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#8526 2013-02-26 05:09:11
- bobbym
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Re: Linear Interpolation FP1 Formula
Yes, the general form for a simple cf is
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8527 2013-02-26 05:16:13
#8528 2013-02-26 05:22:30
- bobbym
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Re: Linear Interpolation FP1 Formula
They are not the same obviously but we might be able to prove they both converge to the same thing.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8529 2013-02-26 05:26:53
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
They said something about evaluating that first fraction at every second term, to get the second fraction...
#8530 2013-02-26 05:28:22
- bobbym
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Re: Linear Interpolation FP1 Formula
And how does that prove they are the same?
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8531 2013-02-26 05:30:13
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
It doesn't but they are claiming there is a way to get the simple CF from the first one...
Or maybe just someone on Wikipedia typed it for the hell of it? Maybe that's why they didn't elaborate on what they meant exactly...
#8532 2013-02-26 05:42:38
- bobbym
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Re: Linear Interpolation FP1 Formula
I do not see any way right now of deriving one from the other.
But I can prove they both converge to the same thing.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8533 2013-02-26 06:17:11
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
What methods of proof do you use?
Also, still nothing from adriana...
#8534 2013-02-26 06:18:57
- bobbym
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Re: Linear Interpolation FP1 Formula
Consider her dead meat.
It is possible to prove what they converge to by algebra and a little trickery.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8535 2013-02-26 06:23:27
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
You may be right... this could venture into F territory. Something is wrong. Just a pity it ended so abruptly.
You mean trying to generate the CF?
#8536 2013-02-26 06:26:51
- bobbym
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Re: Linear Interpolation FP1 Formula
No;
Take a look at the first cf and call it x
Last edited by bobbym (2013-02-26 06:27:21)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8537 2013-02-26 08:56:03
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
Sorry, I had to take my sister to my grandmother's house.
I see, you just get the fraction on its own, invert it, and repeat, and you end up with a quadratic which has a root at x = √3. I imagine the same thing would work for the other CF.
#8538 2013-02-26 09:08:28
- bobbym
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Re: Linear Interpolation FP1 Formula
Hi;
when you solve for x you get √3
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8539 2013-02-26 09:10:47
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
Yes, that is what I meant -- eventually when you follow those steps, you get the original fraction (x again).
#8540 2013-02-26 09:11:58
- bobbym
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Re: Linear Interpolation FP1 Formula
Yes, somewhere in th cf there will be a repeat of x. You just replace it and solve for x.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8541 2013-02-26 09:15:26
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
That seems pretty handy to check if these Wikipedia CFs converge as they're supposed to... but, I suppose this method might be hairy if you have a long period, for large n, say.
#8542 2013-02-26 09:16:40
- bobbym
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Re: Linear Interpolation FP1 Formula
It is easier to use numerical methods and arrive at an experimental result.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8543 2013-02-26 09:26:06
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
We looked at continued fractions and applying them to solve Pell's equation, are there any applications of infinite nested square roots? Those seem interesting too.
#8544 2013-02-26 09:36:55
- bobbym
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Re: Linear Interpolation FP1 Formula
I do not know of any offhand apps but they are solved in the same way as a cf.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8545 2013-02-26 09:42:48
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
Yes, I have read about them. The Wiki article is a lot shorter than the CFs one though.
#8546 2013-02-26 09:46:35
- bobbym
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Re: Linear Interpolation FP1 Formula
They are less useful apparently. CF's are very big in number theory and numerical analysis.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8547 2013-02-26 09:48:28
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
What other things are they used for in number theory?
#8548 2013-02-26 09:53:32
- bobbym
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Re: Linear Interpolation FP1 Formula
Just for the Fermat - Pell equation as far as I know. But these are an important class of diophantine equations. They are more important I think in numerical analysis.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#8549 2013-02-26 10:55:29
- zetafunc.
- Guest
Re: Linear Interpolation FP1 Formula
A girl asked me what I was doing today in chemistry, I was trying to solve a Pell equation using one of my CFs. When I tried to show her what a CF was, she said "I'd rather paint my nails".
23:53, nothing from adriana at all. Sigh...
#8550 2013-02-26 11:02:38
- bobbym
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Re: Linear Interpolation FP1 Formula
Yikes, an intellectual!
I got another method for CF's.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.