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: 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. 
A: Yes, a number is rational if and only if its decimal representation is eventually periodic (including the possibility of a period $\overline 0$)
A formal proof for your specific number requires a formal definition. I assume that your number is
$$\sum_{n=1}^\infty 10^{-\frac{n^2+n}2}.$$
Any decimal representation that has infinitely non-zero digits (which is the case for your number) and has blocks of zeroes of arbitrary size (which is also the case for your number) cannot be eventually periodic: Some late period must lie completely in a sufficiently big block of zeroes, hence the period must be all zeroes, contradicting the fact that some non-zero digit occurs further to the right.

A number with a similar expression can even be shown to be not only irrational, but in fact transcendent:
$$\sum_{n=1}^\infty 10^{-n!}.$$
A: What you have there is a trancedental number; a number for which there is no variable polynomial equation with rational coefficients that has this number as a root. Trancedental numbers are always irrational (but not all irrational numbers are trancedental). Therefore, yes, your number is irrational.
The proof is in the construction; like the well-known trancedental numbers $\pi$ and $e$, your number is the asymptotic limit of the sum of an infinite series; in this case, the sum of a reducing fraction:
$$\sum_{n=1}^{\infty} \dfrac{1}{10^{\dfrac{n(n+1)}{2}}}$$
This is similar to the construction of the Champernowne constant which is proven transcedental. The sum is constructed such that no 2 terms ever modify the value of a decimal place of the same order of magnitude, very much like $C_{10}$, and so the number constantly increases but can never reach a rational sum, unlike the infinite sum of $\frac{1}{2^n}$.
A: 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.
