My All 2020 Mathematics A to Z: Yang Hui


Nobody had particular suggestions for the letter ‘Y’ this time around. It’s a tough letter to find mathematical terms for. It doesn’t even lend itself to typography or wordplay the way ‘X’ does. So I chose to do one more biographical piece before the series concludes. There were twists along the way in writing.

Before I get there, I have a word for a longtime friend, Porsupah Ree. Among her hobbies is watching, and photographing, the wild rabbits. A couple years back she got a great photograph. It’s one that you may have seen going around social media with a caption about how “everybody was bun fu fighting”. She’s put it up on Redbubble, so you can get the photograph as a print or a coffee mug or a pillow, or many other things. And you can support her hobbies of rabbit photography and eating.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Yang Hui.

Several problems beset me in writing about this significant 13th-century Chinese mathematician. One is my ignorance of the Chinese mathematical tradition. I have little to guide me in choosing what tertiary sources to trust. Another is that the tertiary sources know little about him. The Complete Dictionary of Scientific Biography gives a dire verdict. “Nothing is known about the life of Yang Hui, except that he produced mathematical writings”. MacTutor’s biography gives his lifespan as from circa 1238 to circa 1298, on what basis I do not know. He seems to have been born in what’s now Hangzhou, near Shanghai. He seems to have worked as a civil servant. This is what I would have imagined; most scholars then were. It’s the sort of job that gives one time to write mathematics. Also he seems not to have been a prominent civil servant; he’s apparently not listed in any dynastic records. After that, we need to speculate.

E F Robertson, writing the MacTutor biography, speculates that Yang Hui was a teacher. That he was writing to explain mathematics in interesting and helpful ways. I’m not qualified to judge Robertson’s conclusions. And Robertson notes that’s not inconsistent with Yang being a civil servant. Robertson’s argument is based on Yang’s surviving writings, and what they say about the demonstrated problems. There is, for example, 1274’s Cheng Chu Tong Bian Ben Mo. Robertson translates that title as Alpha and omega of variations on multiplication and division. I try to work out my unease at having something translated from Chinese as “Alpha and Omega”. That is my issue. Relevant here is that a syllabus prefaces the first chapter. It provides a schedule and series of topics, as well as a rationale for why this plan.

Was Yang Hui a discoverer of significant new mathematics? Or did he “merely” present what was already known in a useful way? This is not to dismiss him; we have the same questions about Euclid. He is held up as among the great Chinese mathematicians of the 13th century, a particularly fruitful time and place for mathematics. How much greatness to assign to original work and how much to good exposition is unanswerable with what we know now.

Consider for example the thing I’ve featured before, Yang Hui’s Triangle. It’s the arrangement of numbers known in the west as Pascal’s Triangle. Yang provides the earliest extant description of the triangle and how to form it and use it. This in the 1261 Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications). But in it, Yang Hui says he learned the triangle from a treatise by Jia Xian, Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor’s detailed solutions to the Nine Chapters on the Mathematical Art). Jia Xian lived in the 11th century; he’s known to have written two books, both lost. Yang Hui’s commentary gives us a fair idea what Jia Xian wrote about. But we’re limited in judging what was Jia Xian’s idea and what was Yang Hui’s inference or what.

The Nine Chapters referred to is Jiuzhang suanshu. An English title is Nine Chapters on the Mathematical Art. The book is a 246-problem handbook of mathematics that dates back to antiquity. It’s impossible to say when the Nine Chapters was first written. Liu Hui, who wrote a commentary on the Nine Chapters in 263 CE, thought it predated the Qin ruler Shih Huant Ti’s 213 BCE destruction of all books. But the book — and the many commentaries on the book — served as a centerpiece for Chinese mathematics for a long while. Jia Xian’s and Yang Hui’s work was part of this tradition.

Yang Hui’s Detailed Analysis covers the Nine Chapters. It goes on for three chapters, more about geometry and fundamentals of mathematics. Even how to classify the problems. He had further works. In 1275 Yang published Practical mathematical rules for surveying and Continuation of ancient mathematical methods for elucidating strange properties of numbers. (I’m not confident in my ability to give the Chinese titles for these.) The first title particularly echoes how in the Western tradition geometry was born of practical concerns.

The breadth of topics covers, it seems to me, a decent modern (American) high school mathematics education. The triangle, and the binomial expansions it gives us, fit that. Yang writes about more efficient ways to multiply on the abacus. He writes about finding simultaneous solutions to sets of equations. And through a technique that amounts to finding the matrix of coefficients for the equations, and its determinant. He writes about finding the roots for cubic and quartic equations. The technique is commonly known in the west as Horner’s Method, a technique of calculating divided differences. We see the calculating of areas and volumes for regular shapes.

And sequences. He found the sum of the squares of natural numbers followed a rule:

1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{1}{3}\cdot n\cdot (n + 1)\cdot (n + \frac{1}{2})

This by a method of “piling up squares”, described some here by the Mathematical Association of America. (Me, I spent 40 minutes that could have gone into this essay convincing myself the formula was right. I couldn’t make myself believe the (n + \frac{1}{2}) part and had to work it out a couple different ways.)

And then there’s magic squares, and magic circles. He seems to have found them, as professional mathematicians today would, good ways to interest people in calculation. Not magic; he called them something like number diagrams. But he gives magic squares from three-by-three all the way to ten-by-ten. We don’t know of earlier examples of Chinese mathematicians writing about the larger magic squares. But Yang Hui doesn’t claim to be presenting new work. He also gives magic circles. The simplest is a web of seven intersecting circles, each with four numbers along the circle and one at its center. The sum of the center and the circumference numbers are 65 for all seven circles. Is this significant? No; merely fun.

Grant this breadth of work. Is he significant? I learned this year that familiar names might have been obscure until quite recently. The record is once again ambiguous. Other mathematicians wrote about Yang Hui’s work in the early 1300s. Yang Hui’s works were printed in China in 1378, says the Complete Dictionary of Scientific Biography, and reprinted in Korea in 1433. They’re listed in a 1441 catalogue of the Ming Imperial Library. Seki Takakazu, a towering figure in 17th century Japanese mathematics, copied the Korean text by hand. Yet Yang Hui’s work seems to have been lost by the 18th century. Reconstructions, from commentaries and encyclopedias, started in the 19th century. But we don’t have everything we know he wrote. We don’t even have a complete text of Detailed Analysis. This is not to say he wasn’t influential. All I could say is there seems to have been a time his influence was indirect.

I am sorry to offer so much uncertainty about Yang Hui. I had hoped to provide a fuller account. But we always only know thin slivers of life, and try to use those to know anything.


Next week I hope to finish this year’s A-to-Z project. The whole All 2020 A-to-Z should be gathered at this link. And all the essays from every A-to-Z series should be at this link. I haven’t decided whether I’ll publish on Wednesday or Friday. It’ll depend what I can get done over the weekend; we’ll see. Thank you for reading.

How To Multiply Numbers By Multiplying Other Numbers Instead


I do read other people’s mathematics writing, even if I don’t do it enough. A couple days ago RJ Lipton and KW Regan’s Reductions And Jokes discussed how one can take a problem and rewrite it as a different problem. This is one of the standard mathematician’s tricks. The point to doing this is that you might have a better handle on the new problem.

“Better” is an aesthetic judgement. It reflects whether the new problem is easier to work with. Along the way, they offer an example that surprised and delighted me, and that I wanted to share. It’s about multiplying whole numbers. Multiplication can take a fair while, as anyone who’s tried to do 38 times 23 by hand has found out. But we can speed that up. A multiplication table is a special case of a lookup table, a chunk of stored memory which has computed ahead of time all the multiplications someone is likely to do. Then instead of doing them, you just look them up.

The catch is that a multiplication table takes memory. To do all the multiplications for whole numbers 1 through 10 you need … well, not 100 memory cells. But 55. To have 1 through 20 worked out ahead of time you need 210 memory cells. Can we do better?

If addition and subtraction are easy enough to do? And if dividing by two is easy enough? Then, yes. Instead of working out every pair multiplication, work out the squares of the whole numbers. And then make use of this identity:

a \times b = \frac{1}{2}\left( \left(a + b\right)^2 - a^2 - b^2\right)

And that delights me. It’s one of those relationships that’s sitting there, waiting for anyone who’s ever squared a binomial to notice. I don’t know that anyone actually uses this. But it’s fun to see multiplication worked out by a different yet practical way.

The Summer 2017 Mathematics A To Z: Sárközy’s Theorem


Gaurish, of For the love of Mathematics, gives me another chance to talk number theory today. Let’s see how that turns out.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Sárközy’s Theorem.

I have two pieces to assemble for this. One is in factors. We can take any counting number, a positive whole number, and write it as the product of prime numbers. 2038 is equal to the prime 2 times the prime 1019. 4312 is equal to 2 raised to the third power times 7 raised to the second times 11. 1040 is 2 to the fourth power times 5 times 13. 455 is 5 times 7 times 13.

There are many ways to divide up numbers like this. Here’s one. Is there a square number among its factors? 2038 and 455 don’t have any. They’re each a product of prime numbers that are never repeated. 1040 has a square among its factors. 2 times 2 divides into 1040. 4312, similarly, has a square: we can write it as 2 squared times 2 times 7 squared times 11. So that is my first piece. We can divide counting numbers into squarefree and not-squarefree.

The other piece is in binomial coefficients. These are numbers, often quite big numbers, that get dumped on the high school algebra student as she tries to work with some expression like (a + b)^n . They’re also dumped on the poor student in calculus, as something about Newton’s binomial coefficient theorem. Which we hear is something really important. In my experience it wasn’t explained why this should rank up there with, like, the differential calculus. (Spoiler: it’s because of polynomials.) But it’s got some great stuff to it.

Binomial coefficients are among those utility players in mathematics. They turn up in weird places. In dealing with polynomials, of course. They also turn up in combinatorics, and through that, probability. If you run, for example, 10 experiments each of which could succeed or fail, the chance you’ll get exactly five successes is going to be proportional to one of these binomial coefficients. That they touch on polynomials and probability is a sign we’re looking at a thing woven into the whole universe of mathematics. We saw them some in talking, last A-To-Z around, about Yang Hui’s Triangle. That’s also known as Pascal’s Triangle. It has more names too, since it’s been found many times over.

The theorem under discussion is about central binomial coefficients. These are one specific coefficient in a row. The ones that appear, in the triangle, along the line of symmetry. They’re easy to describe in formulas. for a whole number ‘n’ that’s greater than or equal to zero, evaluate what we call 2n choose n:

{{2n} \choose{n}} =  \frac{(2n)!}{(n!)^2}

If ‘n’ is zero, this number is \frac{0!}{(0!)^2} or 1. If ‘n’ is 1, this number is \frac{2!}{(1!)^2} or 2. If ‘n’ is 2, this number is \frac{4!}{(2!)^2} 6. If ‘n’ is 3, this number is (sparing the formula) 20. The numbers keep growing. 70, 252, 924, 3432, 12870, and so on.

So. 1 and 2 and 6 are squarefree numbers. Not much arguing that. But 20? That’s 2 squared times 5. 70? 2 times 5 times 7. 252? 2 squared times 3 squared times 7. 924? That’s 2 squared times 3 times 7 times 11. 3432? 2 cubed times 3 times 11 times 13; there’s a 2 squared in there. 12870? 2 times 3 squared times it doesn’t matter anymore. It’s not a squarefree number.

There’s a bunch of not-squarefree numbers in there. The question: do we ever stop seeing squarefree numbers here?

So here’s Sárközy’s Theorem. It says that this central binomial coefficient {{2n} \choose{n}} is never squarefree as long as ‘n’ is big enough. András Sárközy showed in 1985 that this was true. How big is big enough? … We have a bound, at least, for this theorem. If ‘n’ is larger than the number 2^{8000} then the corresponding coefficient can’t be squarefree. It might not surprise you that the formulas involved here feature the Riemann Zeta function. That always seems to turn up for questions about large prime numbers.

That’s a common state of affairs for number theory problems. Very often we can show that something is true for big enough numbers. I’m not sure there’s a clear reason why. When numbers get large enough it can be more convenient to deal with their logarithms, I suppose. And those look more like the real numbers than the integers. And real numbers are typically easier to prove stuff about. Maybe that’s it. This is vague, yes. But to ask ‘why’ some things are easy and some are hard to prove is a hard question. What is a satisfying ’cause’ here?

It’s tempting to say that since we know this is true for all ‘n’ above a bound, we’re done. We can just test all the numbers below that bound, and the rest is done. You can do a satisfying proof this way: show that eventually the statement is true, and show all the special little cases before it is. This particular result is kind of useless, though. 2^{8000} is a number that’s something like 241 digits long. For comparison, the total number of things in the universe is something like a number about 80 digits long. Certainly not more than 90. It’d take too long to test all those cases.

That’s all right. Since Sárközy’s proof in 1985 there’ve been other breakthroughs. In 1988 P Goetgheluck proved it was true for a big range of numbers: every ‘n’ that’s larger than 4 and less than 2^{42,205,184} . That’s a number something more than 12 million digits long. In 1991 I Vardi proved we had no squarefree central binomial coefficients for ‘n’ greater than 4 and less than 2^{774,840,978} , which is a number about 233 million digits long. And then in 1996 Andrew Granville and Olivier Ramare showed directly that this was so for all ‘n’ larger than 4.

So that 70 that turned up just a few lines in is the last squarefree one of these coefficients.

Is this surprising? Maybe, maybe not. I’ll bet most of you didn’t have an opinion on this topic twenty minutes ago. Let me share something that did surprise me, and continues to surprise me. In 1974 David Singmaster proved that any integer divides almost all the binomial coefficients out there. “Almost all” is here a term of art, but it means just about what you’d expect. Imagine the giant list of all the numbers that can be binomial coefficients. Then pick any positive integer you like. The number you picked will divide into so many of the giant list that the exceptions won’t be noticeable. So that square numbers like 4 and 9 and 16 and 25 should divide into most binomial coefficients? … That’s to be expected, suddenly. Into the central binomial coefficients? That’s not so obvious to me. But then so much of number theory is strange and surprising and not so obvious.

Everything I Learned In Eighth-Grade Math


My title is an exaggeration. In eighth grade Prealgebra I learned many things, but I confess that I didn’t learn well from that particular teacher that particular year. What I most clearly remember learning I picked up from a substitute who filled in a few weeks. It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation. But it’s fun to do, and I liked it a lot, and I’ve never seen it described as a way to factor quadratic expressions. So let me put it on the web and do what I can to preserve its legacy, and get hundreds of people telling me what it actually is and how everybody but the people I know went through a phase of using it.

It’s a method which looks at first like it’s going to be a magic square, but it’s not, and I’m at a loss what to call it. I don’t remember the substitute teacher’s name, so I can’t use that. I do remember the regular teacher’s name, but it wasn’t, as far as I know, part of his lesson plan, and it’d not be fair to him to let his legacy be defined by one student who just didn’t get him.

Continue reading “Everything I Learned In Eighth-Grade Math”