\begin{definition}[Definition:Real Number/Digit Sequence]
Let $b \in \N_{>1}$ be a given natural number which is greater than  $1$.
The set of '''real numbers''' can be expressed as the set of all sequences of digits:
:$z = \sqbrk {a_n a_{n - 1} \dotsm a_2 a_1 a_0 \cdotp d_1 d_2 \dotsm d_{m - 1} d_m d_{m + 1} \dotsm}$
such that:
:$0 \le a_j < b$ and $0 \le d_k < b$ for all $j$ and $k$
:$\ds z = \sum_{j \mathop = 0}^n a_j b^j + \sum_{k \mathop = 1}^\infty d_k b^{-k}$
It is usual for $b$ to be $10$.
\end{definition}