How Bernoullis made his mark

I am referring to Bernoulli’s principle here. It goes something like this: When you have a fluid flowing – water through a pipe, air around an aircraft wing – there is a relationship between the speed at which it flows and the pressure in the fluid. As the speed increases, the pressure decreases.

It’s a physics nugget that doesn’t stay abstract for long. For example, this explains why a tennis ball imparted with topspin – think Rafa Nadal – tends to come down faster than a ball hit without spin. It also helps to explain how the wing of an aircraft generates the force necessary to lift an aircraft into the air. In fact, there are some nuances and arguments about how the theory applies to that lift. Nevertheless, because it is associated with such phenomena, the name has become more widely known than just physics circles.

In any case, the principle is named after Daniel Bernoulli, who discovered the relation in the 1730s. Daniels studied business and mathematics and medicine until the age of 21, completing a PhD in anatomy and botany and an MD in medicine. However, his first love has been mathematics, and in particular its application to fluid mechanics. While he pondered the theory, he found ways to use it in real life, designing a device for measuring blood pressure, for example, that worked because of his theory. While we no longer measure blood pressure that way, you can find essentially the same equipment outside planes. It uses pressure to measure the speed of the aircraft.

Because Bernoulli’s principle had become so well known, students like me assumed without much thought that there was only one Bernoulli. Or at any rate, Daniel was the only Bernoulli to make a mark in science. Both are serious misconceptions. In fact, like Bernoullis, there can never be too many families in math or science, or indeed any field. In the 17th and 18th centuries, this notable Swiss family produced at least eight—perhaps as many as a dozen—scientists and mathematicians who were widely recognized for their work. Daniel was only one of them.

Certainly, he was one of the brightest stars in the family. But there is Daniel’s younger brother, Johann II. He was Professor of Mathematics at the University of Basel. His particular interest was in the way light travels, but there were other subjects in which he achieved great 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 13, and was named Berlin’s “Astronomer Royal” when he was 19. Her youngest brother was Jakob II, who had a love for geometry and learned it from his uncle Daniel. He became a professor of mathematics in St. Petersburg. Tragically, he soon drowned, shortly before his 30th birthday.

However, the earlier Bernoullis were even more iconic. Daniel’s cousin Nicholas I had a keen interest in probability. He dreamed up what mathematicians know as the St. Petersburg Paradox (though it was Daniel who actually gave it that name). It originates from a game of tossing coins which, in theory, offers unlimited returns, but potential players do not see this, and are unwilling to pay more than a token amount to enter the game.

Daniel’s father Johann and uncle Jakob were early pioneers and users of calculus as we know it today, which uses the idea of ​​infinitesimal quantities. Unfortunately, the two became jealous of each other professionally, a feeling that, after Jacob’s death, Johann transferred to his son Daniel. In both cases, the relationship eventually broke down completely.

Jacobs is known for deriving the famous law of large numbers in probability theory: if an experiment has an expected result, and if you do it a large number of times, the results you get will average out to the expected result. , For example, if you toss a coin 10 times and count how many tails you get, you expect five. But for the first time, you might get seven. Then maybe six, four, five, eight—but in the long run, the average of all your 10-toss experiments will get closer and closer to five.

We can also thank Jacobs for discovering “e” (2.71828…) – a number just as ubiquitous and important in mathematics as TT. For just one example, this is fundamental to the idea of ​​compound interest – which is exactly how Jacob found it.

In the early 18th century, he also stumbled upon the so-called Seki–Bernoulli numbers, independently but almost simultaneously with the Japanese mathematician Seki Takakazu. While looking for a formula for the sum of powers of integers, he stumbled upon them by chance. It is a prize that has been chased by mathematicians since ancient times—Archimedes, Aryabhata, Fermat and Pascal, and many others.

The Seki–Bernoulli numbers fall into the branch of mathematics known as “analysis”. I won’t say much about that, but here are some of these numbers, and some comments about them, taken from a 1911 paper by the great Srinivasa Ramanujan. (https://ramanujan.sirinudi.org/Volumes/published/ram01.pdf),

#2: 1/6

#4 and #8: 1/30

#12: 691/2730

#30: 8615841276005/14322

even more.

(They are infinite in number, and every odd numbered Seki–Bernoulli number is 0.)

In that paper, Ramanujan noted some interesting things about these numbers. Two examples:

* All denominators have 2 and 3 prime factors, but each once. 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 ordinal number, the quotient of the result is prime. This works for #12 above, but – and I can hardly believe it – not for #30. This is a known mistake in Ramanujan’s work, although I am surprised that the great man made such a simple mistake.

Besides this puzzle, Ramanujan has much more to say about these numbers. As in I have a lot more to say about Bernoullis. Sadly space only allows so much. Nevertheless, I feel at least equally honored with Ramanujan. Thanks, Bernoulli family.

Dilip D’Souza, once a computer scientist, now lives in Mumbai and writes for his dinner. His Twitter handle is @DeathEndsFun.