Is this number irrational? Is the following (decimal) number irrational?

0.10100100010000100000100000010000000100... etc.


My intuition tells me it is irrational. My informal "proof" is simply that it doesn't contain a repeating set of digits.


For irrationality, is it both a necessary and sufficient condition that the digits never revert to a repeating sequence?
Is there a more formal proof for this case?
A formal justification of your informal proof can be achieved by noting that in the process of long division, the fact that you have a finite number of possible remainders guarantees that eventually a remainder will be repeated. That is, for any rational number its decimal expansion becomes periodic.
The number you gave when you asked the question is indeed an irrational number because it goes on without repeating or ending.  I've used to learn that.  It keeps adding zeroes on and on and on beside the ones.  This never goes into a repeating sequence because it keeps adding on the zeroes.  I don't know if there's a more formal proof for this kind of case, but remember that an irrational number can never be written as a fraction.