Irrational numbers are uncountable while rational numbers are countable. Archimedes theorem says: there exsist a rational number between any two irrational numbers, so there must be rational numbers as much as irrational numbers.So rational numbers must be uncountable like the irrational numbers. Or irrational numbers must be countable like rational numbers.
Natural number is not irretional just directly by definition (remind that a number k is said to be irrational if there are no two natural numbers m and n such that k = m/n). In case k is natural we may just put m = k and n=1 to obtain k = m/n. As to the countability of the set of rational numbers - it would be better if you'l try to find some enumeration of this set by yourself. This exercise is not terribly hard and may bring you a real pleasure.
@Valery Vasil'ev
It is obvious that rational numbers are countable while irrational numbers are uncountable, this is not our question. The question is that: lf we accept Archimedes theorem (there exsist a rational number between any two irrational numbers) this would imply that both of rational and irrational numbers must be countable (or both are uncountable). Otherwise Archimedes theorem is wrong.
First: it is not obvious that rational numbers are countable (please, give a proof, at least, it's main steps). Second, what you call Archimedian theorem DOESN'T IMPLY the conclusion you state (please, give a proof of your assertion if you don't agree).
@Valery Vasil'ev
Cantor gives the proof of the uncountablity of the irrational numbers. so it is well known. Archimedes theorem would numberlize each irrational number by its neigberhood rational number. So both must be countable or both uncountable.
Dear Salah A. Mabkhout Salah, dear Valery Vasil'ev,
I am not in the area and hence may be very incorrect, hence sorry in advance. Only it seems to me that the Archimedes theorem claims that "at least" one rational number exists between any two real numbers. Therefore, the theorem seems not implying that rational and irrational numbers should be both countable or should be both uncountable.
Many thanks for your kind attention, sorry, and best wishes.
Dear Salah A. Mabkhout, dear Valery Vasil'ev,
I am not in the area and hence may be very incorrect, hence sorry in advance. Only it seems to me that the Archimedes theorem claims that "at least" one rational number exists between any two real numbers. Therefore, the theorem seems not implying that rational and irrational numbers should be both countable or should be both uncountable.
Many thanks for your kind attention, sorry, and best wishes.