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Are there really infinite transcendental numbers?

Inverse trigonometric and hyperbolic functions are transcendental for their argument algebraic and nonzero -- a consequence of the Lindemann-Weierstrass theorem

arctan(x)/pi is transcendental for x rational -- a consequence of the Gelfond-Schneider theorem

Transcendentality of zeros of higher dereivatives of functions involving Bessel functions - Lorch - 1995 - International Journal of Mathematics and Mathematical Sciences - Wiley Online Library

Bessel functions J(n,x), their first derivatives J'(n,x) = (d/dx)J(n,x), and one divided by the other, J'(n,x)/J(n,x), are all transcendental for n rational and x algebraic and nonzero.

All nonzero roots of J(n,x) and J'(n,x) are transcendental if n is rational.
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