I would like to share a jotting from my diary (dated: 21-April-2016) which is bothering me:

Is the (decimal) number generated the concatenating all natural numbers like 0.1234567891011…, a transcendental number?

We can see that this number is irrational. Since, if it is a rational number then there exists a natural number such that , which is clearly impossible.

Now, to prove a number is transcendental or not is a difficult question to answer (some open problems). So, either produce a polynomial equation which has this number as a solution or prove that no such polynomial equation exists.