Proof of Irrationality of a non Recurring Decimal

Question

How to prove that 0.101001000100001..... is irrational? There's the fact that it is non recurring but is there any mathematical proof like that we give for square root of two?

Answer 1

Note that your number is nothing but $$0.1+0.001+0.000001+\cdots$$ Assume that the number is rational. Then it must be of the form $$p/q=0.1+0.001+0.000001+\cdots$$ for some integers $p,q$. Then it should hold that $$p.0000\cdots=p=q\cdot(0.1+0.001+0.000001+\cdots),$$ But if $q$ has $n$ digits, then the non zero digits of the summands after the n'th, on the right hand side, occupy different digits of the number, and so they cannot cancel out.

Example: if $q$ were 576 then we would sum $$5.76.+0.576+0.000576+0.0000000576+0.000000000000576\cdots)$$ and you see that from the third summand on, the non zero digits are non-intersecting.