4 ms·

355/113 is the 4th continued fraction of pi, its 4th best approximation. Transcendent numbers have an interesting property: they are the best approximated by ra

by akomtu 4y ago

355/113 is the 4th continued fraction of pi, its 4th best approximation. Transcendent numbers have an interesting property: they are the best approximated by rational numbers, they are the closest to them in a sense; unlike some irrational numbers, such as phi, that are really hard to approximate. There is a curious hypothesis that the world arranges itself by the continued fraction powers of number e, to avoid destructive resonances that would occur if a rational number was used. In other words, the reason period of electron motion or a planet motion is so close to e^(p/q) because that number is very close to a rational fraction, but isn't rational.

klodolph 4y ago
Are you talking about Liouville numbers? Liouville numbers are numbers which can be "best approximated by rational numbers" in a certain sense. A number x is a Liouville number if for every integer n>0, there exist integers (p,q) such that 0<|x-p/q|<1/q^n No algebraic numbers have this property, therefore all of the Liouville numbers are transcendental. However! The reverse is not true; not all transcendental numbers are Liouville. Neither pi nor e are Liouville numbers.
- akomtu 4y ago
  Liouville-ness is a very strict requirement. Instead, we can define "K-liouville numbers, that for all n < K have this property", and then pi is at least a 3-liouville number, which is pretty good.
  - klodolph 4y ago
    Is this the irrationality measure? https://mathworld.wolfram.com/IrrationalityMeasure.html https://mathworld.wolfram.com/IrrationalityMeasure.html I don't think the value for pi is known. The value for e is 2, which is the same value for e.g. sqrt 2.
kbelder 4y ago
>...to avoid destructive resonances that would occur if a rational number was used Hey, thank you. That is obvious upon hearing, and makes perfect intuitive sense, but I might never have thought of it.
codeflo 4y ago
> There is a curious hypothesis that the world arranges itself by the continued fraction powers of number e Sounds intriguing. Do you have any references off the top of your head, or at least some keywords that I can google to learn more?
- akomtu 4y ago
  Iirc, the article was called "physics of transcendent numbers".

by akomtu 4y ago

klodolph 4y ago

Are you talking about Liouville numbers? Liouville numbers are numbers which can be "best approximated by rational numbers" in a certain sense. A number x is a Liouville number if for every integer n>0, there exist integers (p,q) such that 0<|x-p/q|<1/q^n No algebraic numbers have this property, therefore all of the Liouville numbers are transcendental. However! The reverse is not true; not all transcendental numbers are Liouville. Neither pi nor e are Liouville numbers.

akomtu 4y ago
Liouville-ness is a very strict requirement. Instead, we can define "K-liouville numbers, that for all n < K have this property", and then pi is at least a 3-liouville number, which is pretty good.
- klodolph 4y ago
  Is this the irrationality measure? https://mathworld.wolfram.com/IrrationalityMeasure.html https://mathworld.wolfram.com/IrrationalityMeasure.html I don't think the value for pi is known. The value for e is 2, which is the same value for e.g. sqrt 2.

kbelder 4y ago

>...to avoid destructive resonances that would occur if a rational number was used Hey, thank you. That is obvious upon hearing, and makes perfect intuitive sense, but I might never have thought of it.

codeflo 4y ago

> There is a curious hypothesis that the world arranges itself by the continued fraction powers of number e Sounds intriguing. Do you have any references off the top of your head, or at least some keywords that I can google to learn more?

akomtu 4y ago
Iirc, the article was called "physics of transcendent numbers".