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-   -   16-year old Iraqi solves Bernouilli numbers (https://boards.straightdope.com/sdmb/showthread.php?t=519392)

ivylass 05-30-2009 08:06 AM

16-year old Iraqi solves Bernouilli numbers
 
It took him four months to come up with a formula to explain and simplify the Bernouilli numbers.

I predict this kid will with the Nobel prize one day.

Wendell Wagner 05-30-2009 08:33 AM

Follow the links on the story. The equations he discovered had been known before:

http://news.yahoo.com/s/afp/20090528...20090528174251

News stories like this are nearly always exaggerations. Yes, he's pretty smart, but there are a lot of pretty smart people out there. Look, I've spent my whole adult life with mathematicians. There isn't any shortage of first-rate mathematicians. News stories that make it sound like brilliant mathematicians are so rare that every time a bright teenager makes a interesting (but hardly world-shattering) discovery (which wasn't original anyway) we should hail him as our new mathematical savior are merely a waste of newspaper print (or bandwidth).

ivan astikov 05-30-2009 08:41 AM

Unless a mathematician can explain their ideas in good old-fashioned words, they may as well be speaking in Klingonese as far as I'm concerned.

Wendell Wagner 05-30-2009 08:43 AM

In that case, ivan astikov, I would suggest that you not get a job as a science journalist so that you wouldn't be tempted to write a news story as bad as the one in the link in the OP.

ultrafilter 05-30-2009 11:46 AM

Even if it's not an original discovery, the kid's got talent. Let's hope he can do something with it.

Quote:

Originally Posted by ivan astikov (Post 11193515)
Unless a mathematician can explain their ideas in good old-fashioned words, they may as well be speaking in Klingonese as far as I'm concerned.

I'm not quite sure what you're trying to say here.

Boyo Jim 05-30-2009 11:49 AM

Quote:

Originally Posted by ultrafilter (Post 11193880)
Even if it's not an original discovery, the kid's got talent. Let's hope he can do something with it.



I'm not quite sure what you're trying to say here.

Every equation must be explainable in story form, preferably in terms of apples, oranges, and trains leaving the station at certain times. And none of that acceleration crap either!

I hope the kid's math talents extend to figuring out how far away to stay from abandoned cars, to avoid the blast damage.

Wendell Wagner 05-30-2009 01:29 PM

Boyo Jim, did you bother to read the news story in the link? He's an Iraqi immigrant to Sweden. I don't think there have been a lot of bombings in Sweden.

cmyk 05-30-2009 01:45 PM

Quote:

Originally Posted by ivan astikov (Post 11193515)
Unless a mathematician can explain their ideas in good old-fashioned words, they may as well be speaking in Klingonese as far as I'm concerned.

They could probably explain it in Kingon, too.

But seriously, what do you mean? Equations are "words" in the sense they mean something. Just because you can't "speak the language," doesn't mean it's not important.

Johnny L.A. 05-30-2009 02:05 PM

Quote:

Originally Posted by ivan astikov (Post 11193515)
Unless a mathematician can explain their ideas in good old-fashioned words, they may as well be speaking in Klingonese as far as I'm concerned.

Quote:

Originally Posted by cmyk (Post 11194167)
They could probably explain it in Kingon, too.

But seriously, what do you mean? Equations are "words" in the sense they mean something. Just because you can't "speak the language," doesn't mean it's not important.


He's looking for a translation. Without knowing the math, the equations are meaningless. The news article basically says, 'There are these numbers, eh? And a 16-year-old came up with a formula to solve them.' So one might say, 'Interesting. What are these numbers?' And a mathematician says, 'Oh, they're hjürdi-gjürdi hoompty doompty. Woof!'

Wendell Wagner 05-30-2009 02:18 PM

Click on the Wikipedia article on Bernoulli numbers that's linked to in the OP. That's about a simple an explanation as you're going to get, it appears. Does anyone have a link to a simpler explanation of what they are? Furthermore, click on the tab at the top of the Wikipedia page for the Discussion page on Bernoulli numbers. There is a discussion (at the bottom of the page) about whether there is anything interesting in the 16-year-old's rediscovery of a method of generating the Bernoulii numbers.

Skald the Rhymer 05-30-2009 02:48 PM

Quote:

Originally Posted by ivan astikov (Post 11193515)
Unless a mathematician can explain their ideas in good old-fashioned words, they may as well be speaking in Klingonese as far as I'm concerned.

This sentiment is vaguely disturbing, as it sounds exactly like something Glenn Beck would say.

Stealth Potato 05-30-2009 03:02 PM

Quote:

Originally Posted by Johnny L.A. (Post 11194214)
So one might say, 'Interesting. What are these numbers?' And a mathematician says, 'Oh, they're hjürdi-gjürdi hoompty doompty. Woof!'

"...Bork bork bork?"

:D

Markxxx 05-30-2009 03:21 PM

The whole idea is, "it makes good copy." Poor Iraqi boy knows math. Iraq is not a useless country there are people who know stuff. Of course all the people that know stuff, leave :)

Basically it's good copy that reminds everyone Arabs, Muslims and Iraqis aren't any different, and not all are poor and hopeless and out of the ravages of war born is genius.

Convict 05-30-2009 03:49 PM

Quote:

Originally Posted by Wendell Wagner (Post 11194238)
Click on the Wikipedia article on Bernoulli numbers that's linked to in the OP. That's about a simple an explanation as you're going to get, it appears.

When people ask why I majored in history I will give them that link.

As Johnny L.A. pointed out, without the math those equations have no meaning. So let me ask this--what real world application do Bernouilli numbers have? Did they offer a solution to a problem that resulted in a great scientific or technological advance? Or are they simply something that math folks use to show how smart they are?

My brain is not wired for math so any help would be appreciated.

(And if any of my questions could be answered by looking at the wiki page I apologize--all those equations shorted out my brain.)

Scissorjack 05-30-2009 06:17 PM

Quote:

Originally Posted by Wendell Wagner (Post 11194134)
I don't think there have been a lot of bombings in Sweden.

That's probably a great comfort to Olof Palme.

Wendell Wagner 05-30-2009 06:30 PM

What does that have to do with anything? Olof Palme died from a gunshot, not a bombing, and I never said that there were no bombings, just that there weren't a lot of them. My post was in reply to Boyo Jim, who didn't notice that the kid lives in Sweden, not Iraq.

Nicolas Bourbaki 05-30-2009 06:43 PM

Quote:

Originally Posted by Convict (Post 11194457)
As Johnny L.A. pointed out, without the math those equations have no meaning. So let me ask this--what real world application do Bernouilli numbers have? Did they offer a solution to a problem that resulted in a great scientific or technological advance? Or are they simply something that math folks use to show how smart they are?

Why can't they do both?

ETA: They've printed a retraction. He did find an algorithm, but it had been discovered before, and he is not getting early admission to the University of Uppsala.

Captain Carrot 05-30-2009 06:57 PM

Quote:

Originally Posted by Convict (Post 11194457)
When people ask why I majored in history I will give them that link.

As Johnny L.A. pointed out, without the math those equations have no meaning. So let me ask this--what real world application do Bernouilli numbers have? Did they offer a solution to a problem that resulted in a great scientific or technological advance? Or are they simply something that math folks use to show how smart they are?

There is no area of math that does not apply to some problem people face. Prime numbers used to be thought of as the most ivory-tower discipline ever; after all, what use could they possibly be? Enter cryptography.

Wendell Wagner 05-30-2009 07:03 PM

Nicolas Bourbaki, note that that's the same article that I linked to in post #2, although it's on a slightly different webpage.

Commander Keen 05-30-2009 07:09 PM

Quote:

Originally Posted by Convict (Post 11194457)
So let me ask this--what real world application do Bernouilli numbers have? Did they offer a solution to a problem that resulted in a great scientific or technological advance? Or are they simply something that math folks use to show how smart they are?

Yeah, it's not like anyone would ever find pure math interesting, or study it for its own sake. If you're not building rockets or curing cancer, then you're just showing off, right?

ivylass 05-30-2009 07:32 PM

Quote:

Originally Posted by Nicolas Bourbaki (Post 11194862)
ETA: They've printed a retraction. He did find an algorithm, but it had been discovered before, and he is not getting early admission to the University of Uppsala.

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.

Boyo Jim 05-30-2009 07:36 PM

Quote:

Originally Posted by Wendell Wagner (Post 11194134)
Boyo Jim, did you bother to read the news story in the link? He's an Iraqi immigrant to Sweden. I don't think there have been a lot of bombings in Sweden.

Sweden is a death trap! :) No, I read the the thread title, not the link.

Half Man Half Wit 05-30-2009 07:38 PM

Quote:

Originally Posted by Convict (Post 11194457)
When people ask why I majored in history I will give them that link.

As Johnny L.A. pointed out, without the math those equations have no meaning. So let me ask this--what real world application do Bernouilli numbers have? Did they offer a solution to a problem that resulted in a great scientific or technological advance? Or are they simply something that math folks use to show how smart they are?

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?

Boyo Jim 05-30-2009 07:43 PM

Quote:

Originally Posted by Half Man Half Wit (Post 11194971)
...After all, what's the real world application of cubism?

I KNOW! I KNOW! Cubism and jazz lead to teen sex!

Jamaika a jamaikaiaké 05-30-2009 08:12 PM

Quote:

Originally Posted by ivylass (Post 11193472)
16-year old Iraqi solves Bernouilli numbers

What could it mean to "solve" a set of numbers?

Quote:

Originally Posted by Captain Carrot (Post 11194895)
There is no area of math that does not apply to some problem people face. Prime numbers used to be thought of as the most ivory-tower discipline ever; after all, what use could they possibly be? Enter cryptography.

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.

Captain Carrot 05-30-2009 08:14 PM

Well, I guess you would know better than I.

ivylass 05-30-2009 08:14 PM

Quote:

Originally Posted by Jamaika a jamaikaiaké (Post 11195051)
What could it mean to "solve" a set of numbers?

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.

Johnny L.A. 05-30-2009 08:25 PM

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?

Nicolas Bourbaki 05-30-2009 08:30 PM

Quote:

Originally Posted by Wendell Wagner (Post 11194908)
Nicolas Bourbaki, note that that's the same article that I linked to in post #2, although it's on a slightly different webpage.

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.

ivylass 05-30-2009 08:38 PM

Quote:

Originally Posted by Johnny L.A. (Post 11195092)
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?

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.

Nicolas Bourbaki 05-30-2009 08:47 PM

Quote:

Originally Posted by Johnny L.A. (Post 11195092)
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. 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.

ivylass 05-30-2009 08:50 PM

What makes them Bernoulli numbers? It looks like they're decreasing fractions to me.

Marlitharn 05-31-2009 03:20 AM

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.

Indistinguishable 05-31-2009 03:24 AM

Quote:

Originally Posted by ivylass (Post 11195168)
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.

ivylass 05-31-2009 08:46 AM

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? ;)

Johnny L.A. 05-31-2009 08:47 AM

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?

cmyk 05-31-2009 09:32 AM

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.

Wendell Wagner 05-31-2009 11:11 AM

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: X**2 means X squared. X**3 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:

(X**3)/3 - (x**2)/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 (X**3)/3 - (x**2)/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?

Convict 05-31-2009 01:34 PM

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.

Wendell Wagner 05-31-2009 01:40 PM

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."

Wendell Wagner 05-31-2009 01:42 PM

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."

Johnny L.A. 05-31-2009 01:56 PM

Quote:

Originally Posted by Wendell Wagner (Post 11196463)
Is this any help?

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 (X**3)/3 - (x**2)/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...

Marley23 05-31-2009 01:59 PM

Mod Warning
 
Convict, you are way out of line. This is a formal warning: don't do this again.

Tangent 05-31-2009 02:22 PM

Quote:

Originally Posted by Wendell Wagner (Post 11196463)
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, . . .

Wouldn't those sums be 0, 1, 5, 14, 30, 55, 91 ?

Thanks for the explanation!

Half Man Half Wit 05-31-2009 02:52 PM

Quote:

Originally Posted by Convict (Post 11196825)
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.

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.

Indistinguishable 05-31-2009 03:00 PM

Quote:

Originally Posted by Tangent (Post 11196970)
Wouldn't those sums be 0, 1, 5, 14, 30, 55, 91 ?

Yes, they would be, and while we're picking nits,
Quote:

Originally Posted by Wendell Wagner (Post 11196463)

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.

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 0**p + 1**p + ... + (X-1)**p. When p is 0, this comes out to X, rather than X+1. (You could take or leave the 0**p 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).]

But these are just minor points.

Jamaika a jamaikaiaké 05-31-2009 03:04 PM

Quote:

Originally Posted by Johnny L.A. (Post 11195092)
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.

Are you referring to some other thread, maybe?

Indistinguishable 05-31-2009 03:14 PM

Quote:

Originally Posted by ivylass (Post 11196193)
What I don't get are how someone decided that this particular sequence of numbers are Bernoulli numbers.

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.

Indistinguishable 05-31-2009 03:38 PM

Eh, one more, ultra-minor correction to Wendell Wagner's good post:
Quote:

Originally Posted by Wendell Wagner (Post 11196463)
0, 1, 3, 6, 10, 15, 21, 28, 36, . . .

But there's a formula for this:

Y = (X**2)/2 - X/2

(Note: X**2 means X squared. X**3 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.

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]

Wendell Wagner 05-31-2009 04:29 PM

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.


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