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It is often stated that the vast majority of real numbers are irrational. Does it also follow that the vast majority of irrational numbers are transcendental?

Larry Freeman
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    Yes, but in practice we don't know for the vast majority of irrational numbers, whether they are transcendental or not. For example, we know that $\zeta(3)$ is irrational. But is it transcendental? We don't know. – Dietrich Burde May 30 '24 at 19:23
  • Also: https://math.stackexchange.com/q/813287/42969, https://math.stackexchange.com/q/2822919/42969 – Martin R May 31 '24 at 04:42

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There are only countably many algebraic numbers since there are only countably many polynomials with rational coefficients. So yes, almost all real numbers are transcendental.

Ethan Bolker
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    +1 though perhaps worth noting "since there are only countably many polynomials with rational coefficients and they each have a finite number of roots" – Henry May 31 '24 at 09:14