# lo.logic – Non-normal numbers that can be defined without parameters in the language of differential rings with composition

Background: It is currently unknown if $$e$$ is normal. A natural way to approach this question is to find a class to which $$e$$ heard and prove that all members of this class are normal. For example, if we want to know if $$sqrt {2}$$ Normally it makes sense to look at the class of irrational algebraic numbers, but it is still an open problem if every irrational algebraic number is normal. Find a counterexample to this assumption or a similar guess for a corresponding class that contains $$e$$would be very useful for understanding the problem in general.

Differential rings with composition: Containing a natural class $$e$$ Apparently, numbers without parameters in the language of the differential rings are definable with composition, for instance in the range of analytic functions $$mathbb {C}$$, The language of the differential rings with composition is $$(0,1, +, *, partial, circ)$$, from where $$0$$, $$1$$ are the additive and multiplicative identities (constant functions), $$+$$ and $$*$$ are addition and multiplication, $$partial$$ is a derivative (in this case differentiation) and $$circ$$ is composition. We can define the constant function $$e$$ in this language through the formula $$psi (x): ( partial f = f) wedge (f circ 0 = 1) wedge (f circ 1 = x)$$

Question: Is there a non-normal irrational number that can be defined without parameters in the ring of analytic functions? $$mathbb {C}$$ in the language of differential rings with composition?

This is a larger class of numbers than algebraic numbers, so answering a positive question should be fundamentally simpler than asking if there is a non-normal irrational algebraic number, but I think that this is a rather difficult question Answers that slightly modify the question will also be welcome. For example, it may help to generalize to algebraic elements in this differential ring by composition rather than just definable elements or to work in another differential ring $$mathbb {R}$$ embeds.