## No, You Can’t Say What 6/2(1+2) Equals

I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:

$6 \div 2\left(1 + 2\right) =$

Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.

That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:

$6 \div 2(3) =$

Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take $6\div 2$ and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by $2\cdot 3$, in which case the answer is 1?

And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.

I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.

This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.

But you also start to see why mathematicians tend to avoid the $\div$ symbol. We understand, for example, $a \div b$ to mean $a \cdot \frac{1}{b}$. Carry that out and then there’s no ambiguity about

$6 \cdot \frac{1}{2} \cdot 3 =$

I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in $6\div 2(1 + 2)$ also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.

(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)

And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.

And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.

## A Bunch Of Tweets I’d Thought To Save

I’m slow about sharing them is all. It’s a simple dynamic: I want to write enough about each tweet that it’s interesting to share, and then once a little time has passed, I need to do something more impressive to be worth the wait. Eventually, nothing is ever shared. Let me try to fix that.

Just as it says: a link to Leonhard Euler’s Elements of Algebra, as rendered by Google Books. Euler you’ll remember from every field of mathematics ever. This 1770 textbook is one of the earliest that presents algebra that looks like, you know, algebra, the way we study it today. Much of that is because this book presented algebra so well that everyone wanted to imitate it.

An entry in the amusing and novel proofs. This one is John Conway’s candidate for most succinct published mathematics paper. It’s fun, at least as I understand fun to be.

This Theorem of the Day from back in November already is one about elliptic functions. Those came up several times in the Summer 2017 Mathematics A To Z. This day about the Goins-Maddox-Rusin Theorem on Heron Triangles, is dense reading even by the standards of the Theorem of the Day tweet (which fits each day’s theorem into a single slide). Still, it’s worth lounging about in the mathematics.

Elke Stangl, writing about one of those endlessly-to-me interesting subjects: phase space. This is a particular way of representing complicated physical systems. Set it up right and all sorts of physics problems become, if not easy, at least things there’s a standard set of tools for. Thermodynamics really encourages learning about such phase spaces, and about entropy, and here she writes about some of this.

So ‘e’ is an interesting number. At least, it’s a number that’s got a lot of interesting things built around it. Here, John Golden points out a neat, fun, and inefficient way to find the value of ‘e’. It’s kin to that scheme for calculating π inefficiently that I was being all curmudgeonly about a couple of Pi Days ago.

Jo Morgan comes to the rescue of everyone who tries to read old-time mathematics. There were a lot of great and surprisingly readable great minds publishing in the 19th century, but then you get partway through a paragraph and it might as well be Old High Martian with talk about diminishings and consequents and so on. So here’s some help.

As it says on the tin: a textbook on partial differential equations. If you find yourself adrift in the subject, maybe seeing how another author addresses the same subject will help, if nothing else for finding something familiar written in a different fashion.

And this is just fun: creating an ellipse as the locus of points that are never on the fold line when a circle’s folded by a particular rule.

Finally, something whose tweet origin I lost. It was from one of the surprisingly many economists I follow considering I don’t do financial mathematics. But it links to a bit of economic history: Origins of the Sicilian Mafia: The Market for Lemons. It’s 31 pages plus references. And more charts about wheat production in 19th century Sicily than I would have previously expected to see.

By the way, if you’re interested in me on Twitter, that would be @Nebusj. Thanks for stopping in, should you choose to.

## How December 2016 Treated My Mathematics Blog

I’m getting back to normal. And getting to suspect WordPress just isn’t sending out “Fireworks” reports on how the year for my blog went. Fine then; I’ll carry on. Going back to the Official WordPress statistics page and sharing it for whatever value that has we find that … apparently I just held November 2016 all over again. Gads what a prospect.

As ever I exaggerate, and as ever, not by much. There were 956 page views from 589 distinct readers in December. In November there were 923 page views from 575 distinct readers. There were 21 posts in December, compared to 21 posts in November. Both are up from October, 907 page views from 536 visitors, although that was a nice and easy month with only 13 posts published. I’m a little disappointed to fall under a thousand page views for four months running, but, like, I tried posting stuff more often. What else can I do, besides answer comments the same year they’re posted and chat with people on their blogs? You know?

There were 136 pages liked in December, down from November’s 157 and up from October’s 115. Comments were down to 29 from November’s 35, and while that’s up from October’s 24 I should point out some of January’s comments are really me answering December comments. I had a lot of things slurping up time and energy. That doesn’t mean I’m not going to count the comments I wrote in January as anything other than January’s comments, though.

According to Insights, my most popular day for reading is Sunday, with 17 percent of page views coming then. I expected that; Sunday’s been the most popular day the last few months. It’s only slightly most popular, though. 17 percent (18 percent last month) is about what you’d expect for people reading here without any regard for the day of the week. 6 pm was the most popular hour, barely, with 9 percent of page views then. That’s the hour I’ve settled on for posting stuff. But that hour’s down from being 14 percent of page views in November. I don’t know what that signifies.

My roster of countries and the page views from them looks like this. I’m curiously delighted that India’s becoming a regular top-five country.

Country Views
United States 587
United Kingdom 61
India 47
Germany 25
Austria 22
Slovenia 15
Philippines 13
Netherlands 10
Spain 9
Australia 9
Italy 7
Puerto Rico 7
Finland 6
Norway 6
Singapore 6
France 5
Ireland 5
Switzerland 5
Indonesia 4
Sweden 4
Thailand 4
Bahrain 3
Estonia 3
Israel 3
Turkey 3
Chile 2
Greece 2
New Zealand 2
Nigeria 2
Peru 2
Poland 2
Sri Lanka 2
United Arab Emirates 2
Belgium 1
Denmark 1 (*)
Egypt 1
European Union 1
Japan 1 (**)
Kuwait 1
Lebanon 1
Luxembourg 1
Nepal 1
Pakistan 1
Romania 1
Saudi Arabia 1 (**)
Slovakia 1
South Africa 1 (**)

There’s 50 countries altogether that sent me viewers, if we take “European Union” as a country. That’s up from November’s 46. There were 15 single-view countries, the same as in November. Denmark was a single-view country last month. Japan, Saudia Arabia, and South Africa are on three-month single-view streaks. “European Union” is back after a brief absence.

For the second month in a row none of my most popular posts were Reading the Comics essays. They instead were split between the A To Z, some useful-mathematics stuff, and some idle trivia. The most popular stuff in December here was:

There weren’t many specific search terms; most were just “unknown”. Of the search terms that could be known I got this bunch that started out normal enough and then got weird.

• comics strip of production function
• comics of production function theory
• comics about compound event in math
• comics trip math probability
• example of probability comics trip
• population of charlotte nc 1975
• a to z image 2017
• mathematics dark secrets

I, um, maybe have an idea what that last one ought to find.

January starts with my mathematics blog having gotten 44,104 page views total from 18,889 distinct known visitors. That’s still a little page view lead on my humor blog, but that’s going to be lost by the start of February. My humor blog’s been more popular consistently the several months, and the humor blog got some little wave of popularity the past couple days. Why should it have had that? My best guess: I’m able to use that platform to explain what’s going on in Judge Parker, which I can’t quite justify here. Maybe next month.

If you’d like to follow my mathematics blog, please, click the buttons in the upper-right corner of the page to follow the blog on WordPress or by e-mail. You can also find me on Twitter as @nebusj where I try not to be one of those people who somehow has fifty tweets or retweets every hour of the day. But I haven’t done any livetweeting of a bad cartoon in ages. Might change.

## How November 2016 Treated My Mathematics Blog

I didn’t forget about reviewing my last month’s readership statistics. I just ran short on time to gather and publish results is all. But now there’s an hour or so free to review that WordPress says my readership was like in November and I can see what was going on.

Well.

So, that was a bit disappointing. The start of an A To Z Glossary usually sees a pretty good bump in my readership. The steady publishing of a diverse set of articles usually helps. My busiest months have always been ones with an A To Z series going on. This November, though, there were 923 page views around here, from 575 distinct visitors. That’s up from October, with 907 page views and 536 distinct visitors. But it’s the same as September’s 922 page views from 575 distinct visitors. I blame the US presidential election. I don’t think it’s just that everyone I can still speak to was depressed by it. My weekly readership the two weeks after the election were about three-quarters that of the week before or the last two weeks of November. I’d be curious what other people saw. My humor blog didn’t see as severe a crash the week of the 14th, though.

Well, the people who were around liked what they saw. There were 157 pages liked in November, up from 115 in September and October. That’s lower than what June and July, with Theorem Thursdays posts, had, and below what the A To Z in March and April drew. But it’s up still. Comments were similarly up, to 35 in November from October’s 24 and September’s 20. That’s up to around what Theorem Thursdays attracted.

December starts with my mathematics blog having had 43,145 page views from a reported 18,022 distinct viewers. And it had 636 WordPress.com followers. You can be among them by clicking the “Follow” button on the upper right corner. It’s up from the 626 WordPress.com followers I had at the start of November. That’s not too bad, considering.

I had a couple of perennial favorites among the most popular articles in November:

This is the first time I can remember that a Reading The Comics post didn’t make the top five.

Sundays are the most popular days for reading posts here. 18 percent of page views come that day. I suppose that’s because I have settled on Sunday as a day to reliably post Reading the Comics essays. The most popular hour is 6 pm, which drew 11 percent of page views. In October Sundays were the most popular day, with 18 percent of page views. 6 pm as the most popular hour, but then it drew 14 percent of page views. Same as September. I don’t know why 6 pm is so special.

As ever there wasn’t any search term poetry. But there were some good searches, including:

• how many different ways can you draw a trapizium
• comics back ground of the big bang nucleosynthesis
• why cramer’s rule sucks (well, it kinda does)
• oliver twist comic strip digarm
• work standard approach sample comics
• what is big bang nucleusynthesis comics strip

I don’t understand the Oliver Twist or the nucleosynthesis stuff.

And now the roster of countries and their readership, which for some reason is always popular:

Country Page Views
United States 534
United Kingdom 78
India 36
Philippines 22
Germany 21
Austria 18
Puerto Rico 17
Slovenia 14
Singapore 13
France 12
Sweden 8
Spain 8
New Zealand 7
Australia 6
Israel 6
Pakistan 5
Hong Kong SAR China 4
Portugal 4
Belgium 3
Colombia 3
Netherlands 3
Norway 3
Serbia 3
Thailand 3
Brazil 2
Croatia 2
Finland 2
Malaysia 2
Poland 2
Switzerland 2
Argentina 1
Bulgaria 1
Cameroon 1
Cyprus 1
Czech Republic 1 (***)
Denmark 1
Japan 1 (*)
Lithuania 1
Macedonia 1
Mexico 1 (*)
Russia 1
Saudi Arabia 1 (*)
South Africa 1 (*)
United Arab Emirates 1 (*)
Vietnam 1

That’s 46 countries, the same as last month. 15 of them were single-reader countries; there were 20 single-reader countries in October. Japan, Mexico, Saudi Arabia, South Africa, and the United Arab Emirates have been single-reader countries for two months running. Czech has been one for four months.

Always happy to see Singapore reading me (I taught there for several years). The “European Union” listing seems to have vanished, here and on my humor blog. I’m sure that doesn’t signal anything ominous at all.

## How October 2016 Treated My Mathematics Blog

I do try to get these monthly readership review posts done close to the start of the month. I was busy the 1st of the month, though, and had to fit around the End 2016 Mathematics A To Z. And then I meant to set this to post on Thursday, since I didn’t have anything else going that day, and forgot.

The number of page views declined again in October, part of a trend that’s been steady since June. There were only 907 views, down a slight amount from September’s 922 or more significantly from August’s 1002. I’ll find my way back above a thousand in a month if I can. A To Z months are usually pretty good ones, possibly because of all the fresh posts reminding people I exist.

The number of unique visitors dropped to 536. There had been 576 in September, but then there were only 531 unique visitors in August, if you believe that sort of thing. The number of likes was 115, exactly the same as in September and slightly up from August’s 107. The number of comments rose to 24, up from September’s 20 and August’s 16. That’s certainly been helped by people making requests for the End 2016 Mathematics A To Z. But that counts too.

## Popular Posts:

The most popular post of the month was a surprise to me and dates back to September of 2012, incredibly. I suspect someone on a popular web site linked to it and I never suspected. And the Reading the Comics posts were popular as ever.

I’ve been trying to limit these most-popular posts to just five pieces. But How Mathematical Physics Works was the next piece to make the top ten and I am proud of it, so there.

## Listing Countries:

Where did my readers come from in October? All over, but mostly, from 46 particular countries. Here’s the oddly popular list of them:

United States 466
United Kingdom 78
Philippines 55
India 52
Germany 27
Austria 23
Puerto Rico 19
Australia 14
France 12
Slovenia 10
Spain 9
Brazil 7
Netherlands 7
Italy 6
New Zealand 5
Singapore 5
Denmark 4
Sweden 4
Bulgaria 3
Poland 3
Serbia 3
Argentina 2
European Union 2
Indonesia 2
Norway 2
Bahamas 1
Belgium 1
Czech Republic 1 (**)
Estonia 1 (*)
Finland 1
Greece 1
Ireland 1
Israel 1
Jamaica 1
Japan 1
Mexico 1
Portugal 1 (*)
Russia 1
Saudi Arabia 1
Slovakia 1
South Africa 1
Ukraine 1
United Arab Emirates 1
Uruguay 1
Zambia 1

Estonia and Portugal are on two-month streaks as single-read countries. The Czech Republic’s on a three-month streak so. Nobody’s on a four-month streak, not yet.

## Search Term Non-Poetry:

Once again it wasn’t a truly poetic sort of month. But it was one that taught me what people are looking for, and it’s comics about James Clerk Maxwell. Look at these queries:

• comic strips of the scientist maxwell
• comics trip of james clerk maxwell
• comics about maxwell the scientist
• james clerk maxwell comics trip
• log 10 times 10 to the derivative of 10000
• problems with vinyl lp with too many grooves
• comic strip in advance algebra

I admit I don’t know why someone sees James Clerk Maxwell as a figure for a comics trip. He’s famous for the laws of electromagnetism, of course. Also for great work in thermodynamics and statistical mechanics. Also for color photography. And explaining how the rings of Saturn could work. And for working out the physics of truss bridges, which may sound boring but is important. Great subject for a biography. Just, a comic?

November sees the blog start with 42,250 page views, from 17,747 unique visitors if you can believe that. I’m surprised the mathematics blog still has a higher view count than my humor blog has, just now. That one’s consistently more popular; this one’s just been around longer.

WordPress says I started November with 626 followers, barely up from October’s 624. If you have wanted to follow me, there’s a button on the upper-right corner of the blog for that, at least until I change to a different theme. Also if you know a WordPress theme that would work better for the kind of blog I write let me know. I have a vague itch to change things around and that always precedes trouble. Also you can follow me on Twitter, @Nebusj, or check that out to make sure I’m not one of those people who somehow is hard to Twitter-read.

According to the “Insights” tab my readership’s largest on Sundays, which makes sense. I’ve standardized on Sundays for the Reading the Comics essays. That gets 18 percent of page views, slightly more than one in seven views. The most popular hour is again 6 pm, I assume Universal Time. 14 percent of page views come in that hour. That’s the same percentage as last month and it must reflect when my standard posting hour is.

## Some Mathematical Tweets To Read

Can’t deny that I will sometimes stockpile links of mathematics stuff to talk about. Sometimes I even remember to post it. Sometimes it’s a tweet like this, which apparently I’ve been carrying around since April:

I admit I do not know whether the claim is true. It’s plausible enough. English has many variants in England alone, and any trade will pick up its own specialized jargon. The words are fun as it is.

From the American Mathematical Society there’s this:

I talk a good bit about knot theory. It captures the imagination and it’s good for people who like to doodle. And it has a lot of real-world applications. Tangled wires, protein strands, high-energy plasmas, they all have knots in them. Some work by Paul Sutcliffe and Fabian Maucher, both of Durham University, studies tangled vortices. These are vortices that are, er, tangled together, just like you imagine. Knot theory tells us much about this kind of vortex. And it turns out these tangled vortices can untangle themselves and smooth out again, even without something to break them up and rebuild them. It gives hope for power cords everywhere.

Nerds have a streak which compels them to make blueprints of things. It can be part of the healthier side of nerd culture, the one that celebrates everything. The side that tries to fill in the real-world things that the thing-celebrated would have if it existed. So here’s a bit of news about doing that:

I like the attempt to map Sir Thomas More’s Utopia. It’s a fun exercise in matching stuff to a thin set of data. But as mentioned in the article, nobody should take it too seriously. The exact arrangement of things in Utopia isn’t the point of the book. More probably didn’t have a map for it himself.

(Although maybe. I believe I got this from Simon Garfield’s On The Map: A Mind-Expanding Exploration Of The Way The World Looks and apologize generally if I’ve got it wrong. My understanding is Robert Louis Stevenson drew a map of Treasure Island and used it to make sure references in the book were consistent. Then the map was lost in the mail to his publishers. He had to read his text and re-create it as best he could. Which, if true, makes the map all the better. It makes it so good a lost-map story that I start instinctively to doubt it; it’s so colorfully perfect, after all.)

And finally there’s this gem from the Magic Realism Bot:

## Proportional Dice

So, here’s a nice probability problem that recently made it to my Twitter friends page:

(By the way, I’m @Nebusj on Twitter. I’m happy to pick up new conversational partners even if I never quite feel right starting to chat with someone.)

Schmidt does assume normal, ordinary, six-sided dice for this. You can work out the problem for four- or eight- or twenty- or whatever-sided dice, with most likely a different answer.

But given that, the problem hasn’t quite got an answer right away. Reasonable people could disagree about what it means to say “if you roll a die four times, what is the probability you create a correct proportion?” For example, do you have to put the die result in a particular order? Or can you take the four numbers you get and arrange them any way at all? This is important. If you have the numbers 1, 4, 2, and 2, then obviously 1/4 = 2/2 is false. But rearrange them to 1/2 = 2/4 and you have something true.

We can reason this out. We can work out how many ways there are to throw a die four times, and so how many different outcomes there are. Then we count the number of outcomes that give us a valid proportion. That count divided by the number of possible outcomes is the probability of a successful outcome. It’s getting a correct count of the desired outcomes that’s tricky.

## Some Facts For The Day

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

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