
Fractions in the Farey Sequences and the Stern-Brocot Tree - fanf2
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/fareySB.html
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user51442
Don't forget the Calkin-Wilf tree:
[http://www.cs.nott.ac.uk/~psarb2/MPC/RecountingRationalsTwic...](http://www.cs.nott.ac.uk/~psarb2/MPC/RecountingRationalsTwice.pdf)

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mathgenius
Working out the Stern-Brocot tree using modular arithmetic leads to some
finite shapes because after a while the calculation repeats itself. For
example, using mod 5 you get a dodecahedron! These things are called modular
curves (iirc.) It's a little bit tricky to figure this out, one hint is that
every fraction has two names a/b and -a/-b. In modular arithmetic this is
important, eg. 2/3 == -2/-3 == 3/2\. So every node of the tree has two labels.
(Warning: I may not be explaining this very well.) Once you get to mod 7 you
get a three-holed donut shape, called the Klein quartic.

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mtzet
Not a very pretty website, but certainly a pretty subject!

This really is the best way of explaining where continued fractions come from.
The fact that convergents of the continued fraction provide the best rational
approximation always seemed mysterious to me in the classical exposition, but
it is almost obvious when explained through the Stern-Brocot tree.

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jstanley
I don't know about it not being pretty... I thought the website presented the
content in a clear and easy-to-follow way. And it loaded quickly, I didn't
even need to wait for my browser to download the font before I could see the
text. I wish _more_ websites looked like this, instead of plastered with
confusing navigation, adverts, popups asking for your email address,
"elsewhere on the web", "like, share and subscribe", etc.

This is refreshing in its simplicity.

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theoh
Yes, but this kind of HTML authoring is often accompanied by a puritanical
dedication to "structural markup". It is often evidence of quite an
opinionated, reductive, approach to the web, for better or worse.

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btilly
Useless trivia. You know the 20 questions game? You can actually find any
rational through a binary higher/lower search of the Stern-Brocot tree.
Finding n/m this way takes O(log(max(|n|, |m|))).

The logic is slightly tricky. Some day I'll get bored enough to code it up...

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mathgenius
The fractions in the Stern-Brocot tree approximate any real number. There's a
statement about this that is equivalent to the Riemann hypothesis.

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gabcoh
I Remember that some Project Euler question requires you to generate this
series. I guess I now know how to solve it!

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uryga
Could someone better versed in math explain why this is important/useful? I
skimmed the article, and while I find Ford circles and the trees/tables
presented intriguing, I don't really know what to do with the information
presented. Do these techniques have any uses / reveal anything interesting?

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baking
Because the real numbers are a fiction and fractions are all there is. Google
"wilberger."

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mathgenius
But fractions are also fiction: who says that one can be divided into two?
That's just waiting around for inflation to kick in. And negative numbers:
same. It's just hoping for a bigger number to subtract off from.

Only the natural numbers exist. All the rest is fantasy.

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baking
Well, you can describe a negative number or a rational number, but an infinite
sequence of non-repeating decimals is a different matter.

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uryga
Why?

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baking
Because an infinite series of non-repeating decimals . . . can . . . never . .
. stop . . .

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uryga
That's what I can't understand... For example, I can easily write a program
that produces some digits one by one, indefinitely (excluding mechanical
failures etc). I won't ever see the whole sequence – as it never ends – but to
my mind, it seems like the program describes it. (I think that's basically the
computable reals?)

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baking
Sure,there are a large number of "computable reals" such as square roots and
taylor series expansions (pi, e, sine, cosine, etc.) which can be estimated to
greater and greater precision (always with rational numbers) so we can
reasonably talk about them in the abstract (such as use them in equations)
much in the same way we talk about the imaginary numbers, quarternions, and
octonions. But the idea that you need to construct an entire field of
completely undefinable infinite chains of decimals just to allow for a small
handful of irrational numbers is kind of absurd.

Norman Wildberger (among others) has put a lot of effort into building up the
foundations of mathematics using only the rational numbers and it is quite
interesting how far he has gone. He's considered to be a crank by many, but
you have to appreciate his effort.

To go back to your original question, his Math Foundations video series,
especially #14, #95, and #96 might help.

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uryga
> Well, you can describe a negative number or a rational number, but an
> infinite sequence of non-repeating decimals is a different matter.

I'm a bit confused... it sounded like you had objections to defining numbers
with infinite decimal expansions, but you're okay with e, pi, etc? Or did you
just object to uncomputable reals?

Also I don't know much about this whole topic, but I was taught that you can't
fill a continuous line with just the rational numbers – you need the
"undefinable" ones for that... What do you think about this?

Also thanks for the references, I might check them out to see what Wildeberger
has to say.

