Update: The theory here has been checked by Brian M. Scott and with the New Year in mind I am changing the solution-verification tag to recreational-mathematics and adding this question,
Given any integer $p \le 0$, define a natural (using $\text{base-}10$ representations) injective mapping
$\quad \Gamma_p: \Bbb N^{\gt 0} \to \Bbb M$.
Of course this won't be possible unless you've invested some thought and have at least a minimal 'buy-in'. But, to my way of thinking, there is only one answer. I'll be using this 'puzzle piece' in defining addition (and multiplication?!?).
I want to check if the following arguments are valid.
Let $D = \{0,1,2,3,4,5,6,7,8,9\}$, a totally ordered set.
The set all functions $a \in D^{\,\Bbb Z}$,
$\quad \large a =(a_k)_{\,k \in \Bbb Z}$
satisfying the following two conditions
$\tag 1 (\exists\, A \in \Bbb Z) \; [ k \gt A \implies a_k = 0]$
$\tag 2 (\forall \, B \in \Bbb Z) \;(\exists \, k \in \Bbb Z) \; [ k \lt B \land a_k \ne 0]$
will be denoted by the symbol $\Bbb M$.
For $a,b \in \Bbb M$ with $a \ne b$ we can associate the subscript $\delta(a,b) \in \Bbb Z$ where
$\tag 3 \delta(a,b) = \text{max}\bigr(\{k \in \Bbb Z \mid a_k \ne b_k\}\bigr)$
Define a relation $\le$ on $\Bbb M$ by
$\quad a \le b \quad \text{ if } \; \; [a = b] \; \lor \; \large[a \ne b \, \land \, a_{\delta(a,b)} \lt b_{\delta(a,b)}]$
Proposition 1: The relation $\le$ is a total ordering on $\Bbb M$.
Proof
Antisymmetry/Connexity: If $a \ne b$ then upon comparing $a_{\delta(a,b)}$ with $b_{\delta(a,b)}$ one can write
$\quad a \lt b \; \text{XOR} \; b \lt a$
Transitivity: Let $\{a,b,c\} \subset \Bbb M$ satisfy $a \lt b$ and $b \lt c$ and $a \ne c$.
If $\delta(a,b) \ge \delta(b,c)$ then clearly $\delta(a,c) = \delta(a,b)$ and
$\quad a_{\delta(a,c)} = a_{\delta(a,b)} \lt b_{\delta(a,b)} \le c_{\delta(a,b)}= c_{\delta(a,c)}$
If $\delta(a,b) \lt \delta(b,c)$ then clearly $\delta(a,c) = \delta(b,c)$ and
$\quad a_{\delta(a,c)} = a_{\delta(b,c)} = b_{\delta(b,c)} \lt c_{\delta(b,c)}= c_{\delta(a,c)}$ $\quad \blacksquare$
Theorem 2: The totally ordered set $(\Bbb M, \le)$ is complete.
Proof Sketch
Let $M$ be a nonempty subset of $\Bbb M$ bounded above. There is therefore an integer $A$ and an upper bound $b$ of the form
$\quad b_k = 0 \text{ for } k \gt A \; \land \; b_k = 9 \text{ for } k \le A$
We can certainly improve $b$ by defining $b_A$ to be the maximum digit found in the elements of $M$ at the $A$ subscript.
Using recursion, we can continue this technique, reducing $M$ while constructing the 'best' possible upper bound $\hat b$ that will also be the least upper bound for $M$
(verified using induction, recursion's 'twin sister').
$ \quad \blacksquare$
My work
I'm interested in answering the question
$\quad$ How to define operations on Stevin's construction of real numbers?
by, constructing, up to isomorphism, the set
$\quad (0,+\infty) = \{x \in \Bbb R \mid x \gt 0\}$
together with its accompanying binary operation of addition, $+$.
I have not put all the pieces together, but the thinking is that the construction will not be awkward and can be understood with very little set theory (no equivalence relations) or real analysis (no limits/infinite sums).
Note: With this perspective there is no need to worry about $1 = .9999\dots$; we have the element $\hat 1 \in \Bbb M$ defined by
$\quad k \mapsto 9 \; \text{ for } k \lt 0, \text{ else } k \mapsto 0$
This model is abstract, which leads to perhaps an oxymoron,
$\quad$ Stevin's abstract construction
The OP will no doubt need the following 'puzzle pieces' (notation/definitions) to define addition on $\Bbb M$.
Piece #1:
We define the function $\\K : \Bbb N_1 \to \Bbb M$ as follows:
If $n \ge 1$ then set $a = n - 1$ and write $a$ in $\text{base-}10$ format,
$\tag 1 \displaystyle \large a = \sum_{k = 0}^{+\infty} a_k \, \beta^{k} \quad \text{where } 0 \le a_k \le 9$
Defining $a_k = 9$ for $k \lt 0$ completes the specification,
$\tag 2 \LARGE \\K(n) = (a_k)_{\,k \in \Bbb Z}$
Piece #2:
For $a \in \Bbb M$ with $a = (a_k)$ define a function $\\Q: \Bbb M \to \Bbb N$ by writing
$\tag 3 \displaystyle \large \\Q(a) = \displaystyle \sum_{k = 0}^{+\infty} a_k \, \beta^{k}$
Piece #3:
We'll need a function to translate/shift the subscripts.
If $p \in \Bbb Z$ then $\large \Psi_{p}: \Bbb M \to \Bbb M$ is specified as follows.
If $a = (a_k)$ define the bilateral sequence $\large ({a'}_k)_{\,k \in \Bbb Z}$ by writing
$\tag 4 \displaystyle \large {a'}_k = a_{k+p}$
and then
$\tag 5 \displaystyle \large \Psi_{p}(a) = ({a'}_k) \quad \text{(denoted also by } {\hat a}^p \text{)}$