In the same vein, algebraic irrationals (e.g., √2) all have an irrationality measure of two, but the proof of this is fiendishly difficult and netted its discoverer the Fields Medal back in 1958.
It's worth noting that this is a standard method of proving that a value is transcendental -- just show that it has better rational approximations than any algebraic number can have.
cperciva•8m ago
It's worth noting that this is a standard method of proving that a value is transcendental -- just show that it has better rational approximations than any algebraic number can have.