Question

Is 9.521521521 rational or irrational

Answer 1

If the numbers after the decimal terminate, yes, it's rational.

9.521521521 = 9,521,521,521 / 1,000,000,000

If they don't terminate, but the pattern continues (which I suspect is the case here), yes, it's still rational.

If x = 9.521521521…, then

1000x = 9521.521521521…

Subtract x from this to eliminate the fractional part:

1000x - x = 9521.521521521… - 9.521521521…

999x = 9512

x = 9512/999

If they don't terminate, but the pattern does not continue, meaning the next few digits could be something random like

9.52152152119484929271283583457…

then the number would be irrational.