# Proof of Irrationality of a non Recurring Decimal

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?

• There is a proof in the sense that you prove that $p/q$ is always has a recurring decimal expansion, and your number does not, but that does not seem to be what you're looking for. Can you be more precise? Commented Jan 12, 2017 at 12:48
• I've been told this can be proved by using binomials, though I can't see how. How can we prove your first statement? Thanks. Commented Jan 12, 2017 at 12:52
• @user406333 it suffices to show that every $q$ divides a number of the form $99\cdots900\cdots0$, and that any fraction with such a number in the denominator will eventually repeat. Commented Jan 12, 2017 at 12:55

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.