How to prove that $x=0.1234567891011\dots $ is irrational?

Question

I'm in $9th$ class and I was wondering how to solve this problem. I only know how to prove that $0.1010010001\dots$ is irrattional.

Answer 1

It has many proofs. Okay I can give you one hint this is much interesting any rational number can have only finite decimal representation or recurrence decimal representation.

For start just take $\frac pq$ where $gcd(p,q)=1$ then see after using division algorithm what are the remainders that will come.

Do it then you will get generalised statement.

Answer 2

The decimal representation of a rational ends by repeating the same digits periodically.

The number 0.1234567891011… includes arbitrarily large sequences of zeroes (it inculdes 10, 100, 1000, 10000, ...). But it also includes arbitrarily large sequences of 1's (it includes 1, 11, 111, 1111, ...). This is contradictory with the fact that it ends with a finite period.