Transcendental numbers - definition and etymology?

Etymology: not sure about the historical roots, but I believe it comes from very similar words in Latin (say "transcendentem") meaning "transcendent" or "rising above" or similar. The sense is that algebraic numbers are roots of polynomials of finite degree, but transcendental numbers take this degree "to infinity". A more rigorous version of this is to consider the degree of the field extension obtained by adjoining an algebraic or transcendental number--it's finite in the former case and infinite in the latter.

I suppose you could take the last sentence of the preceding paragraph as another definition of transcendental numbers, though you might then say defining something to be infinite is also saying that thing is not finite which is negative rather than positive. There are a few other theorems that you could turn into definitions of transcendental numbers (Thue-Siegel-Roth comes to mind), but I don't know of any that seem any more likely to satisfy you than the above.

Transcendental Numbers Definition

The best explanation I have heard is that these numbers "transcend algebra." That is that they go beyond the set of algebraic numbers.

Perhaps since it is so hard to explain what transcendental numbers are (as opposed to what they are not) It makes it so difficult to prove that any number is in fact a transcendental number.