Wow, thanks for the update. The article made it sound like it was the discovery of the century. I admit, looking at the equations in the Wiki article made me squiggle-eyed. I’m more English and history, and while math and science fascinate me, you might as well be talking the way adults do in Charlie Brown cartoons for all the sense I can make of it.
Sweden is a death trap! 
No, I read the the thread title, not the link.
Even if no use would ever manifest itself, this line of reasoning just doesn’t work out – thankfully, it’s not like all human endeavour has to be utilitarian in order to be meaningful. There’s no use to a painting, a novel or a pop song, yet individuals excelling in those sectors rarely have their contribution questioned the way people sometimes question the utility of mathematics or similarly esoteric subjects. After all, what’s the real world application of cubism?
I KNOW! I KNOW! Cubism and jazz lead to teen sex!
What could it mean to “solve” a set of numbers?
Okay, that proves it for some facts about prime numbers. What about the rest of pure mathematics? What about the study of large cardinals? I totally disagree with your assertion.
Well, I guess you would know better than I.
I was abbreviating because of the lack of space in the title field. In my OP I said he came up with a formula (apparently not a new discovery) to explain and simplify the numbers.
May I paraphrase the discussion?
Math-type: Hey, look what this kid did!
Non-Math-type: Great! What are Bernoulli Numbers?
Math-type: Erm… Did you bother to read the Wiki link? :rolleyes:
Non-Math-type: Of course, I did. But I’m a Non-Math-type. Those equations are meaningless to me. Will you explain what Bernoulli Numbers are, in such a way that I can understand them?
Math-type: No. I’m not going to try to explain them to you. Another Non-Math-type wanted to know what they’re good for, and I’d rather mock him for not seeing the beauty of Mathematics.
Is that where this thread is going?
Whoops. Sorry I didn’t notice that.
On the topic of usefulness, I’d say the length and breadth of that Wikipedia article is as good an argument as any. With that many connections in places they weren’t looking in, it’s a safe bet that these numbers are being used by someone building iPhones or a Sharper Image air ionizers. Whether ‘useful’ is a question we should bother asking, I’ll turn the mic over to my man G.H. Hardy.
“I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”
This is a man who spent his whole life in number theory, to the exclusion of almost everything else, and still thought it a life well spent. His work doesn’t need any justification to be studied and generalized and whatnot, outside of its own intrinsic beauty and “is eternal because the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years.”. Of course, (supporting my conjecture that only British people can use the word ‘ironic’ correctly) his Mona Lisa, the Hardy-Ramanujan asymptotic formula for partitions of an integer, is used heavily by physicists to hash out Feynman amplitudes for the subatomic doodads they find. And as Captain Carrot mentioned, typing your credit card number in on Amazon would be about as secure as writing it on a bathroom wall without Hardy’s work on the primes.
Let’s find out.
Ahem:
I don’t understand what Bernoulli numbers are and cannot make sense of the equations in the Wiki link. Would you mind explaining them to me in a way a Non-Math person can grasp? In return I’d be happy to discuss poetry with you.
Whoops. Posted before I saw this. It’s an honest question, and I should give it an honest answer. Bernoulli Numbers. 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510 . . . If you can find a pattern in that, you’re probably pretty smart. If you can, at age 16, find a function that produces them (that is, you have an arithmetical formula where plugging in the number n in your formula spits out the nth number), then you’re hella smart. I’ll try to explain where these numbers come from a priori, but I wanted to post this quick, because my post was everything you were complaining about to the nth degree, and I don’t want to look like a jagoff.
What makes them Bernoulli numbers? It looks like they’re decreasing fractions to me.
Wait, I though Bernoulli had something to do with why the shower curtain flies in and sticks to your legs?
Oh, wait, Cecil just thought it did. Never mind.
Well, they are a sequence of fractions which trends broadly downwards in magnitude. But it’s not just any such sequence; it’s a very special one.
Part of why Bernoulli numbers are interesting is because they have many different applications, and so there are many different ways of defining them, which all end up equivalent. I’ll just give one.
Consider the series 0 + 1 + 2 + 3 + … + (X-1). As you may or may not know, this always adds up to X[sup]2[/sup]/2 - X/2. That is, there some fixed polynomial which tells us the answer to the question “What’s the sum of the first X many numbers?”.
And how about the series 0[sup]2[/sup] + 1[sup]2[/sup] + 2[sup]2[/sup] + 3[sup]2[/sup] + … + (X-1)[sup]2[/sup]? (I.e., the sum of the first X many square numbers). Well, this always adds up to X[sup]3[/sup]/3 - X[sup]2[/sup]/2 + X/6. I don’t expect you to know that right off the bat; I’m just telling you that it happens to be the case. Once again, there is a fixed polynomial which gives the result of summing up this series.
And, indeed, as it turns out, a similar thing happens for any power, and uniquely so: there is some unique polynomial which gives the sum of the first X many cubes, there is some unique polynomial which gives the sum of the first X many 4th powers, etc.
So this gives us a sequence of polynomials. How does this relate to Bernoulli numbers? Well, from each of these polynomials, we can pull out the coefficient of X [so, from the polynomial for adding up the first X many numbers, we would pull out the coefficient -1/2; from the polynomial for add up the first X many square numbers, we would pull out 1/6; and so on]. The sequence of numbers this gives us is, let us say by definition, the Bernoulli numbers.
Now, I haven’t yet explained why these are useful or particularly interesting; just hinted at the fact that these happen to come up in many different contexts. But if you just want to know some simple definition for what, at least, the Bernoulli numbers are, well, there you go.
Okay, please be patient with me. I’m trying to understand.
I got 1 + 2 + 3 = 6. I also got that there’s a formula that you can use to plug in any list of sequential numbers (right?) and get the same answer.
What I don’t get are how someone decided that this particular sequence of numbers are Bernoulli numbers. I think you explain it in your 6th paragraph, but I need you to dumb it down some more. Sorry.
Emily Dickinson…quite the writer, eh? 
Nicolas Bourbaki said that he didn’t want to appear to be a jagoff. I was going to reply to him last night that I wasn’t offended, as I was only trying to clarify questions that others posted. But now I’m getting interested.
So here’s a dumb question: Jakob Bernoulli studied the numbers, and they were named after him. Where did the numbers come from? That is, wasn’t there an algorithm to generate the numbers in the first place?
The study or creation of anything can be said to increase posterity, no matter how useless it tends to look on the face of it. Especially those things that seem pointless. There’s no telling what may come of it, until the works been done. This has proven useful for the entire history of humanity.
Let’s start over in the explanation of Bernoulli numbers:
Look at the following sorts of sums:
0 + 1 + 2 + 3 + 4 + 5 + 6 + . . .
The sequence of sums of the nonnegative integers is the following:
0, 1, 3, 6, 10, 15, 21, 28, 36, . . .
But there’s a formula for this:
Y = (X**2)/2 - X/2
(Note: X2 means X squared. X3 means X cubed. X**4 means X to the fourth power. And so on.)
Try it yourself. If you put 1 in for X, you get 1 for Y. If you put 2 in for X, you get 3 for Y. If you put 3 in for X, you get 6 for Y. So this means that to find out the sum of all the first X nonnegative integers, you don’t have to add them up separately, you just put X into that formula and solve for Y.
Then there is the sums of the squares of the nonnegative integers:
0 + 1 + 4 + 9 + 16 + 25 + 36 + . . .
The sums are as follows:
0, 1, 5, 15, 25, 50, 86, . . .
There’s a formula for this:
(X3)/3 - (x2)/2 + X/6
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 multiple 1 by the number three times. The square of a number (the second power of a number) is what you get when you multiple 1 by the number two times. The first power of a number is what you get when you multiple 1 by the number one time. So the zero-th power of a number is what you get when you multiple 1 by the number zero times.
So the sequence of zero-th powers is:
1, 1, 1, 1, 1, 1, . . .
The sequence of first powers is:
0, 1, 2, 3, 4, 5, . . .
The sequence of squares is:
0, 1, 4, 9, 16, 25, . . .
The sequence of cubes is:
0, 1, 8, 27, 64, 125, . . .
O.K., so then the sum of the sequence of zero-th powers is:
1, 2, 3, 4, 5, 6, . . .
The formula for this is X + 1.
Now we’re going to create another sequence. Look at the coefficient for the sum of the zero-th powers. The formula was X + 1. The coefficient for X was 1. The coefficient means the number that the variable is multiplied by in the formula.
Look at the coefficient of the X term in the formula for the sum of the first powers. The formula was Y = (X**2)/2 + X/2, so the coefficient of X was -1/2.
Look at the coefficient of the X term in the formula for the sum of the second powers. The formula was (X3)/3 - (x2)/2 + X/6. The coefficient for X is 1/6.
You can create an formula for any sum of the powers of integers. It turns out that the coefficient for X in the formula for cubes is 0, the coefficient for X in the formula for fourth powers is -1/30, etc.
So we have a sequence that goes like this:
1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, . . .
These numbers are the Bernoulli numbers. There are other ways of defining them, but this is the easiest way to explain it. Take this as being the definition of Bernoulli numbers.
Is this any help?
For those that have attempted to explain Bernouilli numbers to those of us who are Non-Math Types, I thank you. You have fought ignorance and won.
For those who decided to mock those of us for not understanding and wanting to understand, for those who mocked the ways we are attempting to understand (Half Man Half Wit and Commander Keen in particular), well, you can take your fucking Bernouilli numbers, roll them up into a tight little ball and shove them up your ass. Thanks for fighting ignorance but your efforts are no longer needed.
I don’t know why I try anymore.
Go fuck yourselves. I hope an AIDS infested faggot rapes you and you die as a result. Cunts.
Please substitute this for one of my paragraphs:
Now we’re going to create another sequence. Look at the coefficient of X for the sum of the zero-th powers. The formula was X + 1. The coefficient for X was 1. The coefficient means the number that the variable is multiplied by in the formula.
I forgot the words “of X.”