Question

Justify 2.010010001.....is an irrational number

Answer 1

No its rational....................

Answer 2

A decimal expansion of a rational number either terminates or follows a repeating pattern of a *finite* sequence of digits.

Neither of these is the case here. The decimal expansion is obviously non-terminating, nor is the sequence of digits finite.

Writing the number as

$2.010010001\ldots=1+1.010010001\ldots$

and only considering the second number, you have the following sequence of digits: $\{1,0,1,0,0,1,0,0,0,1,\ldots\}$, where the $n$th term, starting with $n=0$ corresponds to the number $10^{-n}$. The sequence can be described recursively by the recurrence

$\begin{cases}a_0=1\\a_{n+1}=a_n+n+2&\text{for }n\ge1\end{cases}$

and explicitly by $a_n=\dfrac{n(n+3)}2$.

This sequence is not periodic, and indeed diverges to $\infty$ as $n\to\infty$. This means the number cannot be rational.