Is the number $0.1234567891011121314\ldots$ a rational or irrational number?

Question

Is the number $0.1234567891011121314\ldots$ a rational or irrational number?

The number has a very clear pattern but however in order for the number to be a rational number it would have to be written as a/b. The normal tricks of writing it as an equation and solving

$3.3333333... = x \\ 10*3.333333 = 10x\\ 33.333333 = 10x\\ 30 + x = 10x\\ 30=9x\\ x=30/9 \\= 10/3$

does not seem to work here

Answer 1

This is the Champernowne constant in base 10, which is in fact transcendental.

In order to be capable to write it as a fraction, you would need to have a repeating block of digits. Evidently, there does not exist such a repeating block, and thus it is irrational.

Answer 2

The Champernowne constant $$C = 0.12345678910111213141516\dots$$ is a transcendental real number, so it is also irrational.