For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its coefficients. Or consider all the valuations of $\alpha$, etc. In this context, the height is supposed to be some kind of measure of complexity.


**Question.** Is there a reasonable definition of the "height" of a transcendental number.


I'm not sure what such a height would mean though in this context.


If there isn't any reasonable definition, is there any reasonable explanation for why this isn't possible?



Is there a reasonable definition of the height of a transcendental number
It does'nt quite answer your question, but maybe there is a resaonable definition of the height of a [period](http://en.wikipedia.org/wiki/Ring_of_periods). The ring of period is a countable over-ring of the field of algebraic numbers, sharing (at least conjecturally) many properties with them. This ring contains most of interesting transcendental numbers.


