Do we always have an uncountable number of transcendental numbers between any two different real numbers?

Question

If $a$ and $b$ are real numbers and $a

Answer 1

Yes, there are.

Any (nonempty) interval $(a, b)$ is uncountable, and the union of two countable sets is countable. So all you need now is the fact that the set of algebraic numbers is countable. HINT: how many polynomials with rational coefficients are there? (If you want more details see this question.)

Answer 2

Of course, it is true. The set of all algebraic numbers is countable, while the whole interval is uncountable.