For example, if you change the length of your "unit scale" or basis for numbers to $\sqrt{2}$, then you may represent all fractional multiples of $\sqrt{2}$ as "rational numbers" in the new basis system. Is there any complicated transformation on numbers possible like basis/"base" changes, etc mapping rationals to irrationals and vice-versa. If not, what is the reason for this "bias" between rationals and irrationals?
2025-01-13 02:16:14.1736734574
Is there any basis transformation under which all irrational numbers are rationals and vice-versa?
1.2k Views Asked by Vineeth Bhaskara https://math.techqa.club/user/vineeth-bhaskara/detail At
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It is unclear exactly what kind of "basis transformations" you want to allow, but presumably any such thing would be a bijection $\mathbb{R}\to\mathbb{R}$. But there is no bijection $\mathbb{R}\to\mathbb{R}$ which exchanges the irrationals and the rationals, because it would restrict to a bijection between the rationals and the irrationals, and the rationals are countable and the irrationals are uncountable.