Please substitute this for another of my paragraphs:
I have to define something that’s slightly confusing now - the zero-th powers of numbers. The cube of a number (the third power of a number) is what you get when you multiply 1 by the number three times. The square of a number (the second power of a number) is what you get when you multiply 1 by the number two times. The first power of a number is what you get when you multiply 1 by the number one time. So the zero-th power of a number is what you get when you multiply 1 by the number zero times.
I made myself look like an idiot who doesn’t know how to spell the word “multiply.”
It’s been 30 years since I’ve taken Calc, and longer than that for Algebra; so I’m very, very rusty.
Still, the way you’ve written the equations is more understandable to me that the equations on the Wiki page. I’ll still need some help seeing the formula for the Bernoulli numbers in (X3)/3 - (x2)/2 + X/6 format, but I’m beginning to see the light.
I think I’ll see if I can get some refresher training at the local college…
Woah. I don’t even know what to say to this. I didn’t even intend to mock. And such an outburst is the result? A wish for death by AIDS infested faggot rape? This is so over the top that I can’t even get outraged about it. I hope you don’t have a temper like this in real life.
Technically, to be consistent with your indexing conventions everywhere else, the formula you want here is just X. [That is, we’ve been using the formulas for 0p + 1p + … + (X-1)p. When p is 0, this comes out to X, rather than X+1. (You could take or leave the 0p term, and it wouldn’t matter so far as our purposes go, but even if you left it off here, the p = 0 case would come out to X - 1, rather than X+1).]
Well, to know whether something is or is not the Bernoulli numbers, you need to first have some definition in mind of what “the Bernoulli numbers” means. So what we’ve presented in this thread is a particular definition of the term “the Bernoulli numbers”. There isn’t really any question “But how do you know that that sequence (which Indistinguishable and Wendell Wagner brought up) is the Bernoulli numbers?”, because, by definition, it is.
That is, there’s nothing fancier involved in deciding that this particular sequence of numbers is the Bernoulli numbers than in deciding that the particular sequence of letters D-O-G is the name for a particular domesticated caniform. It’s just a definition.
Eh, one more, ultra-minor correction to Wendell Wagner’s good post:
You have the right polynomial, but your explanation of plugged-in values is off by one: If you put 1 in for X, you get 0 for Y. If you put 2 in for X, you get 1 for Y. If you put 3 in for X, you get 3 for Y. You have to put 4 in for X to get 6 for Y.
That is, the square-summing formula of concern is the one where, whatever you plug in for X, the result will be the sum of the consecutive squares starting from the square of 0 and going up to, but not including, the square of X. [And the same for the formulas for summing any other power]
Could someone go through my explanation and fix all the mistakes and then write an even clearer and fuller explanation of the Bernoulli numbers? It would be nice if you could go beyond what I said and explain some of the other ways that Bernoulli numbers are used in mathematics. Please post that explanation in full in this thread. My explanation was written in haste, and I don’t actually know that much about the Bernoulli numbers.
I cannot stand people who pride themselves on ignorance and cannot understand why such people look at TSD.
I have known a number of prodigies in my life and most of them (not all, but most) never amounted to much. A fact that made me note that if you don’t graduate from HS, you probably won’t graduate from college. The first one I knew went from 11B in HS to college, taking a graduate math course along the way. Although he later awarded himself a PhD (probably with more justification than one from U Phoenix), he never proved a theorem. And I’ve known others. I hope for the best for this kid, but I will not hold my breath. There is, incidentally, no Nobel prize in math. Nobel distrusted theoreticians. Although theoreticians have won the physics prize it was only by ignoring Nobel’s will. There is an Abel prize given out by Norway that is supposed to make up for it.
Hari Seldon, I can’t figure out what you’re talking about. What is TSD? What do you mean by “he awarded himself a PhD”? You can’t award yourself a doctorate. What is 11B? I presume that it means the second half of the eleventh grade. I presume then that this prodigy skipped the last year of high school. It would really help is you didn’t assume that everyone else knew the same abbreviations that you do.
I don’t have the citations at the moment, but I don’t think that it’s true that most prodigies burn out. It’s my observation that most of them do quite well. The only thing is that they don’t really do that much better than people who go through high school, college, and grad school at the normal rate. I’m not sure if I would call this Iraq-born Swedish kid a prodigy. He’s pretty smart, but we don’t have any information about how well he’s done in general.
I’ve known a lot of first-rate mathematicians, and very few of them have been prodigies, in the sense of going through the educational system faster than average. In general, they got through school at the normal rate. In general they did quite well at each level, but even that isn’t always true. The notion that mathematicians are constantly looking to find teenage mathematicians who discover important new mathematics is ridiculous. Sometimes mathematicians will publish their first paper while they are still undergraduates. Rarely it will be an important paper. Even first-rate mathematicians aren’t generally expected to publish papers until they are in grad school. The notion that there is any significant number of mathematical prodigies out there who are making important mathematical discoveries while they are still teenagers is ridiculous. In mathematics, like in any field, you have to learn a large body of present knowledge to be able to contribute to the field.
Indistinguishable’s explanation was superb. Likewise, net of minor typos, Wendell’s. Now we know what they are.
Can we get a similar explanation of what they’re useful for? Or at least where else they are found in math?
As a comparison, it’s nice to know that log[sub]e[/sub] x= y means that e[sup]y[/sup] = x. That helps explain how to compute, at least in principle, the log[sub]e[s/sub] of any number. But what’s *useful *about logarithms is they let you perform multiplication by using just addition.
Alas, I should say, despite the typo-nitpicking on your post, the same applies to me. I can give the basic introductions, but so far as the actual in-depth applications go, all I can do is digest the Wikipedia.
The Bernoulli numbers are one of those things that everyone’s familiar with, but only a very few people can say exactly why. Even Knuth covers them in Concrete Mathematics without really explaining what the purpose of covering them is. The short version of the Wikipedia article (and everything else I’ve read on the topic) is that having the Bernoulli numbers around lets you write down some other formulas more simply than you would be able to if you didn’t have them. Note that this also means that, if you have an efficient algorithm for computing Bernoulli numbers, you can apply that to simplify the computation of all of those formulas.
Okay, I read ultrafilter’s post, albeit too late. And you will note he’s no help whatsoever.
I’ve written an inefficient (if you can call a program running on a 1mhz computer that listed them faster than one could read them “inefficient”) program to list them. Did the same with an Sieve of Eratosthenes program for a ZX81 that was (barely) faster than doing it in my head. As an incompetent programmer I’m unimpressed by your protege.
They wouldn’t change at all. Most things don’t; only things that explicitly deal with notation change depending on notation; everything else stays the same, regardless of how you choose to write it down.
To pre-empt further misunderstandings, I don’t intend mockery, like, at all. I can certainly sympathise with finding some concepts of mathematics esoteric, or simply not caring much about it. It’s a perfectly valid stance to take – like, for instance, I don’t care terribly much about wine, while to some there is evidently a whole world of experience hidden in nuances I lack the taste buds to detect, and have no interest in developing them.
My question is as honest as it is possible for me to be – why does everything in mathematics need a use (beyond the use it has as simply being a piece of mathematics)? A mathematician might spend months working out a sublime proof to a theorem that isn’t really anything beyond a beautiful piece of mathematics. Similarly, a painter might spend a considerable amount of time producing a painting that isn’t really anything beyond a beautiful picture. It might be something abstract, something I haven’t developed a taste for, or even something I consider to be not very artful – but I won’t question it with regards to its use. I don’t expect a painting to be useful, so why would I expect a use of every random piece of mathematics? Both the painting and the theorem (and the fine wine, or a novel, or heck, even professional sports performances) have at least one thing in common, namely that they entertain (in the broadest sense) and interest a certain subset of people – those with a taste for art, or mathematics (or wine, literature, football…). That a lot of mathematics eventually does turn out to be useful in some way doesn’t even necessarily enter the discussion, and neither does the fact that independently of its usefulness, every piece of mathematics increases the totality of our knowledge. Mathematics – like painting, writing, sport, mountain climbing, what have you – is worth doing for its own sake, or at least that’s my take on it.
