Is 1.0227 repeated a rational number

Yes, it is 10226/9999.

You can calculate this by doing saying n is the number (1.022702270227...)
Then shift it to the left and subtract the fractional part:

So 10000*n - (n-1) = 10227,
so 9999n = 10226, n=10226/9999

Think of it as five or above give it a shove. change the 5 or above to the right to a zero and add one to the number on the left.