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=\frac{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.