## Paul Dirac discussed on the In Our Time podcast

It’s a touch off my professed mathematics focus. Also off my comic strips focus. But Paul Dirac was one of the 20th century’s greatest physicists, this in a century rich in great physicists. Part of his genius was in innovative mathematics, and in trusting strange implications of his mathematics.

This week the BBC podcast In Our Time, a not-quite-hourlong panel show discussing varied topics, came to Paul Dirac. It can be heard here, or from other podcast sources. I get it off iTunes myself. The discussion is partly about his career and about the magnitude of his work. It’s not going to make anyone suddenly understand how to do any of his groundbreaking work in quantum mechanics. But it is, after all, an hourlong podcast for the general audience about, in this case, a physicist. It couldn’t explain spinors.

And even if you know a fair bit about Dirac and his work you might pick up something new. This might be slight: one of the panelists mentioned Dirac, in retirement, getting to know Sting. This is not something impossible, but it’s also not a meeting I would have ever imagined happening. So my week has been broadened a bit.

The web site for In Our Time doesn’t have a useful archive category for mathematics, at least that I could find. But many mathematical topics are included in the archive of science subjects, including important topics like the kinetic theory of gases and the work of Emmy Noether.

## Reading the Comics, September 1, 2017: Getting Ready For School Edition

In the United States at least it’s the start of the school year. With that, Comic Strip Master Command sent orders to do back-to-school jokes. They may be shallow ones, but they’re enough to fill my need for content. For example:

Bill Amend’s FoxTrot for the 27th of August, a new strip, has Jason fitting his writing tools to the class’s theme. So mathematics gets to write “2” in a complicated way. The mention of a clay tablet and cuneiform is oddly timely, given the current (excessive) hype about that Babylonian tablet of trigonometric values, which just shows how even a nearly-retired cartoonist will get lucky sometimes.

Dan Collins’s Looks Good On Paper for the 27th does a collage of school stuff, with mathematics the leading representative of the teacher-giving-a-lecture sort of class.

Olivia Walch’s Imogen Quest for the 28th uses calculus as the emblem of stuff that would be put on the blackboard and be essential for knowing. It’s legitimate formulas, so far as we get to see, the stuff that would in fact be in class. It’s also got an amusing, to me at least, idea for getting students’ attention onto the blackboard.

Tony Carrillo’s F Minus for the 29th is here to amuse me. I could go on to some excuse about how the sextant would be used for the calculations that tell someone where he is. But really I’m including it because I was amused and I like how detailed a sketch of a sextant Carrillo included here.

Jim Meddick’s Monty for the 29th features the rich obscenity Sedgwick Nuttingham III, also getting ready for school. In this case the summer mathematics tutoring includes some not-really-obvious game dubbed Integer Ball. I confess a lot of attempts to make games out of arithmetic look to me like this: fun to do but useful in practicing skills? But I don’t know what the rules are or what kind of game might be made of the integers here. I should at least hear it out.

Michael Cavna’s Warped for the 30th lists a top ten greatest numbers, spoofing on mindless clickbait. Cavna also, I imagine unintentionally, duplicates an ancient David Letterman Top Ten List. But it’s not like you can expect people to resist the idea of making numbered lists of numbers. Some of us have a hard time stopping.

Patrick Roberts’s Todd the Dinosaur for the 1st of September mentions a bunch of mathematics as serious studies. Also, to an extent, non-serious studies. I don’t remember my childhood well enough to say whether we found that vaguely-defined thrill in the word “algebra”. It seems plausible enough.

## Any Requests?

I’m thinking to do a second Mathematics A-To-Z Glossary. For those who missed it, last summer I had a fun string of several weeks in which I picked a mathematical term and explained it to within an inch of its life, or 950 words, whichever came first. I’m curious if there’s anything readers out there would like to see me attempt to explain. So, please, let me know of any requests. All requests must begin with a letter, although numbers might be considered.

Meanwhile since there’s been some golden ratio talk around these parts the last few days, I thought people might like to see this neat Algebra Fact of the Day:

People following up on the tweet pointed out that it’s technically speaking wrong. The idea can be saved, though. You can produce the golden ratio using exactly four 4’s this way:

$\phi = \frac{\cdot\left(\sqrt{4} + \sqrt{4! + 4}\right)}{4}$

If you’d like to do it with eight 4’s, here’s one approach:

And this brings things back around to how Paul Dirac worked out a way to produce any whole number using exactly four 2’s and the normal arithmetic operations anybody knows.

## Reading the Comics, March 31, 2015: Closing Out March Edition

It’s been another week of Comic Strip Master Command supporting my most popular regular feature around here. As sometimes happens there were so many comics in a row that I can’t catch them all up in a single post. Actually, there were enough just on the 29th of March to justify another Reading The Comics post, but I didn’t want to overload what was already a pretty busy month with more postings. This is a Gocomics.com-heavy entry, so I’m afraid folks have to click the links to see images. I hope you’ll be all right.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (March 29) is a bit of a geography joke built around the idea that a circle hasn’t got a side. Whether it does or not — besides “inside” and “outside”, source for another joke — requires thinking carefully what you mean by a shape’s side: does it have to be straight? If it can be curved, can it curve so sharply that it looks like it’s a corner? For that matter, can you tell a circle apart from, for example, the chiliagon, a regular polygon with a thousand equal sides? (If you can, then, how about a regular polygon with a million, or a billion, or more equal sides, to the point that you can’t tell the difference?) If you can’t, then how do you know a circle was in the story at all?

## How Dirac Made Every Number

A couple weeks back I offered a challenge taken from Graham Farmelo’s biography (The Strangest Man) of the physicist Paul Dirac. The physicist had been invited into a game to create whole numbers by using exactly four 2’s and the normal arithmetic operations, for example:

$1 = \frac{2 + 2}{2 + 2}$

$2 = 2^{2 - \left(2 \div 2\right)}$

$4 = 2^2 \div 2 + 2$

$8 = 2^{2^{2}} \div 2$

While four 2’s have to be used, and not any other numerals, it’s permitted to use the 2’s stupidly, as every one of my examples here does. Dirac went off and worked out a scheme for producing any positive integer from them. Now, if all goes well, Dirac’s answer should be behind this cut and it hasn’t been spoiled in the reader or the mails sent out to people reading it.

## Can You Be As Clever As Dirac For A Little Bit

I’ve been reading Graham Farmelo’s The Strangest Man: The Hidden Life of Paul Dirac, which is a quite good biography about a really interestingly odd man and important physicist. Among the things mentioned is that at one point Dirac was invited in to one of those number-challenge puzzles that even today sometimes make the rounds of the Internet. This one is to construct whole numbers using exactly four 2’s and the normal, non-exotic operations — addition, subtraction, exponentials, roots, the sort of thing you can learn without having to study calculus. For example:

$1 = \left(2 \div 2\right) \cdot \left(2 \div 2\right)$
$2 = 2 \cdot 2^{\left(2 - 2\right)}$
$3 = 2 + \left(\frac{2}{2}\right)^2$
$4 = 2 + 2 + 2 - 2$

Now these aren’t unique; for example, you could also form 2 by writing $2 \div 2 + 2 \div 2$, or as $2^{\left(2 + 2\right)\div 2}$. But the game is to form as many whole numbers as you can, and to find the highest number you can.

Dirac went to work and, complained his friends, broke the game because he found a formula that can any positive whole number, using exactly four 2’s.

I couldn’t think of it, and had to look to the endnotes to find what it was, but you might be smarter than me, and might have fun playing around with it before giving up and looking in the endnotes yourself. The important things are, it has to produce any positive integer, it has to use exactly four 2’s (although they may be used stupidly, as in the examples I gave above), and it has to use only common arithmetic operators (an ambiguous term, I admit, but, if you can find it on a non-scientific calculator or in a high school algebra textbook outside the chapter warming you up to calculus you’re probably fine). Good luck.