In this question, I define an "elementary irrational number" as an irrational number which is built up of a finite combination of integers, field operations (addition, multiplication, division, and root extractions -- the elementary operations) and exponential and trigonometric functions and their inverses under repeated compositions, as if the definition of elementary function (I made up the name since I couldn't find it on the Internet). The numbers $\pi=\arccos(-1)$, $\mathrm e=\exp(1)$ and $\phi=\frac{1+\sqrt5}2$ are "elementary irrational numbers", while $1$, $\frac32$ and $1.2121121112\dots$ are not.
It's easy to find either a rational number or an irrational one without a certain digit in its decimal presentation, for example, $1.2121121112\dots$ doesn't contain digit $0$. However, among "elementary irrational numbers", it seems hard to find such number, despite proving there is no such number is not easy either.
I can come up with some possible ways to solve the problem:
- Most constructed numbers aren't "elementary", while there does exists possibility to find one.
- To disprove the existence of such number, we may prove through all the operations to build up the elementary functions that if its arguments are "elementary number(s)" which is/are either rational or contains every kind of digit, then the result satisfies the same property, but it seems to be quite difficult to prove it.
- Or maybe the problem is connected to some existed open problems in mathematics (normal numbers might be an example), then we may be convinced with the difficulty of the problem.
I'd appreciate it if you could give me any advice on the problem.