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#8501 2013-02-25 00:17:43

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

repeat step 2.

Last edited by bobbym (2013-02-25 00:18:28)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8502 2013-02-25 00:19:07

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Doing it by hand, I mean.

#8503 2013-02-25 00:26:04

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

Hi;

I do not know of a hand method. I think I remember one though but do not remember where I saw it.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8504 2013-02-25 00:26:51

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Do you remember what it was?

#8505 2013-02-25 00:27:10

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Also, why does this method solve Pell equations?

#8506 2013-02-25 00:29:29

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

For that we would have to ask Mr. Fermat and Mr. Legendre.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8507 2013-02-25 00:31:09

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

For which question?

#8508 2013-02-25 00:37:56

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

How taking the cf and convergents solves a Fermat-Pell equation.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8509 2013-02-25 00:38:28

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

It's an unsolved problem?

#8510 2013-02-25 00:46:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

I do not think so. This is all covered in number theory books that do a little computation.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8511 2013-02-25 04:42:42

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I will see if I can find out about it somewhere, then...

Nothing from adriana yet, 16:41. Usually on Mondays she contacts me while she is at school.

#8512 2013-02-25 04:57:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

The method I showed is very important because it finds a simple continued fraction. This simple continued fraction's convergents are the best rational approximations of the constant. Other cf's do not have that property.

Look here:

http://en.wikipedia.org/wiki/Continued_fraction

Last edited by bobbym (2013-02-25 05:01:36)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8513 2013-02-25 05:12:48

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

I have skimmed that page for days but I've still not been able to find a method for finding a simple continued fraction that differs from yours (I believe they use your method to find the continued fraction for pi). However, I did notice that on the Wiki page for sqrt(3), they said this:

The square root of 3 can be expressed by generalised continued fractions such as [2; -4, -4, -4, ...] which is identical to [1; 1, 2, 1, 2, 1, 2, ...] evaluated at every second term.

The trouble is, I don't understand what they mean by that... this seems to be the important bit because they can get the simple CF from a generalised one. Do you know what they mean by 'evaluated at every second term'?

#8514 2013-02-25 05:19:10

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

I think that is a misprint.

Generalized ones a generated using other methods some of them tricky, than the floor method.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8515 2013-02-25 05:21:07

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

But, are the two continued fractions I mentioned identical?

They do both converge to root 3, but I am unsure how I show that they're actually the same fraction...

#8516 2013-02-25 05:26:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

I think that is a misprint. The convergents of the two are different.

There is no 30 / 17 in the second group of convergents.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8517 2013-02-25 05:29:16

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Hmm... so, do you remember the other way to get the simple continued fraction? Every page I go to seems to use your method but it is difficult to do by hand...

#8518 2013-02-25 05:31:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

First things first.

does not converge to the √3


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8519 2013-02-25 05:35:49

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes it does... I evaluated it after a few terms and I got root 3 with an error of only 0.0000274 %.

#8520 2013-02-25 05:39:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

I am getting convergence to something close to 1.76393202250021


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8521 2013-02-25 05:44:59

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

And, you are using the CF

?

I am getting root 3...

#8522 2013-02-25 05:53:17

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

Yes that cf is correct but is that

{2,-4,-4,-4,-4,-4...}?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8523 2013-02-25 05:54:38

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Yes, they are just distributing the negative through the denominator, I think.

#8524 2013-02-25 05:58:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: Linear Interpolation FP1 Formula

I do not think that is correct.

{2,-4,-4,-4,-4,-4...} is

which is not converging to the √3

Last edited by bobbym (2013-02-25 05:59:28)


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#8525 2013-02-25 06:05:44

zetafunc.
Guest

Re: Linear Interpolation FP1 Formula

Hmm, I agree, that one does not converge...

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