By abandoning the reals and ability to do arithmetic with them such as $$\left( \sqrt{2} \right)^2 =2$$, you can have a consistent theory of
positive rational numbers and quasi-decimal representations and quasi-infinitesimals if you limit yourself to one base.
Rather than the standard definition of $$ { 0.a_1 a_2 a_3 a_4 a_5 .... }_b = \lim_{n\to \infty} \sum_{k=1}^n \frac{a_k}{b^k}$$, this model of the rational numbers adopts $$ { 0.a_1 a_2 a_3 a_4 a_5 .... }_b = \sum_{k=1}^{\tilde{\omega}} \frac{a_k}{b^k}$$ where $$\tilde{\omega}$$ is defined as an integer large enough to be divisible by all relevant figures.
Thus we have the equality $$ \frac{p}{q} = \lim_{n\to \infty} \sum_{k=1}^n \frac{a_k}{b^k} = \sum_{k=1}^{\tilde{\omega}} \frac{a_k}{b^k} + \frac{ \sum_{k=\tilde{\omega}+1}^{\tilde{\omega}+\ell} \frac{a_k}{b^k} }{\sum_{k=\tilde{\omega}+1}^{\tilde{\omega}+\ell} \frac{b-1}{b^k} } \epsilon_{b,\tilde{\omega}}$$ where $$\ell$$ is the repetition length of the rational number.
So we have
$$
\begin{eqnarray}
1 & = & {0.111...}_2 + \epsilon_{2,\tilde{\omega}} & = & {0.222...}_3 + \epsilon_{3,\tilde{\omega}} & = & {0.444...}_5 + \epsilon_{5,\tilde{\omega}} & = & {0.666...}_7 + \epsilon_{7,\tilde{\omega}} & = & {0.999...}_{10} + \epsilon_{10,\tilde{\omega}}
\frac{1}{2} & = & {0.1}_2 & = & {0.111...}_3 + \frac{1}{2} \epsilon_{3,\tilde{\omega}} & = & {0.222...}_5 + \frac{1}{2} \epsilon_{5,\tilde{\omega}} & = & {0.333...}_7 + \frac{1}{2}\epsilon_{7,\tilde{\omega}} & = & {0.5}_{10}
\frac{1}{3} & = & {0.010101...}_2 + \frac{1}{3} \epsilon_{2,\tilde{\omega}} & = & {0.1}_3 & = & {0.131313...}_5 + \frac{1}{3} \epsilon_{5,\tilde{\omega}} & = & {0.222...}_7 + \frac{1}{3} \epsilon_{7,\tilde{\omega}} & = & {0.333...}_{10} + \frac{1}{3} \epsilon_{10,\tilde{\omega}}
\frac{1}{5} & = & {0.001100110011...}_2 + \frac{1}{5} \epsilon_{2,\tilde{\omega}} & = & {0.012101210121...}_3 + \frac{1}{5} \epsilon_{3,\tilde{\omega}} & = & {0.1}_5 & = & {0.125412541254...}_7 + \frac{1}{5} \epsilon_{7,\tilde{\omega}} & = & {0.2}_{10}
\frac{1}{7} & = & {0.001001...}_2 + \frac{1}{7} \epsilon_{2,\tilde{\omega}} & = & {0.010212...}_3 + \frac{1}{7} \epsilon_{3,\tilde{\omega}} & = & {0.032412...}_5 + \frac{1}{7} \epsilon_{5,\tilde{\omega}} & = & {0.1}_7 & = & {0.142857...}_{10} + \frac{1}{7} \epsilon_{10,\tilde{\omega}}
\frac{1}{210} & = & \frac{53}{105} - \frac{1}{2} & = & \frac{47}{70} - \frac{2}{3} & = & \frac{17}{42} - \frac{2}{5} & = & \frac{13}{30} - \frac{3}{7} & = & \frac{19}{21} - \frac{9}{10}
\frac{1}{210} & = & {0.000000010011100000010011...}_2 + \frac{53}{105} \epsilon_{2,\tilde{\omega}} & = & {0.000010110201200010110201...}_3 + \frac{47}{70} \epsilon_{3,\tilde{\omega}} & = & {0.000244200244200244...}_5 + \frac{17}{42} \epsilon_{5,\tilde{\omega}} & = & {0.0014301430143014...}_7 + \frac{13}{30} \epsilon_{7,\tilde{\omega}} & = & {0.004761904761904761...}_{10} + \frac{19}{21} \epsilon_{10,\tilde{\omega}}
\end{eqnarray}
$$
But since $$\tilde{\omega}$$ isn't infinite (it's just very large compared to any number we choose to think of), to be consistent you need particular rules to handle addition and multiplication. In particular since $$\tilde{\omega}$$ isn't infinite there is a "last digit" which eats carries from the quasi-infinitesimal.
Example:
$$\frac{3}{210} = 3 \times \frac{1}{210} = 3 \times ( \frac{19}{21} - \frac{9}{10} ) = \frac{57}{21} - \frac{27}{10} = \frac{15}{21} - \frac{7}{10} = \frac{5}{7} - \frac{7}{10}
\frac{3}{210} = 3 \times \left( {0.004761904761904761...}_{10} + \frac{19}{21} \epsilon_{10,\tilde{\omega}} \right) = 3 \times {0.004761904761904761...}_{10} + 2 \epsilon_{10,\tilde{\omega}} + \frac{15}{21} \epsilon_{10,\tilde{\omega}} = {0.014285714285...}_{10} + \frac{15}{21} \epsilon_{10,\tilde{\omega}}$$
But, this does not give you a math system more powerful than the rational numbers. It does not have any of the benefits of the real numbers and irrational numbers don't have a representation. It's just a wasteful way to think about the rational numbers
unless you use logic and mathematical rigor to bridge the gap between numbers too large to think of and the infinite.
That is the benefit of the real numbers. They obey the axioms common to other number systems but with the concept of continuity so that numbers defined as limits or boundaries of sets actually exist.
| $$0 \neq 1$$ | |
| $$0 \in \mathbb{X}$$ | $$1 \in \mathbb{X}$$ |
| $$p + q \in \mathbb{X}$$ | $$p \times q \in \mathbb{X}$$ |
| $$p + 0 = p$$ | $$p \times 1 = p$$ |
| $$p + q = q + p$$ | $$p \times q = q \times p$$ |
| $$(p + q) + r = p + ( q + r )$$ | $$(p \times q) \times r = p \times ( q \times r )$$ |
| $$\exists s \in \mathbb{X} \quad p + s = 0$$ | $$p \neq 0 \quad \Rightarrow \quad \exists s \in \mathbb{X} \quad p \times s = 1$$ |
| $$p \times ( q + r ) = ( p \times q ) + ( p \times r )$$ | |
| $$p \lt q \quad \Leftrightarrow \quad p \neq q \; \textrm{and} \; p \not \gt q$$ | |
| $$p \lt q \; \textrm{and} \; q \lt r \quad \Rightarrow \quad q \lt r$$ | |
| $$p \lt q \quad \Rightarrow \quad r + p \lt r + q$$ | $$0 \lt p \; \textrm{and} \; 0 \lt q \quad \Rightarrow \quad 0 \lt p \times q$$ |
It doesn't matter if $$\mathbb{X}$$ stands for the rationals or the reals, these rules apply to them all. This is the burden of claiming one has a system of infinitesimal numbers -- one has to play by the same rules as any other number. The quasi-infinitesimals above play by these rules and turn out to be just rational numbers by a different name.
