Well put, HMHW; I agree entirely. So if it’s AIDS-infested faggot rape for you, well, then I guess it has to be AIDS-infested faggot rape for me too.
But… perhaps the posters you are replying to are not so much questioning “What is the extra-mathematical application of Bernoulli numbers?” as “What is the interest, even within a pure mathematics context, of Bernoulli numbers? What makes this sequence interesting as opposed to any of the millions of other arbitrary ones one could come up with?”. Which I think would indeed be a worthwhile question to ask (though, unfortunately, no one in this thread has been able to answer in any more specific detail than “Well, I’m told they come up a lot”).
I’m going to put it into my own words to see if I’ve got it right:
So there are formulae for how to work out the square of a number, the cube of a number, or the answer when you do that number to the power of four, then the power of five, and so on - when these numbers are in a sequence, that is.
These formulae always include a variable (the number that’s in the sequence) that’s multiplied by another number, the coefficient. What that coefficient is depends on whether you’re working out the square, cube, fourth power or whatever.
‘Bernoulli numbers’ means all those coeffiecents listed in order.
I suppose it doesn’t have to be useful for anything, but, for us non-maths types, the level of maths we study does always have practical purposes. It’s kinda like someone not understanding fiction because they’ve only ever been exposed to factual books.
Also, if you don’t understand the maths, then you don’t see the beauty that mathematicians claim is there. It’s pretty easy for anyone to look at a painting and see that it’s beautiful, or at least have a personal opinion on whether or not it’s beautiful, but expecting me to see the beauty in theoretical maths is like expecting me to appreciate the subtle wordplay in a Nepali poem written backwards in doctor’s handwriting and shown to me in the dark.
Yes, well, I did go into a sort of general purpose rant there; I wasn’t intending to single out or attack the posters I quoted in any way, I just took their posts as a jumping-off place to respond to a point that maybe wasn’t really present in the generality that I answered it. It’s just that the attitude I reacted to is such a prevalent one that I didn’t think myself to be out of line in presenting my own take on it.
As for the Bernoulli numbers, all I could hint at from a user-sided perspective is that they crop up in some Taylor expansions (which, to those unfamiliar with the term, are a means to approximate a function that would otherwise be difficult to impossible to evaluate exactly – in physics, sometimes you don’t need the exact function, all the input data being experimental, and thus subject to measurement error, anyway, so you can work with something that gets as close as you need it to be and is simpler to handle in calculations, or even only makes them possible in the first place), or in Euler’s summation formula, which allows numerical integration by relating the value of an integral with that of an infinite sum over supporting nodes (? not sure if this is the right word; I need to get a good scientific/math dictionary some time), which for instance makes them evaluable by computer. So they make some calculations easier sometimes, and an easier way to calculate them in turn potentially makes those calculations easier still, however, I doubt that any reasonable calculation ever goes into territory where one would have to individually evaluate the relevant Bernoulli numbers; rather, one could probably look them up somewhere (the first 10,000 here, for example).
I’ve read through this thread, and the linked Wikipedia article, I still have no clear idea what a Bernoulli number is, why it’s special, or why their decipherment is a newsworthy event beyond “They’re really hard mathematical equations.”
Somehow, I suspect that’s a bit like describing the Second World War as “lots of people shooting at each other”. Basically correct, but completely missing the context that enables the casual observer to understand it.
I’m completely useless with mathematics beyond arithmetic, so all those equations in the Wikipedia article might as well be in Martian. Hell, Egyptian Hieroglyphics make more sense to me than those equations. I get that they’re “Pure Mathematics”, in the sense that (at the present) they have no practical application, but frankly it’s a bit rude for so many people to be sniffily dismissing those of us who just don’t get numbers when we say “Can someone explain this, in English, without using mathematical equations, since I don’t understand them?”
There’s another reason why I can’t give you a long detailed explanation of what Bernoulli numbers are with complete, accurate details of how they are generated and a long exposition about all the ways that they are useful both within and outside of mathematics, making sure that all of this is at a level appropriate for people without a lot of mathematical experience: I don’t have the time. It took me a long time to write the posts above. If I had to spend the time to write everything that you want to know about Bernoulli numbers, it would take a couple solid days of work to look up all the things you want to know and to write a coherent explanation of them. Do you want to pay me for my time?
I think the newsworthiness derives itself more from the fact that the guy who came up with this algorithm is only 16; that’s extraordinary in the same way that such a young guy winning the Australian Open or running a marathon in under two hours is.
I mean, the only things I discovered with 16 were alcohol and weed, and the only mathematics I did with that was figuring out how many beers it took for me to loose my dinner and fall into a drunk stupor, so well… I’m impressed.
With respect (and I appreciate your efforts so far), you’re being way too technical. We don’t want to know everything there is to know about Bernoulli Numbers. We just want to know, in broad terms what they are (They’re a kind of mathematical “riddle”, it seems) and why someone solving them is a big deal (They’re really difficult, as far as it could tell.) If that’s the only way to explain them without using mathematical equations, then fine. Just say that, instead of pulling out more and more equations and integers and variables and other concepts we haven’t used since 5th Form Maths.
The approaches so far seem to be a bit like trying to explain the Scramble For Africa by first insisting everyone learn Swahili- useful for an in-depth study of the subject, but far too complicated for The Edited Highlights.
The explanations given so far have defined Bernoulli numbers. That’s all we can say without a chance to do more research ourselves. You got to understand that Bernoulli numbers aren’t something that every mathematician has at the tip of his tongue. What we’ve said so far is everything we can talk about without doing extensive research ourselves. If you want more detail, it would take days of work to research it. You seem to have the idea that a clear explanation takes little time. On the contrary, the clearer an explanation is, the more time it takes to prepare it.
Fair enough. Several people have thanked you for the explanation you’ve already given, though. Trust me, we don’t want a long, detailed explanation!
Can we try this?
*Bernoulli numbers are some of the numbers that are used when you’re working out the cube of a number in a sequence (or the fourth power of it, etc).
They’re useful in physics when you’re trying to work out the effects of some types of expansion and stuff like that.
You can’t just look at the Bernoulli number for ‘to the power of 73million’ and know what the Bernoulli number for ‘to the power of 73million and 1’ is - it’s much more complicated than that. That’s why it’s impressive that this kid, independently, came up with a way of working out Bernoulli numbers.
*
If that’s all correct (and I’m not certain that it is), then that’s the level of explanation I need right now.
To be exact, Bernoulli numbers are related to how you calculate the sum of the n-th powers of the first m numbers.
Once again, the problem is not providing enough detail. If all I had to do was provide detail, I could just type out the explanation given in a book about Bernoulli numbers. You wouldn’t understand any of it. The problem is clearness. It is very difficult to be clear in any such discussion. It takes a long time to prepare a discussion of such topics at a level that would be useful to you.
For some reason, clicking on quote takes me to a blank reply box that pretends the last two posts don’t even exist, and this is the third time I’ve tried to write this reply because the page keeps refreshing and deleting everything I’ve written. Aaargh! So I’m going to post this just to see if I can post a proper reply afterwards.
So I should rephrase my first paragaph as:
*
Bernoulli numbers are some of the numbers that are used when you’re working out the answer to ‘what do you get if you cube (or square, or whatever) a number in a sequence and then add it to the cubes (or squares, or whatever) of all the numbers that went before it?’ *
No, the lack of detail’s not the problem - it’s the opposite, actually; too much detail has been provided. Clearness is the problem, you’re right - sometimes, the better you understand something, the harder it is for you to explain it to laymen, because you’ve forgotten that said layman will be blinded by words like ‘polynomial’.
Basically, something like that. However, the way you’re saying ‘effects of […] expansions’ makes me think that you believe this to be some sort of physical phenomenon – it’s not, it’s merely a mathematical technique to approximate a function at (or around) a given value, basically by calculating a polynomial (a sum of powers of ‘x’) that ‘comes close’ to the actual function. For instance, an exponential e[sup]x[/sup] can be approximated by a Taylor polynomial of the form 1 + x + [sup]x[sup]2[/sup][/sup]/[sub]2![/sub] + [sup]x[sup]3[/sup][/sup]/[sub]3![/sub] + … + [sup]x[sup]n[/sup][/sup]/[sub]n![/sub] = Σ[sub]n[/sub][sup]x[sup]n[/sup][/sup]/[sub]n![/sub], which at first looks rather messy and not like you’ve actually won anything; but, for practical purposes, it is often enough to only use a couple of the terms in this expansion, so for instance you might ‘break off’ the calculation after the third term and you’re left with a simple quadratic equation, which is often easier to handle (or makes the calculation possible in the first place). For better visualization, here’s a graphic that compares the exponential with the fourth-order Taylor polynomial; as you can see, in a certain range, there is virtually no difference.
ETA:
Oh, well, in that case my explanation is probably not going to help matters. And I was so proud of the coding!
Yeah, sorry, after the word ‘approximate’ I started to feel like the dog in the Scot Adams cartoons. It’s not for lack of trying, though - I just don’t understand the explanations I’ve seen of ‘polynomials,’ and that was part of the ‘basically’ bit!
I’m still not sure, by the way, whether I should thank you for supporting my position, or apologize for getting you into what has to be one of the worst ‘in for a penny, in for a pound’-deals ever…
Redacted for my own sanity. Suffice to say that the math snobbery in this thread is pretty disgusting. I wonder how many of the pooh-poohing mathematicians could have done that at 16.
You’re a scientist. You understand as well as anyone that being a successful scientist is much more about perseverance and dedication rather than brilliance. Why do you find it difficult to believe that mathematics is the same?
I don’t. I had a lot of drive to learn about the world around me when I was 16, but I doubt that I could have given you a treatise on, oh, say, the finer points of DNA topology back then. This kid shows the signs of having both the intellect AND the drive at a very young age.
