Talk:Bernoulli number

No neutral point of view - removed paragraphs

There is no place for such remarks:

So Pavlyk could have saved his employer Wolfram a lot of resources had he used free and open source software. Pavlyk wrote: "The numerator begins with 571642756... and ends with ...597039303." This result is false as a quick check with the asymptotic formula in the next section reveals. Interestingly, none of the math-professors in the newsgroup seem to have noticed.

The following removed paragraphs are personal opinions of Peter Luschny, see an essay of Luschny The Bernoulli Confusion. This guy (is he an expert on number theory or a graduate mathematician at all? I don't think so!) rigidly claims, against the consensus of experts, to change the definition of Bernoulli numbers so that B(1) = 1/2. For both cases, B(1) = 1/2 or B(1) = -1/2, there are advantages and disadvantages. There is no favored viewpoint! One can start with the summation of powers up to n-1 (simple) or the Riemann zeta function on the negative x-axis (advanced).

This convention is looked at as an unhappy choice by many mathematicians as it is not in accordance with the viewpoint suggested by the Riemann zeta function. Therefore more and more modern writers depart from it. John H. Conway for example uses the convention B_n = B_n(1) in his books. He remarked that this convention also makes "certain formulas more aesthetically pleasing to our eyes".

and is often preferred by number theorists

In conclusion: The Bernoulli numbers B_n admit a variety of different representations. Since not all of these agree with one another conventions are introduced. Which among them should be adopted may depend not only on mathematical insight. But mathematical insight, simplicity and aesthetics suggest that the convention supporting the concordance is the best one in almost all of the cases.

Indeed for many mathematicians there is no valid alternative to the above concordance as they agree with G. H. Hardy: "The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

Help for the perplexed?

I came here looking for an explanation of what a Bernoulli number is that a non-mathematician could understand -- which I hope is the goal of every lead paragraph of mathematical articles -- but failed to find one. (I will admit that I have found this kind of explanation in other mathematical articles.) After reading a bit further into the article -- specifically the start of the History section -- I had some vague idea that Bernoulli numbers have something to do with calculating the sum of a given sequence of numbers, & the relationship of that sum to the sum of the squares of those numbers, the sum of their cubes, & so forth. Which is not satisfactory.

Is it possible to explain in plain English without any jargon what a Bernoulli number is, & why it is important, so that a person with some college education can understand? -- llywrch (talk) 15:51, 6 October 2020 (UTC)

Great point raised here. I've modified the first section so that it doesn't jump straight into the math so much, and expanded upon what they're useful for. -- Bespeak (talk) 15:36, 22 July 2024 (UTC)

Knuth now accepts PLUS 1/2 for $B_1$

According to [1] he now accepts what Luschny has been on about for over a decade [2]:

"Luschny's webpage cites, for example, recent treatments of the subject by leading mathematicians such as Terence Tao. And his most compelling argument, from my personal perspective, is the way he unveils the early publications: I learned from him that my own presentation of the story, in The Art of Computer Programming and much more extensively in Concrete Mathematics, was a violation of history! I had put words and thoughts into Bernoulli and Euler's minds that were not theirs at all. This hurt, because I've always tried to present the evolution of ideas faithfully; in this case I'd fooled myself, by trying to conform what they wrote to what I'd learned.
By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.
Therefore I changed the definition of $B_1$ in all printings of The Art of Computer Programming during the latter half of 2021. And the new (34th) printing of Concrete Mathematics, released in January 2022, contains the much more extensive changes that are needed to tell a more comprehensive story."

And I agree, we need a simpler definition in the first paragraph that is understandable for non-mathematicians. --WiseWoman (talk) 15:18, 6 January 2022 (UTC)

I am really that to see that WiseWoman is sharing this here. Thank you! I am eager to see how this gets incorporated into the Wikipedia entry for the Bernoulli numbers. MDW333 (talk) 11:49, 8 January 2022 (UTC)

Cross-references with no clear referents

There are a couple of points in the article where a promise is made to explain something, but this promise is not kept.

1. "Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on)"

2. "to the modern form (more on different conventions in the next paragraph)"

In case (1) it is not clear where the nonstandard notation is explained, and in case (2) the different conventions are not covered in the next paragraph.

This kind of internal cross-reference does not work in Wikipedia as it is not robust against editing. Explanations need to be in the same paragraph or in footnotes. 08:23, 30 July 2022 (UTC) Gdr 08:23, 30 July 2022 (UTC)

Should the re-typeset image of Bernoulli's book be replaced by a scan of the original?

A scan of Bernoulli's book compared to the re-typeset version currently in use.

It seems to me that a high-quality scan of the original Ars Conjectandi would be preferable to the (not entirely accurately) retypeset version currently on this page. The Smithsonian made a nice scan and hosted it on the internet archive at https://archive.org/details/jacobibernoulli00bern/page/97

In particular, the beige color seems significantly better than the bright yellow, the scan gives a better idea of what the original notation actually looked like, and there are no typos accidentally introduced by the modern typesetter.

Should we replace this? If so, I'm not sure if it would be best to crop like I have done here, or just to include the full page. We don't have too many images near this section of the article, so it would even be possible to stack two full pages vertically there (possibly with the margins cropped)

Thoughts? –jacobolus (t) 11:08, 26 February 2023 (UTC)