Firstly the title may not explain my question well but i cant think of a logical title for my question as its been a long time since i did math.
i know that the set of rational numbers is countably infinite (have done the proof before its a pain and way to long to write so im going to take this one as a fact)
Now i also know a proof to prove that between every rational number there exists an irrational number. ( i swear i used to know at least 3 ways to prove this from algebra to calculus)
i also used to know that the irrational numbers are unaccountably infinite.
it seems to me that this means there exists irrational numbers such that there is no rational number between them.
1)is there a way to prove that this pair exists?
2)is there anyway to express 2 such irrational numbers
Edit: so the answer im receiving are causing me more confusion if my assumption isn't true and there is a rational between any two irrational numbers then it seems like they should have the same cardinality.
Best Answer
Between every two real numbers (in particular between two irrational numbers), there are rational numbers (in fact infinitely many of them).
Depending on which definition of real numbers you are using, this is more or less obvious. For example, if you define real numbers as equivalence classes of rational Cauchy sequences, it is clear that the rational numbers are dense in $\Bbb{R}$.