# Is there a reasonable definition of the height of a transcendental number

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?

• For what it's worth - the Kolmogorov complexity of the number in question. Does it help in any way ? probably not. Oct 12, 2011 at 8:25
• There is Mahler's classification en.wikipedia.org/wiki/… which roughly speaking distinguishes in terms of approximation properties by algebraic numbers. That's possibly rather coarser than what you are asking for though.
– dke
Oct 12, 2011 at 10:21
• The (logarithmic) height of a rational number $x$ is roughly the number of digits needed to write $x$. Given an arbitrary real number $x$, one could try to define the height'' of $x$ as the minimal number of symbols needed to write to $x$. There are two problems with this definition : 1) it is not precise - what expressions are allowed ? 2) it is very ineffective... Oct 12, 2011 at 17:56
• In my answer mathoverflow.net/questions/53724/… to a similar question, I describe several commonly used hierarchies for measuring the complexity of transcendental real numbers. Oct 13, 2011 at 22:38

Here is one potentially reasonable explanation for why such an invariant shouldn't exist. One property of height that can be useful is that the height of an algebraic number is invariant under all automorphisms of all rings that contain that number. For any algebraically independent pair of transcendental complex numbers, one may choose a ring-theoretic automorphism of $\mathbb{C}$ that exchanges them. If we want the same sort of invariance as in the algebraic setting, then all heights of transcendental numbers must be equal.