Prove that 2.101001000100001... is an irrational number.

My try: This number is non-terminating and non-repeating, so this is an irrational number.

But how do I prove it more formally in a more mathematically rigorous way?

Solution 1:

Let $x = 2 + \sum\limits_{k=1}^\infty10^{-k(k+1)/2}$ be the number at hand. If $x$ is rational, say $x = \frac{p}{q}$ for some positive integers $p,q$, we can pick a $n > 1$ such that $10^n > q + 1$. It is clear $$qx \times 10^{n(n-1)/2} = p \times 10^{n(n-1)/2}$$ is also an integer. However, the fractional part of this number is equal to

$$ \left\{ q \times 10^{n(n-1)/2} \left(2 + \sum_{k=1}^\infty 10^{-k(k+1)/2}\right)\right\} = \left\{ q \times \sum_{k=1}^\infty 10^{-k(k+2n-1)/2} \right\} $$ which belongs to $(q \times 10^{-n}, (q+1)\times 10^{-n} ) \subset (0,1)$. Since $(0,1)$ doesn't contain any integer, this leads to a contradiction and hence $x$ is irrational.

Solution 2:

Hint:

2 is rational
$0.1010010001\ldots$ is easy to show irrational (proof in first chapter of Rudin IIRC)

rational + irrational = irrational