Talk:Bernoulli number

message: 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)
thread_title:  Help for the perplexed? 

message: According to  he now accepts what Luschny has been on about for over a decade : 
 "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)
replies: text: 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)
thread_title:  Knuth now accepts PLUS 1/2 for $B_1$ 

message: 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)
thread_title:  Cross-references with no clear referents 

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