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Talk:Irrational numbers

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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 
n
of an algebraic number 
ξ
, the larger the denominator 
q
needs to be, such that

Since

and

where 
r
q
is the reduced form of 
r
q
for which 
q
is much smaller than 
q
for some small 
s
coprime to 
q
, would that explain why? — Daniel Forgues 22:18, 16 March 2016 (UTC)
For transcendental numbers, the denominator 
q
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 
    0
    of an uncountable set with cardinality 
    20
    ) 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 
    0
    of the real line is marked);
  • mark all the quadratic numbers of the complex plane (thus an everywhere dense subset with cardinality 
    0
    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 
    0
    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 
    0
    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 
    0
    of quintic numbers);
  • (...)
  • finally (can you?)*, mark all the transcendental numbers of the complex plane (thus between any pair of those 
    0
    algebraic numbers, you marked an uncountable infinity 
    20
    of transcendental numbers) and now all the 
    20
    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 
0
of numbers!)

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