May 20 While doing some other reading last week, I found my way to one of Srinivasa Ramanujan's earliest (the earliest? I don't know) papers, "Some Properties of Bernoulli's Numbers". It reminded me of something I've had at the back of my mind for a while: to write about the remarkable Bernoulli family, so many of whom made their mark in mathematics and physics. And coming full circle, I wanted to end with a little bit about Bernoulli's numbers, which have some interesting properties ... which, of course, is why Ramanujan wrote about them.
So here's the thing. While writing this column, I was stuck for an hour or two right at the end, because I stumbled on an error in Ramanujan's paper. I simply could not believe, one, that he had made an error; two, that I had found it pretty easily. I don't know, I felt an immediate connection to the great man. Take a look at my column (Mint, Friday May 19) and let me know your thoughts - about the Bernoullis and about Ramanujan's error. I've described it briefly in the column, with a little more below, and I've attached an image of the relevant part of Ramanujan's paper. The statement #14 is the one in question. Swiss Family Bernoulli, https://www.livemint.com/opinion/columns/the-remarkable-bernoulli-family-scientists-mathematicians-and-discoverers-of-infinitesimal-quantities-and-probability-paradoxes-11684430595408.html (I was more than a little disappointed that the folks at Mint changed my title to "How Bernoullis made their mark". Why do you think I preferred "Swiss Family Bernoulli"?) cheers, dilip In the attached image, B12, B26, B30 etc are Bernoulli's numbers. Checking Ramanujan's statement marked 14, I tried B24 and B30, and was astonished that #14 doesn't apply to either. B30, for example, is 5 x 1721 x 1001259881 / 2 x 3 x 7 x 11 x 31. Divide this by 30, and you get 1721 x 1001259881 / 2 x 3 x 7 x 11 x 31 x 2 x 3. Since both factors in the numerator (1721, 1001259881) are prime, it is in its lowest terms - yet the numerator clearly isn't prime. Maybe easier is B20, whose numerator 174611 is 283 x 617, and both those are prime. So when you divide B20 by 20 (according to statement #14), the resultant numerator is still 174611, which is clearly not prime. --- Swiss Family Bernoulli Once upon a time, there lived in Switzerland a family named Bernoulli. I mention that name because most once-students of science know of it from physics in high school or early college years. Students like me, for example. I refer here to Bernoulli's Principle. It goes something like this: when you have a fluid that's flowing - water through a pipe, air around the wing of an aircraft - there's a relationship between the speed at which it flows and the pressure in the fluid. As the speed increases, the pressure decreases. This is a physics nugget that doesn't stay abstract long. For example, it explains why topspin imparted to a tennis ball - think Rafa Nadal - makes it descend faster than a ball that's hit without spin. It also helps explain how an aircraft's wing generates the force needed to lift a plane into the air. Well actually, there is some nuance and argument about how exactly the Principle applies to that lift. Still, because it's been connected to phenomena like these, the name is known more widely than just in physics circles. In any case, the Principle is named after Daniel Bernoulli, who discovered this relationship in the 1730s. Daniel studied business and mathematics and medicine, finishing a PhD in anatomy and botany and a MD in medicine by the age of 21. His first love seems to have been mathematics, though, and particularly its applications to fluid mechanics. When he happened upon the Principle, he found ways to use it in real life, devising an instrument to measure blood pressure, for example, that worked because of his Principle. While we no longer measure blood pressure that way, you can find essentially the same instrument on the outside of planes. It uses pressure to measure the speed of the plane. Because Bernoulli's Principle became so well-known, students like me tended to assume without much thought that there was just one Bernoulli. Or at any rate, that Daniel was the only Bernoulli to make a mark in science. Both are serious misconceptions. In fact, there can't be too many families in mathematics or science, or really in any field, quite like the Bernoullis. In the 17th and 18th Centuries, this remarkable Swiss family produced at least eight - maybe as many as a dozen - scientists and mathematicians who found widespread recognition for their work. Daniel was only one of those. To be sure, he was one of the brightest stars in the family. But there's Daniel's younger brother, Johann II. He was a professor of mathematics at the University of Basel. His particular interest was the way light moves, but there were other subjects he dabbled in too, with much success: he won prizes for his work from the Academy of Sciences in Paris. Johann II had four sons. The eldest was Johann III. This Bernoulli was considered a child prodigy: he completed his PhD at just thirteen, and was named Berlin's "Astronomer Royal" when he was nineteen. His youngest brother was Jakob II, who loved geometry and learned it from his uncle Daniel. He became a professor of mathematics in St Petersburg. Sadly, he drowned soon after, short of his 30th birthday. The earlier Bernoullis, though, were even more eminent. Daniel's cousin Nicolaus I had a deep interest in probability. He dreamed up what mathematicians know as the St Petersburg paradox (though it was Daniel who actually gave it that name). This arises from a game of flipping coins that, in theory, offers unlimited returns - but potential players don't see that and are unwilling to pay more than a token amount to enter the game. Daniel's father Johann and uncle Jacob were early pioneers and users of calculus as we know it today, that uses the idea of infinitesimal quantities. Unfortunately the two grew professionally jealous of each other, a feeling that, after Jacob died, Johann transferred to his son Daniel. In both cases, the relationships eventually broke down altogether. Jacob is known for deriving the well-known law of large numbers in probability theory: if an experiment has an expected result, and if you perform it a large number of times, the average of the results you get will tend to approach the expected result. For example, if you toss a coin 10 times and count how many tails you get, you expect 5. But the first time, you may get 7. Then maybe 6, 4, 5, 8 ... but in the long run, the average of all your 10-toss experiments will get closer and closer to 5. We can also thank Jacob for discovering "e" (2.71828...) - a number just as ubiquitous and important in mathematics as π is. For just one example, it is fundamental to the idea of compound interest - which is actually how Jacob found it. In the early 18th Century, he also stumbled on the so-called Seki-Bernoulli numbers, independently of but almost simultaneously with the Japanese mathematician Seki Takakazu. They ran into them by chance, while looking for a formula for the sums of powers of integers. This is a prize that mathematicians since ancient times - Archimedes, Aryabhata, Fermat and Pascal, among many others - have chased. The Seki-Bernoulli numbers turn up in the branch of mathematics called "analysis". I won't say more about that, but here are some of these numbers, and some remarks about them, taken from a 1911 paper by the great Srinivasa Ramanujan ( http://ramanujan.sirinudi.org/Volumes/published/ram01.pdf): #2: 1/6 #4 and #8: 1/30 #12: 691/2730 #30: 8615841276005/14322 etc. (They are infinite in number, and every odd-numbered Seki-Bernoulli number is 0.) In that paper, Ramanujan noted some intriguing things about these numbers. Two examples: * All the denominators have 2 and 3 as prime factors, but only once each. Thus all denominators are divisible by 6, but not by 4 (2 x 2) or 9 (3 x 3). * They are fractions, and if you divide one by its serial number, the numerator of the result is prime. This works for #12 above, but - and I can hardly believe this - not for #30. This is a known mistake in Ramanujan's work, though I am flummoxed that the great man made such a simple mistake. That conundrum aside, Ramanujan has plenty more to say about these numbers. Just as I have plenty more to say about the Bernoullis. Sadly, space only allows this much. Still, I'm honoured to have at least that in common with Ramanujan. Thank you, Bernoulli family. -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. 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