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Old 05-31-2009, 03:24 AM
Indistinguishable is offline
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Quote:
Originally Posted by ivylass View Post
What makes them Bernoulli numbers? It looks like they're decreasing fractions to me.
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 X2/2 - 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 02 + 12 + 22 + 32 + ... + (X-1)2? (I.e., the sum of the first X many square numbers). Well, this always adds up to X3/3 - X2/2 + 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.

Last edited by Indistinguishable; 05-31-2009 at 03:28 AM.