See User_talk:Alonso_del_Arte#Irrational_numbers. — Daniel Forgues 16:27, 16 March 2016 (UTC)

## The higher the degree of an algebraic number, the faster the sequence of convergents converges

Is that because the higher the degree

of an algebraic number

, the larger the denominator

needs to be, such that

- ${\begin{array}{l}\displaystyle {\left\vert \xi -{\frac {p}{q}}\right\vert <\epsilon ,\quad (p,q)=1.}\end{array}}$

Since

- ${\begin{array}{l}\displaystyle {\left\vert \xi -{\frac {p}{q}}\right\vert =\left\vert \xi -\left({\frac {r+s}{q}}\right)\right\vert =\left\vert \xi -\left({\frac {r}{q}}+{\frac {s}{q}}\right)\right\vert =\left\vert \xi -\left({\frac {r^{\prime }}{q^{\prime }}}+{\frac {s}{q}}\right)\right\vert ,\quad r+s=p,\quad (s,q)=1,}\end{array}}$

and

- ${\begin{array}{l}\displaystyle {\left\vert \xi -{\frac {r^{\prime }}{q^{\prime }}}\right\vert <\epsilon +\left\vert {\frac {s}{q}}\right\vert ,\quad (s,q)=1,}\end{array}}$

where

is the reduced form of

for which

is much smaller than

for some small

coprime to

, would that explain why? —

Daniel Forgues 22:18, 16 March 2016 (UTC)

For transcendental numbers, the denominator

would need to be larger than for any algebraic number, which would explain why they are the easiest to approximate with rational numbers. —

Daniel Forgues 22:18, 16 March 2016 (UTC)

## About densities

All the following statements are true (am I correct?)

- rational numbers are everywhere dense but have density 0 among the up to [real] quadratic numbers of the real line;
- up to quadratic numbers are everywhere dense but have density 0 among the up to cubic numbers of the complex plane;
- up to cubic numbers are everywhere dense but have density 0 among the up to quartic numbers of the complex plane;
- up to quartic numbers are everywhere dense but have density 0 among the up to quintic numbers of the complex plane;
- (...)
- algebraic numbers are everywhere dense but have density 0 (worst than that, they are a countable subset with cardinality of an uncountable set with cardinality ) among the complex numbers.

Am I correct? — Daniel Forgues 22:18, 16 March 2016 (UTC)

## Marking up the numbers of the complex plane

You start with a complex plane with all numbers unmarked, and then (am I correct?)

- mark all the rational numbers of the complex plane (thus an everywhere dense subset with cardinality of the real line is marked);
- mark all the quadratic numbers of the complex plane (thus an everywhere dense subset with cardinality of the complex plane is marked);
- mark all the cubic numbers of the complex plane (thus between any pair of up to quadratic numbers, you marked a countable infinity of cubic numbers);
- mark all the quartic numbers of the complex plane (thus between any pair of up to cubic numbers, you marked a countable infinity of quartic numbers);
- mark all the quintic numbers of the complex plane (thus between any pair of up to quartic numbers, you marked a countable infinity of quintic numbers);
- (...)
- finally (can you?)*, mark all the transcendental numbers of the complex plane (thus between any pair of those algebraic numbers, you marked an uncountable infinity of transcendental numbers) and now all the numbers of the complex plane are marked.

* I don't think you can: the transcendental numbers are a non denumerable set (you can't go about marking a first one, a second one, ... because this will only mark a countable infinity

of numbers!)

Am I correct? — Daniel Forgues 22:18, 16 March 2016 (UTC)