Inspired by normal numbers I created the simple following problem:
First take a rational number, for example $\frac{3}{4}$ which is equal to $0.75$; now add the first digit after the decimal separator to the digit $4$ we get $\frac{3}{4.7}$ which is equal $0.63829\cdots$. Now take the second digit after the dot of the new number $0.63829\cdots$ and add it to the number $4.7$ we get $\frac{3}{4.73}$. We can repeat the process an infinity of times.
Now, I conjecture that if we make the process on the numerator there exists a period of digits which is repeated infinitely many times i.e it's a rational number.We have diffrerent category of number.Well take $\frac{5}{4}$ and we recognize something like a divisibility .There is an other example wich is a counter-example I think it's $\frac{729}{653}$ it's a special example because the digit after the dot of the numerator corresponds to the digits after the dot of the result .Well furthermore $653$ is a prime number so maybe we can find a criteria to distinguish rational and irrational number (always with numerator) with the prime number. On the other hand as in my example I conjecture that it's not the case for the denominator i.e it's a irrational number. I have tested all the rational numbers $\frac{a}{b}$ such that $0<a\leq b\leq 100$ for my second conjecture.
To prove it I have tried to build something similar to the following paper (without success) https://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf
My question :
Can we find a criteria or a method to know if we have a rational or a irrational number making the process describe above ?
I think this is the kind of problem which is either trivial, either not for this century (remember a quote of Paul Erdös).
Update :
Well making a Google research I found this Can there be an Irrational Numbers Hotel? .In fact my little algorithm is not new see the Cantor's diagonal argument .So I'm a little bit sad .Moreover I think there is a link with prime gaps .good day :-(
Any help is greatly appreciated...
...Thanks a lot for all your contributions.