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Is 1 Prime, and Does It Matter? |
If you ask a person on the street whether 1 is a prime number, they’ll probably pause, try to remember what they were taught, and say “no” (or “yes” or “I don’t remember”). Or maybe they’ll cross the street in a hurry. On the other hand, if you ask a mathematician, there's a good chance…
How Pi Almost Wasn’t |
Since you’re reading this essay, you probably already know about the mathematical holiday called Pi Day held on March 14th of each year in honor of the mystical quantity π = 3.14.... Pi isn’t just a universal constant; it’s trans-universal in the sense that, even in an alternate universe with a different geometry than ours,…
The Real Line versus the Fakes |
There’s a pretty thought experiment that’s sometimes attributed to Democritus though it’s actually due to a later popularizer of the atomic hypothesis1 and it goes like this: Suppose we use the world’s sharpest knife to cut a block of cheese in half, leaving two small blocks where before there was one large one. If cheese…
When 1+1+1 Equals 1 |
There are mathematical operations of all kinds with the property that doing the operation twice is tantamount to not doing anything at all. Such operations are called involutions, and you can find them all over the place in math: taking the negative of a number, taking the reciprocal of a number, rotating an object by…
Can Math Save Your Soul? |
"Nobody knew math could be so complicated."-- nobody ever The most truthful -- and, to me, the most infuriating -- thing a certain public figure1 uttered over the course of his ongoing career was "It is what it is." Given that he was referring to the deaths of over a hundred thousand people -- deaths that,…
Industrious Dice |
I’m a professional mathematician. That means somebody pays me to do math. I’m also a recreational mathematician. That means you might have to pay me to get me to stop. Wearing my recreational mathematician hat – quite literally, as you’ll see – I gave a talk earlier this year on some newfangled dice that do…
Our Fractional Universe |
(by this month's guest-columnist, Jeff Glibb) An esoteric branch of math called fraction theory may hold the answers to science's deepest mysteries You may think you know what numbers are. Chances are, you learned to count before you entered kindergarten, and number-names like "one", "two", and "three" were among the first words you learned. But…
Math for English Majors and Everybody Else |
Ben Orlin’s charming new book Math for English Majors: A Human Take on the Universal Language is a welcome addition to the growing fold of books about math for non-mathematicians – though I have to say, speaking here as an ally of English majors everywhere, that I vehemently protest the cover art’s implication that majoring…
Math’s Mutable Rules |
John McWhorter, one of my favorite public intellectuals, writes (in his recent essay “Lets chill out about apostrophes”), “Writing does not entail immutable rules in the way that mathematics does.” I think he’d be happy to know that some of the rules that govern mathematical formulas are just as mutable as the rules of punctuation.…
Zero-to-the-Zero and the Do-Nothing Machine |
I’m sure you know how to add and multiply counting numbers, but did you ever add or multiply sets of things? Did you ever raise one set of things to the power of another set of things? If you never did, you should. These things are fun, and fun is good. And, if you’ve ever wondered…
“Jewish Mathematics”? |
Quick math-personality quiz: What is seven-and-one-fourth minus three-fourths, expressed as a mixed number (a whole number plus a proper fraction)? What matters isn’t what answer you get but how you arrive at it; your thought-process will reveal what kind of thinker you are. So please stop reading now and continue once you’ve found the answer.…
Nine Years of Blogging About Math |
[This month's post is a transcript of a talk I gave on March 13, 2024, as part of a UMass Lowell "Conversation Starter" event on the topic of scientific literacy and communication. I was addressing other members of the Kennedy College of Sciences, so I treated mathematics as a subfield of the sciences, though I…
Plus and Times Set Free |
This essay chronicles the liberation of plus and times from narrow notions about what sorts of things can be added and multiplied — and, relatedly, what sorts of things numbers should be. With the advent of abstract notions of fields and rings in the late 19th and early 20th centuries, algebraists took addition and subtraction…
What “a 96 percent chance” doesn’t mean |
When people find out I'm a mathematician, they assume I'm into numbers. I find this assumption frustrating (and a little sad) since there's so much more to math than numbers, but the truth is that I am into numbers -- so into them that I'm writing a book about them. It's also the case (though…
Vectors From Leibniz to Einstein |
And how naive to have imagined that the series ended at this point, in only three dimensions! − 2001, A Space Odyssey, by Arthur C. Clarke The German philosopher-scientist Gottfried Leibniz dreamed of a universal language and a method of calculation to go with it, so that, if disputes arose on any subject, a disputant…
Numbers Far Afield |
"To learn all that is learnable; to deliver all collected data to the Creator on the third planet. That is the programming." (Vejur, née Voyager 6, in Star Trek: The Motion Picture) Imagine a vast growing sphere centered on the Earth with a radius of fifteen billion miles, the distance a beam of light travels…
Marvelous Arithmetics of Distance |
No reckoning allowed save the marvelous arithmetics of distance (from Smelling the Wind by Audre Lorde) Suppose a child comes up to you and says “I know 1 is odd and 2 is even, but I think 4 is more even than 2, and 1/2 is more odd than 1.” You might be tempted to…
When Five Isn’t Prime |
How far would you go to save a theorem? Would you invent a new kind of number? That’s what the mid-19th century German mathematician Ernst Eduard Kummer did, and while he was partly driven by the hope of proving Fermat’s Last Theorem, that wasn’t actually the theorem he was trying to save. Most readers of…
Math for Your Ear |
I didn't have time to compose an essay this month1, so I can't offer any of my own writing for you to read (unless this 500-word trifle counts). But I feel I should offer you something, so I decided I'd tell you what I've been reading lately in a pop-math vein. Or rather, what I've…
The Triumphs of Sisyphus |
To err is human; we all make mistakes. But some mistakes have worse consequences than others. According to Greek myth, King Sisyphus of Ephyra made the especially big mistake of cheating Hades, the God of Death. Twice. Hades said "Fool me once, shame on you; fool me twice, you have to roll a huge boulder…
The Triumphs of Sisyphus |
To err is human; we all make mistakes. But some mistakes have worse consequences than others. According to Greek myth, King Sisyphus of Ephyra made the especially big mistake of cheating Hades, the God of Death. Twice. Hades said "Fool me once, shame on you; fool me twice, you have to roll a huge boulder…
What is a Matrix? |
"What is The Matrix?" -- Morpheus, The Matrix (1999) A matrix is a rectangular array of numbers, but a rectangular array of numbers only becomes a matrix when you think of it in the right way. We’ll get back to Morpheus’ rhetorical question1 in a bit. Matrices are everywhere in science and technology, from ecology…
Hamilton’s Quaternions, or, The Trouble with Triples |
In 1853, the mathematician and physicist William Rowan Hamilton paid one last call on Catherine Barlow, whom he had once loved passionately and for whom he still had great affection. She had won his heart three decades earlier when he was a first-year student at Trinity College, Dublin and she was Miss Catherine Disney –…
Seekers of the One-Stone |
There are some who would begin the story this way: Long before Earth was formed, before any planet or star existed, there was the One-Stone. Not an actual stone, of course – just an idealized shape that certain two-legged, one-headed inhabitants of Earth would later call “the hat”. It existed in the realm of Pure…
Unlimited Powers |
Today’s mathematical journey will go from India to Europe and back, starting with Madhava of Sangamagrama’s invention of infinite series and culminating in Srinivasa Ramanujan’s discovery of the most potent piece of mathematical clickbait of all time: the outrageous assertion that 1 + 2 + 3 + 4 + ... is equal to −1/12. TWO…
Things, Names, and Numbers |
Happy January 48th, everyone! (More about that strange date later.) Mathematician Henri Poincaré once wrote “Mathematics is the art of giving the same name to different things,” and he wasn’t wrong, exactly. He was thinking about the way mathematics advances by generating new concepts that unify old ones. For instance, mathematicians noticed that adding 0…
Denominators and Doppelgängers |
It hapneth also some times, that the Quotient cannot be expressed by whole numbers, as 4 divided by 3 in this sort, whereby appeareth, that there will infinitly come from the 3 the rest of 1/3 and in such an accident you may come so neere as the thing requireth, omitting the remaynder... -- Simon…
Beneath (and Beyond) |
Tens of thousands of years ago, long before humankind hit on the the nifty trick of preserving language with marks on clay or papyrus, our ancestors notched tally marks on animal bones to count ... things. We don’t know what the things were. Take for instance the 40,000-year-old Lebombo bone found fifty years ago in…
The Clatter of the Primes |
The trouble began, as trouble often does, with a rivalry between friends. It took place during the Big Before, when numbers and operations were new and still figuring themselves out, and none of them had any idea what a universe was or whether having one would be a good idea. Plus said to Times "No…
Twisty Numbers for a Screwy Universe |
If new kinds of numbers were like new consumer products, mathematicians would have every right to fire the marketing company that came up with the names "complex numbers" and "imaginary numbers". I mean, what kind of sales pitch goes with that branding? "Psst: wanna buy a number? It's really hard to understand, and best of all, it…
Good Shurik Grothendieck |
I used to tell people that the title character of the film Good Will Hunting didn't strike me as very believable -- not because of the self-taught janitor's ability to do cutting-edge research, but because of his contempt for his own work. At one point in the movie, having shown his mentor a proof he's…
Teaching with Magic Paper |
I know that the sentence “The year is 2022” is just a bland statement of fact, but it hits my ear like a voice-over in a trailer for a bad science fiction movie made in the 1900s. Blame Walter Cronkite; I grew up watching his TV series The Twenty-First Century (1967-1969) and came to indelibly…
The Infinite Stairway |
I’m sure you’ve counted (“One, two, three, . . . ”) on too many occasions to count. The process can be boring (counting sheep), exciting (counting your winnings at a casino), or menacing (“If you kids aren't at the dinner table by the time I reach ten, I’ll ...”). But one thing counting is not is liberating.…
The Remarriage of Math and Physics |
I've long been a fan of comedies of remarriage ("It Happened One Night", "The Philadelphia Story", "His Girl Friday", etc.), and one of the greatest comedies of remarriage is the story of Math and Physics (or "Phyz", as Math likes to call her). You could say that the source of their eventual breakup was present…
Tricks of the Trade |
In the 1950s, a Scottish mathematician named C. Dudley Langford looked at a stack of six blocks his young son had assembled (see Endnote #1) and noticed something interesting that would lead him to the mathematical discovery he's remembered for today. Langford noticed that between the two red blocks was one block, between the two…
Breaking Pi |
I love working with others to discover new mathematics, but there's a kind of research I'd love even more: helping decode a Message from an extraterrestrial civilization. The chance to do that would make me drop all my mathematical projects — though in a way it wouldn't, since decoding the Message would almost certainly involve…
Let x Equal x |
Dedicated to the memory of Herb Wilf Mathematicians celebrate the French thinker René Descartes for inventing Cartesian coordinates.1 But we should also remember him as the person who tilted the terrain of Europe's mathematical alphabet, using early letters of the alphabet to signify known quantities and imbuing later letters (especially x) with the pungent whiff of…
What Lovelace Did: From Bombelli to Bernoulli to Babbage |
I want to tell you about difference tables for polynomials, not only because they're fun but also because they'll give us a chance to see how polynomials played a role in the dawn of the computer age through the work of computer pioneers Charles Babbage and Ada Lovelace. But first, where did polynomials come from?…