For instance, if the ‘silver ratio’ of a transcendental number is algebraic, how would someone classify that number? …. perhaps a special subset of transcendental numbers?
Also, perhaps related: are there classifications of transcendental numbers such as the set of this category of numbers which can be expressed as infinite product or arithmetic series, and the set of those that cannot be?
Thank you for any thoughts on this.
If $f(x)$ is a non-constant polynomial with integer coefficients, or with rational coefficients, or with algebraic coefficients, or a rational function with algebraic coefficients, or an algebraic function with algebraic coefficients, and $t$ is a transcendental number, then $f(t)$ is a transcendental number. If Ramanujan said otherwise, then Ramanujan was wrong ––– but, more likely, someone else misunderstood what Ramanujan was trying to say.