of this page to have it appear in your Reader regularly. If you have an RSS reader — and a free account at Dreamwidth or Livejournal will give you one, by the way — you can add the feed at https://nebusresearch.wordpress.com/feed/ to it. And, as I’ve warned, my Twitter account @Nebusj posts announcements of things as they’re published. But I haven’t had the spoons to figure out how to actually get at it myself, so it’s not a way to chat with me. I’m not sure what is a way to chat with me. Ask my parents what it’s like.

Nevertheless, thank you for reading, however it is you do it, and however often you do it.

Reading the Comics, January 4, 2020: The Little Things Edition


Today’s essay is just to mention the comic strips which, last week, said mathematics but in some incidental way. Or some way that I can’t write a reasonable blog entry for.

Gary Larson’s The Far Side reruns for the 30th of December, 2019, included this classic about curiosity killing cats. This 1985 strip rates a mention because a blackboard of mathematical symbols gets used to represent their intellectual inquiries.

Bill Amend’s FoxTrot for the 29th, a Sunday and thus new strip, is some wordplay based on the Disney+ line of entertainment product.

Jim Meddick’s Monty for the 29th has the time-travelling Professor Xemit (get it?) show a Times Square Ball Drop of the future. The ball gets replaced with a “demihypercube”, the idea being that the future will have some more complicated geometry than a mere “ball”. There is no such thing as “a” demihypercube, in the same way there is not “a” pentagon. There is a family of shapes, all called demihypercubes. There’s a variety of ways to represent them. A reasonable one, though, is a roughly spherical shape made of pointy triangles all over. It wouldn’t look absurd. There are probably time ball drops that use something like a demihypercube already.

Ruben Bolling’s Super-Fun-Pak Comix rerun for the 1st of January, 2020 features a Comics For The Elderly speaking of the advantages an abacus has over a spreadsheet.

Neal Rubin and Rod Whigham’s Gil Thorp for the 2nd has one of the student athletes working on calculus. And coach Mimi Thorp is doing the mathematics of studying athlete performance. If this strip makes you curious, too, my other blog should this Sunday recap what’s going on in Gil Thorp.

Also this coming Sunday I should look at more mathematically-themed comic strips. That should appear at this link, unless something urgent commands my attention first. Thank you.

Reading the Comics, January 4, 2020: Representations Edition


The start of the year brings me comic strips I can discuss in some detail. There are also some that just mention a mathematical topic, and don’t need more than a mention that the strip exists. I’ll get to those later.

Jonathan Lemon’s Rabbits Against Magic for the 2nd is another comic strip built on a very simple model of animal reproduction. We saw one late last year with a rat or mouse making similar calculations. Any calculation like this builds on some outright untrue premises, particularly in supposing that every rabbit that’s born survives, and that the animals breed as much as could do. It also builds on some reasonable simplifications. Things like an average litter size, or an average gestation period, or time it takes infants to start breeding. These sorts of exponential-growth calculations depend a lot on exactly what assumptions you make. I tried reproducing Lemon’s calculation. I didn’t hit 95 billion offspring. But I got near enough to say that Lemon’s right to footnote this as ‘true’. I wouldn’t call them “baby bunnies”, though; after all, some of these offspring are going to be nearly seven years old by the end of this span.

Eight-Ball; 'I'm starting an online movement against the stereotype that rabbits breed like ... rabbits.' Weenus: 'In a 7-year breeding life span, a singe mother can, mathematically, be responsible for 95 billion baby bunnies.' Eight-Ball: 'Hashtag guilty as charged.'
Jonathan Lemon’s Rabbits Against Magic for the 2nd of January, 2020. This and other essays featuring Rabbits Against Magic should be at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd justifies why “mathematicians are no longer allowed to [sic] sporting events” with mathematicians being difficult. Each of the signs is mean to convey the message “We’re #1”. The notations are just needlessly inaccessible, in that way nerds will do things.

0.\bar{9} first. The bar over over a decimal like this means to repeat what is underneath the bar without limit. So this is the number represented by 0.99999… and this is another way to write the number 1. This sometimes makes people uncomfortable; the proof is to think what the difference is between 1 and the number represented by 0.999999 … . The difference is smaller than any positive number. It’s certainly not negative. So the difference is zero. So the two numbers have to be the same number.

0^0 is the controversial one here. The trouble is that there are two standard rules that clash here. One is the rule that any real number raised to the zeroth power is 1. The other is the rule that zero raised to any positive real number is 0. We don’t ask about zero raised to a negative number. These seem to clash. That we only know zero raised to positive real numbers is 0 seems to break the tie, and justify concluding the number-to-the-zero-power rule should win out. This is probably what Weinersmith, or Weinersmith’s mathematician, was thinking. If you forced me to say what I think 0^0 should be, and didn’t let me refuse to commit to a value, I’d probably pick “1” too. But.

Three people holding up signs: 'We're #0.9' (bar above the 9). 'We're #0^0$'. 'We're #e^{pi/2} i^i$. Caption: 'Mathematicians are no longer allowed to sporting events.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd of January, 2020. I am always finding reasons to write about this strip. The essays including Saturday Morning Breakfast Cereal discussions should be at this link.

The expression x^x exists for real-valued numbers x, and that’s fine. We can look at \lim_{x \rightarrow 0 } x^x and that number’s 1. But what if x is a complex-valued number? If that’s the case, then this limit isn’t defined. And mathematicians need to work with complex-valued numbers a lot. It would be daft to say “real-valued 0^0 is 1, but complex-valued 0^0 isn’t anything”. So we avoid the obvious daftness and normally defer to saying 0^0 is undefined.

The last expression is e^{\frac{\pi}{2}} \imath^{\imath} . This \imath is that famous base of imaginary numbers, one of those numbers for which \imath^2 = -1 . Complex-valued numbers can be multiplied and divided and raised to powers just like real-valued numbers can. And, remarkably — it surprised me — the number \imath^{\imath} is equal to e^{-\frac{\pi}{2}} . That’s the reciprocal of e^{\frac{\pi}{2}} .

There are a couple of ways to show this. A straightforward method uses the famous Euler formula, that e^{\imath x} = \cos(x) + \imath\sin(x) . This implies that e^{\imath \frac{\pi}{2}} = \imath . So \imath^{\imath} has to equal (e^{\imath \frac{\pi}{2}})^{\imath} . That’s equal to e^{\imath^2 \frac{\pi}{2}}) , or e^{- \frac{\pi}{2}}) . If you find it weird that an imaginary number raised to an imaginary number gives you a real number — it’s a touch less than 0.208 — then, well, you see how weird even the simple things can be.

Locomotive engineer, speaking to Abraham Lincoln, who's making notes: 'Well, Mr President, let's see. Carry the one, take away three, carry the two ... that would be four score and seven years ago.'
Gary Larson’s The Far Side for the 4th of January, 2020. While there are only a few so far, I’m sure the number of essays based on something from The Far Side at this link will grow. This strip originally ran in 1980, if the copyright date is correct.

Gary Larson’s The Far Side for the 4th references Abraham Lincoln’s famous use of “four score and seven” to represent 87. There have been many ways to give names to numbers. As we’ve gotten comfortable with decimalization, though, most of them have faded away. I think only dozens and half-dozens remain in common use; if it weren’t for Lincoln’s style surely nobody today would remember “score” as a way to represent twenty. It probably avoids ambiguities that would otherwise plague words like “hundred”, but it does limit one’s prose style. The talk about carrying the one and taking away three is flavor. There’s nothing in turning eighty-seven into four-score-and-seven that needs this sort of arithmetic.


I hope later this week to list the comic strips which just mentioned some mathematical topic. That essay, and next week’s review of whatever this week is mathematical, should appear at this link. Thanks for reading.

Reading the Comics, December 28, 2019: Running Out The 2010s Edition


And here’s the last four comic strips from the final full week of 2019. I have already picked a couple strips for the end of December to say at least something about. Those I intend to wait for Sunday to review, though. And, as with the strips from this past Sunday, these are too slight for me to write much about. That’s all right. I don’t need the extra workload of thinking this week.

Doug Savage’s Savage Chickens for the 26th uses a blackboard of mathematics (as part of “understanding of particle physics”) as symbolic of intelligence. I’m not versed enough in particle physics to say whether the expressions make sense. I’m inclined toward it, since the first line has an integral of the reciprocal of the distance between a point x and a point x’. That looks to me like a calculation of some potential energy-related stuff.

Dana Simpson’s Phoebe and her Unicorn for the 27th uses “memorizing multiplication tables” as the sort of challenging and tedious task that a friend would not put another one through. The strip surprised me; I would have thought Phoebe the sort of kid who’d find multiplication tables, with their symmetry and teasing hints of structure (compare any number on the upper-left-to-lower-right diagonal to the numbers just up-and-right or down-and-left to it, for example), fascinating enough to memorize on their own.

Leigh Rubin’s Rubes for the 27th has a rat-or-mouse showing off one of those exciting calculations about how many rats-or-mice could breed in a year if absolutely nothing limited their growth. These sorts of calculations are fun for getting to big numbers in pretty little time. They’re only the first, loosest pieces of a model for anything’s population, though.

Pab Sungenis’s New Adventures of Queen Victoria for the 28th gets into the question about whether the new decade starts in 2020 or 2021. I wasn’t aware people were asking the question until a few weeks ago, when my father asked me for an authoritative answer. He respects my credentials as a mathematician and a calendar freak. The only answer I can defend, though, is to say of course a new decade starts in 2020. A new decade also starts in 2021. There’s also a decade starting in 2022. There’s a new decade starting five minutes from the moment you read this sentence. If you hurry you just might make it.

If you want to make any claims about “the” new decade, you have to say what you pick “the” to signify. Complete decades from the (proleptically defined) 1st of January, 1, is a compelling choice. “Years starting the 1st of January, 2020” is also a compelling choice. Decide your preference and you’ll decide your answer.

Thank you for reading, this essay and this whole year. 2020 is, of course, a leap year, or “bissextile year” if you want to establish your reputation as a calendar freak. Good luck.

Reading the Comics, December 25, 2019: Running Out The Year Edition


The last full week of the year had, again, comic strips that mostly mention mathematics without getting into detail. That’s all right. I have a bit of a cold so I’m happy not to have to compose thoughts about too many of them.

John Zakour and Scott Roberts’s Maria’s Day for the 22nd has Maria finishing, and losing, her mathematics homework. I suppose the implication’s that she couldn’t hope to reconstruct it before class. It’s not like she could re-write a short essay for history, though.

Percy Crosby’s Skippy for the 23rd has Skippy and Sookie doing the sort of story problem arithmetic of working out a total bill. The strip originally ran the 11th of August, 1932.

Cy Olson’s Office Hours for the 24th, which originally ran the 14th of October, 1971, comes the nearest to having enough to talk about here. The secretary describes having found five different answers in calculating the profits and so used the highest one. The joke is on incompetent secretaries, yes. But it is respectable, if trying to understand something very complicated, to use several different models for what one wants to know. These will likely have different values, although how different they are, and how changes in one model tracks changes in another, can be valuable. We’re accustomed to this, at least in the United States, by weather forecasts: any local weather report will describe expected storms by different models. These use different ideas about how much moisture moves into the air, how fast raindrops will form (a very difficult problem), how winds will shift, that sort of thing. It’s defensible to make similar different models for reporting the health of a business, particularly if company owns things with a price that can’t be precisely stated.

Marguerite Dabaie and Tom Hart’s Ali’s House for the 24th continues a story from the week before in which a character imagines something tossing us out of three-dimensional space. A seven-dimensional space is interesting mathematically. We can define a cross product between vectors in three-dimensional space and in seven-dimensional space. Most other spaces don’t allow something like a cross product to be coherently defined. Seven-dimensional space also allows for something called the “exotic sphere”, which I hadn’t heard of before either. It’s a structure that’s topologically a sphere, but that has a different kind of structure. This isn’t unique to seven-dimensional space. It’s not known whether four-dimensional space has exotic spheres, although many spaces higher than seven dimensions have them.

Gordon Bess’s Redeye for the 25th of December has Pokey asking his horse Loco to do arithmetic. There’s a long history of animals doing, or seeming to do, arithmetic. The strip originally ran the 23rd of August, 1973.

I’ll have some more comic strips to close out the year, I expect, which should appear at this link, most like on Tuesday. Thanks for reading.

Reading the Comics, December 21, 2019: My Favorite Kind Of Explanation Edition


And here’s the other half of last week’s comic strips that name-dropped mathematics in such a way that I couldn’t expand it to a full paragraph. We’ll likely be back to something more normal next week.

David Malki’s Wondermark for the 20th is built on the common idiom of giving more than 100%. I’m firmly on the side of allowing “more than 100%” in both literal and figurative uses of percent, so there’s not much more to say.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers rerun for the 20th has a wall full of mathematical scribbles and plays on the phrase “calculating killer”. The strip originally ran the 7th of January, 2011.

Samson’s Dark Side of the Horse for the 19th is wordplay on “the thought that counts”. The joke demands Horace be pondering arithmetic, as we see.

Maria Scrivan’s Half Full for the 20th is the Venn Diagram joke for this week.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses Big Numbers as the sort of thing that need a down-to-earth explanation. The strip is about explanations that don’t add clarity. It shows my sense of humor that I love explanations that are true but explain nothing. The more relevant and true without helping the better. Right up until it’s about something I could be explaining instead.

Tom Batiuk’s vintage Funky Winkerbean for the 21st is part of a week of strips from the perspective of a school desk. It includes a joke about football players working mathematics problems. The strip originally ran the 8th of February, 1974, looks like.

Thaves’s Frank and Ernest for the 21st is the anthropomorphic-numerals (and letters) joke for the week.


And there we go; thank you for looking over a quick list of things. I should be back with more comic strips on Sunday, barring surprises.

Just reminding you that you could watch Arthur Christmas


Folks who’ve been with me a long while know one of my happy Christmastime traditions is watching the Aardman Animation film Arthur Christmas. The film also gave me a great mathematical-physics question. You should watch the movie, but you might also consider the questions it raises.

First: Could `Arthur Christmas’ Happen In Real Life? There’s a spot in the movie when Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. What does a straight line on the surface of the Earth mean?

Second: Returning To Arthur Christmas. From here spoilers creep in and I have to discuss, among other things, what kind of straight line the reindeer might move in. There is no one “right” answer.

Third: Arthur Christmas And The Least Common Multiple. If we suppose the reindeer move in a straight line the way satellites move in a straight line, we can calculate how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

Fourth: Six Minutes Off. Waiting for the reindeer to get back becomes much harder if Arthur and Grand-Santa are not on the equator. This has potential dangers for saving the day.

Fifth and last: Arthur Christmas and the End of Time. We get to the thing that every mathematical physics blogger really really wants to get into. This is the paradox that conservation of energy and the fact of entropy seem to force us into some weird conclusions, if the universe can get old enough. Maybe; there’s some extra considerations, though, that can change the conclusion.

Reading the Comics, December 17, 2019: Mathematics In The Home Edition


As I referenced on Sunday while there were a good number of comic strips mentioning mathematics last week, there weren’t many touching deeply enough for me to make real essays about them. But you may enjoy seeing the strips anyway. So here’s the first half of this roster.

Dan Thompson’s Brevity for the 15th is a spot of wordplay about “the odds”. There’s a similar wordplay used in Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 16th, and it’s a repeat. I saw and even commented on it a bit over a year ago.

Lincoln Peirce’s Big Nate:First Class for the 16th reprints a strip from 1994 with Nate not doing his mathematics homework.

Greg Cravens’s The Buckets for the 16th has Toby discovering a personal need for arithmetic. Accounting doesn’t get much praise from us mathematics majors, but it’s deserving of attention.

Carol Lay’s Lay Lines for the 16th feels to me like a narrative version of the liar’s paradox. On studying the story I’m not sure I can justify that. But it feels like it to me.

Terry Beatty’s Rex Morgan, M.D. for the 17th has young Sarah Morgan not wanting to do the mathematics of counting days to Christmas. Or working out the number of days to Christmas. (And for those who don’t know, I regularly do recaps of the plot in Rex Morgan, M.D. on my other blog. I plan to get to this strip the first week in January, but the older essay will catch you up to October and it’s not like “family at Christmas” needs a lot of backstory even in the story strips.)

Bud Blake’s vintage Tiger for the 19th (a rerun from February 1967, looks like) has an improbably long string of coin tosses all go the same way.

Marguerite Dabaie and Tom Hart’s Ali’s House for the 19th builds on a character worrying about the dimensions of space he occupies. I don’t know where he’s going with that, though.


Thanks for reading that much. There’ll be a second half to this at this link later in the week, trusting that all goes well.

Reading the Comics, December 16, 2019: The Far Side Is Back Edition, Part I


As will sometimes happen I write this without having read Saturday’s comic strips. Press of time and all that. But it has been a week of only casual mentions of mathematics, not enough to need much detail. There were a lot of strips with this kind of casual mention. But one is of special interest.

So, yes. Gary Larson’s The Far Side has an official online home, and is reprinting strips from the classic 80s-to-early-90s comic strip. I’m glad for this, not just to reacquaint myself with an old friend. The strip was a pioneer in the good sort of nerd humor. Jokes about topics of narrow, specific interest, but — generally — not told in an exclusionary way. One might not understand why a particular joke should be funny, but only because you don’t happen to know something in the background. I’m thinking here of a desert-island strip that Larson, in one of his collections, said went over almost everybody’s head. The characters remarked on their good luck that the island was covered with mussels (or something), so at least they wouldn’t get hungry. The thing that makes this funny is that the mussels (or whatever) only grow places that get covered in water every day; that is, the island sinks with the tides.

Man at Heaven's Gate. An angel with a clipboard and pencil asks: 'OK, now listen up. Nobody gets in here without answering the following question: A train leaves Philadelphia at 1:00 pm. It's traveling at 65 miles per hour. Another train leaves Denver at 4:00 ... say, you need some paper?' Caption: 'Math phobic's nightmare.'
One of Gary Larson’s The Far Side reruns for the 19th of December, 2019. So this is not a new tag, to my own surprise. To see why it’s not, and to see this and any future essays mentioning topics raised by The Far Side please use this link. Also, I’m curious if the math phobic person here tries to hold to ‘You need some paper?’ being the question that must be answered.

Anyway, the first official online Far Side is, as you can see, your generic mathematics anxiety joke, using a story problem — with trains leaving stations, even — as the premise. And I admit this particular strip might not convince a young reader to today that The Far Side was anything special. This is the fate of many pioneers. If you look at it and think, well, that could run in Bizarro or The Argyle Sweater or Brevity or F Minus or Non Sequitur a dozen other comics, it’s because those are comic strips that want to be like this.

I’m sorry to say that, as best I can tell, there isn’t a lasting archive of strips on the new site. This particular rerun was one of the selection printed the 19th of December, but when I go to the link that should have shown that day’s strips I get bounced to the front page. This is vexing to someone who hopes to use the strips to lead conversations about mathematics topics. I’ll have to deal with that in one way or another.

Well, so be it. Later this week I’ll carry on with the roster of comic strips mentioning mathematical topics. For now I am still enjoying seeing the comics back in a mass media.

Reading the Comics, December 9, 2019: It’s A Slow Week Edition, Part II


And here’s the rest of last week’s mathematically-themed comic strips. On reflection, none of them are so substantially about the mathematics they mention for me to go into detail. Again, Comic Strip Master Command is helping me rebuild my energies after the A-to-Z wrapped up. I appreciate it, folks, but would like, you know, two or three strips a week I can sink my teeth into.

Charles Schulz’s Peanuts rerun for the 11th sees Sally Brown working out metric system unit conversions. The strip originally ran the 13th of December, 1972, a year when people in the United States briefly thought there might ever be a reason to use the prefix “deci-” for something besides decibels. “centi-” for anything besides “centimeter” is pretty dodgy too.

Rick Detorie’s One Big Happy for the 13th is a strip about percentages, and the question of whether a percentage over 100 can be meaningful. I’m solidly in the camp that says “of course it can be”.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 13th is titled “Do Not Date A Mathematician”. This seems personal. The point here is the mathematician believing her fiancee has “demonstrated a poor understanding of probability” by declaring his belief in soulmates. The joke seems to be missing some key points, though. Just declaring a belief in soulmates doesn’t say anything about his understanding of probability. If we suppose that he believed every person had exactly one soulmate, and that these soulmates were uniformly distributed across the world’s population, and that people routinely found their soulmates. But if those assumptions aren’t made then you can’t say that the fiancee is necessarily believing in something improbable.

Lincoln Peirce’s Big Nate: First Class sees Nate looking for help with his mathematics homework. The strip originally ran the 16th of December, 1994.


And that covers the comic strips of last week! I figure on Sunday to have a fresh Reading the Comics post at this link. And I’m thinking whether, or what, to have later this week. Thanks for reading.

Reading the Comics, December 9, 2019: It’s A Slow Week Edition, Part I


Comic Strip Master Command decided to respect my need for a writing break. At least a break around here. So here’s the first half of last week’s comic strips that mention mathematics. None of them get into material substantial enough that I feel justified including pictures. Some of them are even repeats, at least to my Reading the Comics essays.

Richard Thompson’s Richard’s Poor Almanac for the 7th reprints the part of the Christmas Tree guide with the Platonic Fir. It’s drawn as a geometric illustration. I’ve discussed this before.

Ed Allison’s Unstrange Phenomena for the 8th riffs on mathematics-formula rules of thumb. Here, by presenting a complicated expression for the Woolly Bear Caterpillar’s judgement.

Neil Kohney’s The Other End for the 9th uses mathematics as the subject for its quiz. In this case, apparently, simplifying algebraic expressions. I discussed it back in February.

Lincoln Peirce’s Big Nate: First Class for the 9th has Nate and Francis working on some arithmetic problem. The strip originally ran the 10th of December, 1994.

John Kovaleski’s Bo Nanas rerun for the 9th shows someone proclaiming himself the superhero “Perpendicular Man”. Then, “Parallel Man”. It’s basically wordplay with implied slapstick.


This does not exhaust all the comic strips run the past week that at least mention mathematics. I’ll pick up the rest in a post at this link, likely on Tuesday.

Reading the Comics, December 2, 2019: Laconic Week Edition


You know, I had picked these comic strips out as the ones that, last week, had the most substantial mathematics content. And on preparing this essay I realize there’s still not much. Maybe I could have skipped out on the whole week instead.

Bill Amend’s FoxTrot for the 1st is mostly some wordplay. Jason’s finding ways to represent the counting numbers with square roots. The joke plays more tightly than one might expect. Root beer was, traditionally, made with sassafras root, hence the name. (Most commercial root beers don’t use actual sassafras anymore as the safrole in it is carcinogenic.) The mathematical term root, meanwhile, derives from the idea that the root of a number is the thing which generates it. That 2 is the fourth root of 16, because four 2’s multiplied together is 16. That idea. This draws on the metaphor of the roots of a plant being the thing which lets the plant grow. This isn’t one of those cases where two words have fused together into one set of letters.

Jason, pouring pop: 'Sqrt(9) ounces .. sqrt(16) ounces ... sqrt(81) ounces ... sqrt(144) cold, delicious ounces!' Paige: 'Weirdo.' Jason: 'I take my root beer pouring seriously.'
Bill Amend’s FoxTrot for the 1st of December, 2019. Essays mentioning either the reprint or Sunday-only new issues of FoxTrot appear at this link.

Jef Mallett’s Frazz for the 1st is set up with an exponential growth premise. The kid — I can’t figure out his name — promises to increase the number of push-ups he does each day by ten percent, with exciting forecasts for how many that will be before long. As Frazz observes, it’s not especially realistic. It’s hard to figure someone working themselves up from nothing to 300 push-ups a day in only two months.

Also much else of the kid’s plan doesn’t make sense. On the second day he plans to do 1.1 push-ups? On the third 1.21 push-ups? I suppose we can rationalize that, anyway, by taking about getting a fraction of the way through a push-up. But if we do that, then, I make out by the end of the month that he’d be doing about 15.863 push-ups a day. At the end of two months, at this rate, he’d be at 276.8 push-ups a day. That’s close enough to three hundred that I’d let him round it off. But nobody could be generous enough to round 15.8 up to 90.

Kid: 'I'm going to do one push-up today. And I'm going to keep doing push-ups every day for a month. And: I'm going to increase the number of push-ups by a modest 10 percent each day. Know how many push-ups I'll do on the last day of the month? 90! And if I keep it up one more month, I'll be up to 300 push-ups at a time!' Frazz: 'Well-intended, if not especially realistic.' Kid: 'Also by then, the world will have completely forgotten about this history assignment I'm avoiding right now.' Frazz: 'Realistic, if not especially well-intended.'
Jef Mallett’s Frazz for the 1st of December, 2019. Essays which mention something from Frazz should be at this link.

An alternate interpretation of his plans would be to say that each day he’s doing ten percent more, and round that up. So that, like, on the second day he’d do 1.1 rounded up to 2 push-ups, and on the third day 2.2 rounded up to 3 push-ups, and so on. Then day thirty looks good: he’d be doing 94. But the end of two months is a mess as by then he’d be doing 1,714 push-ups a day. I don’t see a way to fit all these pieces together. I’m curious what the kid thought his calculation was. Or, possibly, what Jef Mallett thought the calculation was.

Kid: 'I'm not gonna be an accountant like you, dad! [Holding guitar] I'll become a musician so I don't have to work a real job!' [In front of computer, in suit.] 'I can just sit with my guitar, optimizing search results and maximizing click velocity and ... ' [ Realizing he's studying spreadsheets, clicks-per-ad-dollar; curses himself ]
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 2nd of December, 2019. There are a lot of essays that get into Saturday Morning Breakfast Cereal, and those essays are gathered here.

Zach Weinersmith’s for the 2nd has a kid rejecting accounting in favor of his art. But, wanting to do that art with optimum efficiency … ends up doing accounting. It’s a common story. A common question after working out that someone can do a thing is how to do it best. Best has many measures, yes. But the logic behind how to find it stays the same. Here I admit my favorite kinds of games tend to have screen after screen of numbers, with the goal being to make some number as great as possible considering. If they ever made Multiple Entry Accounting Simulator none of you would ever hear from me again.


Which may be some time! Between Reading the Comics, A to Z, recap posts, and the occasional bit of filler I’ve just finished slightly over a hundred days in a row posting something. That is, however, at its end. I don’t figure to post anything tomorrow. I may not have anything before Sunday’s Reading the Comics post, at this link. I’ll be letting my typing fingers sleep in instead. Thanks for reading.

Why does the Quantum Mechanics Momentum Operator look like that?


I don’t know. I say this for anyone this has unintentionally clickbaited, or who’s looking at a search engine’s preview of the page.

I come to this question from a friend, though, and it’s got me wondering. I don’t have a good answer, either. But I’m putting the question out there in case someone reading this, sometime, does know. Even if it’s in the remote future, it’d be nice to know.

And before getting to the question I should admit that “why” questions are, to some extent, a mug’s game. Especially in mathematics. I can ask why the sum of two consecutive triangular numbers a square number. But the answer is … well, that’s what we chose to mean by ‘triangular number’, ‘square number’, ‘sum’, and ‘consecutive’. We can show why the arithmetic of the combination makes sense. But that doesn’t seem to answer “why” the way, like, why Neil Armstrong was the first person to walk on the moon. It’s more a “why” like, “why are there Seven Sisters [ in the Pleiades ]?” [*]

But looking for “why” can, at least, give us hints to why a surprising result is reasonable. Draw dots representing a square number, slice it along the space right below a diagonal. You see dots representing two successive triangular numbers. That’s the sort of question I’m asking here.

From here, we get to some technical stuff and I apologize to readers who don’t know or care much about this kind of mathematics. It’s about the wave-mechanics formulation of quantum mechanics. In this, everything that’s observable about a system is contained within a function named \Psi . You find \Psi by solving a differential equation. The differential equation represents problems. Like, a particle experiencing some force that depends on position. This is written as a potential energy, because that’s easier to work with. But it’s the kind of problem done.

Grant that you’ve solved \Psi , since that’s hard and I don’t want to deal with it. You still don’t know, like, where the particle is. You never know that, in quantum mechanics. What you do know is its distribution: where the particle is more likely to be, where it’s less likely to be. You get from \Psi to this distribution for, like, particles by applying an operator to \Psi . An operator is a function with a domain and a range that are spaces. Almost always these are spaces of functions.

Each thing that you can possibly observe, in a quantum-mechanics context, matches an operator. For example, there’s the x-coordinate operator, which tells you where along the x-axis your particle’s likely to be found. This operator is, conveniently, just x. So evaluate x\Psi and that’s your x-coordinate distribution. (This is assuming that we know \Psi in Cartesian coordinates, ones with an x-axis. Please let me do that.) This looks just like multiplying your old function by x, which is nice and easy.

Or you might want to know momentum. The momentum in the x-direction has an operator, \hat{p_x} , which equals -\imath \hbar \frac{\partial}{\partial x} . The \partial is partial derivatives. The \hbar is Planck’s constant, a number which in normal systems of measurement is amazingly tiny. And you know how \imath^2 = -1 . That – symbol is just the minus or the subtraction symbol. So to find the momentum distribution, evaluate -\imath \hbar \frac{\partial}{\partial x}\Psi . This means taking a derivative of the \Psi you already had. And multiplying it by some numbers.

I don’t mind this multiplication by \hbar . That’s just a number and it’s a quirk of our coordinate system that it isn’t 1. If we wanted, we could set up our measurements of length and duration and stuff so that it was 1 instead.

But. Why is there a -\imath in the momentum operator rather than the position operator? Why isn’t one \sqrt{-\imath} x and the other \sqrt{-\imath} \frac{\partial}{\partial x} ? From a mathematical physics perspective, position and momentum are equally good variables. We tend to think of position as fundamental, but that’s surely a result of our happening to be very good at seeing where things are. If we were primarily good at spotting the momentum of things around us, we’d surely see that as the more important variable. When we get into Hamiltonian mechanics we start treating position and momentum as equally fundamental. Even the notation emphasizes how equal they are in importance, and treatment. We stop using ‘x’ or ‘r’ as the variable representing position. We use ‘q’ instead, a mirror to the ‘p’ that’s the standard for momentum. (‘p’ we’ve always used for momentum because … … … uhm. I guess ‘m’ was already committed, for ‘mass’. What I have seen is that it was taken as the first letter in ‘impetus’ with no other work to do. I don’t know that this is true. I’m passing on what I was told explains what looks like an arbitrary choice.)

So I’m supposing that this reflects how we normally set up \Psi as a function of position. That this is maybe why the position operator is so simple and bare. And then why the momentum operator has a minus, an imaginary number, and this partial derivative stuff. That if we started out with the wave function as a function of momentum, the momentum operator would be just the momentum variable. The position operator might be some mess with \imath and derivatives or worse.

I don’t have a clear guess why one and not the other operator gets full possession of the \imath though. I suppose that has to reflect convenience. If position and momentum are dual quantities then I’d expect we could put a mere constant like -\imath wherever we want. But this is, mostly, me writing out notes and scattered thoughts. I could be trying to explain something that might be as explainable as why the four interior angles of a rectangle are all right angles.

So I would appreciate someone pointing out the obvious reason these operators look like that. I may grumble privately at not having seen the obvious myself. But I’d like to know it anyway.


[*] Because there are not eight.

Reading the Comics, December 6, 2019: The Glances Edition


Although I’m out of the A to Z sequence, I like the habit of posting just the comic strips that name-drop mathematics for the Sunday post. It frees up so much of my Saturday, at the cost of committing my Sunday. So here’s last week’s casual mentions of some mathematics topic.

Wayno and Piraro’s Bizarro for the 3rd of December has a kid doing badly in arithmetic and blaming forces beyond their control.

Bill Holbrook’s On The Fastrack for the 5th has the CEO of Fastrack, Inc, disappointed in what analytics can do. Analytics, here, is the search for statistical correlations, traits that are easy to spot and that indicate greater risks or opportunities. The desire to find these is great and natural. Real data is, though, tantalizingly not quite good enough to answer most interesting questions.

Ruben Bolling’s Super-Fun-Pak Comix for the 5th repeats A Voice From Another Dimension, Bolling’s riff on the Flatland premise.

Tauhid Bondia’s Crabgrass for the 6th uses a background panel of calculus work as part of illustrating deep thinking about something, in this case, how to fairly divide chocolate. One of calculus’s traditional strengths is calculating the volumes of interesting figures.

Richard Thompson’s Richard’s Poor Almanac for the 6th reprints the Christmas Tree guide with a Cubist Fir that “no longer inhabits Euclidean space”.

The World And The Way It Would Be If Numbers Never Existed. Man looking at an elevator control panel: 'This is why I don't like elevators. It's always pot luck.'
Joe Martin’s Mr Boffo for the 6th of December, 2019. The occasional essay mentioning Mr Boffo is put at this link.

Joe Martin’s Mr Boffo for the 6th is a cute joke on one of the uses of numbers, that of being a convenient and inexhaustible index. The strip ran on Friday and I don’t know how to link to the archives in a stable way. This is why I’ve put the comic up here.

And that’s enough comics for just now. Later this week I’ll get to the comics that inspire me to write more.

Here Are All My Past A To Z Sequences


While I’m not necessarily going to continue highlighting old A to Z essays every Friday and Saturday, it is a fact I’ve now got six pages listing all the topics for the six A to Z’s that I have completed. So let me share them here. This may be convenient for you, the reader, to see what kinds of things I’ve written up. It’s certainly convenient for me, since someday I’ll want all this stuff organized. The past A to Z sequences have been:

  • Summer 2015. Featuring anzatz, into, and well-posed problem.
  • Leap Day 2016. With continued fractions, polynomials, quaternions, and transcendental numbers.
  • End 2016. Featuring the Fredholm alternative, general covariance, normal numbers, and the Monster Group.
  • Summer 2017. Starring Benford’s Law, topology, and x.
  • Fall 2018. Featuring Jokes, the Infinite Monkey Theorem, the Sorites Paradox, and the Pigeonhole Priciple.
  • Fall 2019. With Buffon’s Needle, Versine, the Julia Set, and Fourier Series.