Here are the rules:
- You are given a decimal irrational number.
- You may (only if you want) insert ONE SINGLE digit between each two digits of the given irrational number.
- Your task is to make it rational.
For example: If he given irrational is: 0.99090090009...
You can insert 9's and 0's to make it rational: 0.90909090909...
Is it always possible?
Conclusion: No.
Counterexaple: 0.0112223333...
Credits to: @lulu
To summarize the discussion in the comments:
If the base is bigger than $2$, you can't guarantee being able to do this.
A counterexample is $\alpha= .011222000011111222222\cdots$ where the digits cycle forever in blocks that increase by one each time.
To see that this is a counterexample, suppose that you had successfully made this rational, and that the length of the periodic block of the rational is $n$. Then fix a digit $d\in \{0,1,2\}$ and look at one of the blocks of the digit $d$ of length $>n$ in $\alpha$. In your rationalized version, the image of that string must contain a periodic block. Now, you might have inserted a different digit in between two consecutive $d's$, but at leaast half of that periodic block must be $d's$. However, that can't hold for all three digits simultaneously, so we have a contradiction.
In binary it's different. If your number starts with a $1$, say, then you can always produce $.\overline {10}$.