Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Yes, the general form for a simple cf is

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**zetafunc.****Guest**

But how do you show that

and

are the same?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**zetafunc.****Guest**

They said something about evaluating that first fraction at every second term, to get the second fraction...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

And how does that prove they are the same?

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

Offline

**zetafunc.****Guest**

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...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

What methods of proof do you use?

Also, still nothing from adriana...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

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?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

No;

Take a look at the first cf and call it x

*Last edited by bobbym (2013-02-25 07:27:21)*

**In mathematics, you don't understand things. You just get used to them.**

Offline

**zetafunc.****Guest**

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.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Hi;

when you solve for x you get √3

**In mathematics, you don't understand things. You just get used to them.**

Offline

**zetafunc.****Guest**

Yes, that is what I meant -- eventually when you follow those steps, you get the original fraction (x again).

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

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.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

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.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

Yes, I have read about them. The Wiki article is a lot shorter than the CFs one though.

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

What other things are they used for in number theory?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

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.**

Offline

**zetafunc.****Guest**

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...

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 89,124

Yikes, an intellectual!

I got another method for CF's.

**In mathematics, you don't understand things. You just get used to them.**

Offline