# (Roughly) Daily

## “He told me that in 1886 he had invented an original system of numbering”*…

The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with.

The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools, like most of calculus.

Mathematicians usually solve this problem by arranging the rationals in a line and filling the gaps with irrational numbers to create a complete number system that we call the real numbers.

But there are other ways of organizing the rationals and filling the gaps: the p-adic numbers. They are an infinite collection of alternative number systems, each associated with a unique prime number: the 2-adics, 3-adics, 5-adics and so on.

The p-adics can seem deeply alien. In the 3-adics, for instance, 82 is much closer to 1 than to 81. But the strangeness is largely superficial: At a structural level, the p-adics follow all the rules mathematicians want in a well-behaved number system…

“We’re all on Earth and we work with the reals, but if you went [anywhere] else, you’d work with the p-adics,” [University of Washington mathematician Bianca] Viray explained. “It’s the reals that are the outliers.”

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory… which is itself at the heart of computer science, numerical analysis, and cryptography: “An Infinite Universe of Number Systems.”

* Jorge Luis Borges, Labyrinths

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As we dwell on digits, we might send carefully-calculated birthday greetings to Klaus Friedrich Roth; he was born on this date in 1925. After escaping with his family from Nazi Germany, he was educated at Cambridge, then taught mathematics first at University College London, then at Imperial College London. He made a number of important contribution to Number Theory, for which he won the De Morgan Medal and the Sylvester Medal, and election to Fellowship of the Royal Society. In 1958 he was awarded mathematics’ highest honor, the Fields Medal, for proving Roth’s theorem on the Diophantine approximation of algebraic numbers.

source

Written by (Roughly) Daily

October 29, 2020 at 1:01 am

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## “Beauty is the first test: there is no permanent place in the world for ugly mathematics”*…

Long-time readers will know of your correspondent’s admiration and affection for Martin Gardner (c.f., e.g., here and here).  So imagine his delight to learn from @MartyKrasney of this…

Martin wrote about 300 articles for Scientific American between 1952 and 1998, most famously in his legendary “Mathematical Games” column starting in Jan 1957. Many of those articles are now viewed as classics, from his seminal piece on hexaflexagons in Dec 1956—which led to the offer to write a regular column for the magazine—to his breakthrough essays on pentomnoes, rep-tiles, the Soma cube, the art of Escher, the fourth dimension, sphere packing, Conway’s game of Life, Newcomb’s paradox, Mandelbrot’s fractals, Penrose tiles, and RSA cryptography, not forgetting the recurring numerological exploits of his alter ego Dr. Matrix, and the tongue-in-cheek April Fool column from 1975.

Many of those gems just listed were associated with beautiful graphics and artwork, so it’s no surprise that Martin scored some Scientific American covers over the years, though as we’ll see below, there’s surprisingly little overlap between his “greatest hits” and his “cover stories.”

It’s worth noting that, just as the magazine editors selected the titles under which his original articles appeared—he generally ditched those in favor of his own when he republished them in the spin-off books—artwork submitted was often altered by Scientific American staff artists…

The full dozen, replete with the cover art, at “A Gardner’s Dozen—Martin’s Scientific American Cover Stories.”

* G.H. Hardy

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As we agree with G.K Chesterton that “the difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head,” we might send carefully calculated birthday greetings to John Charles Fields, he was born on this date in 1863.  A mathematician of accomplishment, he is better remembered as a tireless advocate of the field and its importance– and best remembered as the founder of the award posthumously named for him:  The Fields Medal, familiarly known as “the Nobel of mathematics.”

Written by (Roughly) Daily

May 14, 2017 at 1:01 am

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## The Music of the Spheres…

From the redoubtable Roger Ebert, who observed, “now all I need to know is: (1) How to remember the song; (2) How to play the piano.”

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As we hum along, we might send carefully-calculated birthday greetings to Richard Ewen Borcherds; he was born on this date in 1959.  A chess prodigy in line to become Grand Master, he forsook the board (feeling that higher levels of play were more about the competition than the chess) for mathematics.  A specialist in in lattices, number theory, group theory, and infinite-dimensional algebras, he is best known for solving/proving the so-called “Moonshine conjecture,” which had been formulated in the late ’70s by John Conway and Simon Norton (and was so named as the proposition seemed so outlandish).  More recently, Borcherds has been working to develop a mathematically-rigorous construction of quantum field theory.  Among his many prizes, he has been awarded the “Nobel of Math,” the Fields Medal.

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November 29, 2012 at 1:01 am

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