We develop definitions and a theory for convergent series that have terms of the form $1/a_j$ where $a_j$ is an integer greater than one and the series convergence point is less than one. These series have terms with denominators that can be used as number bases. The series for $e-2$ and $z_n=\zeta(n)-1$ are of this type. Further, both series yield number bases that can represent all possible rational convergence points as single digits. As partials for these series are rational numbers, all partials can be given as single decimals using some $a_j$ as a base. In the case of $e-2$, the last term of a partial yields such a base and partials form systems of nesting inequalities yielding a proof of the irrationality of $e-2$. Using limits in an unusual way we are able to give a second proof for the irrationality of $e-2$. A third proof validates the second using Dedekind cuts. In the case of $z_n$, using the $z_2$ case we determine that such systems of nesting inequalities are not formed, but we discover partials require bases greater than the denominator of their last term. We prove this property for the general $z_n$ case and, using the unusual limit style proof mentioned, prove $z_n$ is irrational. We once again validate the proof using Dedekind cuts. Finally, we are able to give what we consider a satisfying proof showing why both $e-2$ and $z_n$ are irrational.