This is a exploration of some difficulties I have with dealing with the multiplication of two irrational numbers (algebraic or transcendental) or two rational numbers or one of each type; the only rule is that they have to be formed from the bounded infinite summation of an ordered sequence of rational numbers, $a_n$ (for the first number $A$) and $b_n$ (for the second number $B$).
Consider the generalised approximate multiplication of two numbers $A$ and $B$.
$A$ is defined (for this purpose) by
$$A=\sum_{k=0}^{n} a_k+\sum_{k=n+1}^{\infty} a_k\tag{1}$$
where $n$ is a finite positive integer, $A_n=\sum_{k=0}^{n} a_k$ is always rational, and the remainder $A(R)_{n+1}=\sum_{k=n+1}^{\infty} a_k$ can be rational or irrational.
Analogous definitions apply for $B$, $B_m$ and $B(R)_{m+1}$ using the integer $m$
Now let us calculate $A\times B$ i.e. the multiplication of $A$ and $B$
$$A \times B= \left( A_n\right)\left( B_m\right)+\left( A_n\right)\left( B(R)_m\right)+\left( B_m\right)\left( A(R)_n\right)+\left( A(R)_n\right)\left( B(R)_m\right)\tag{2}$$
Dividing through by the rational number $\left( A_n\right)\left( B_m\right)$ to normalize the equation we have exactly
$$\frac{A \times B}{\left( A_n\right)\left( B_m\right)}= 1+\frac{\left( A(R)_n\right)}{\left( A_n\right)}+\frac{\left( B(R)_m\right)}{\left( B_m\right)}+\frac{\left( A(R)_n\right)\left( B(R)_m\right)}{\left( A_n\right)\left( B_m\right)}\tag{2}$$
If we define the first order (rational) approximation as $$\frac{A \times B}{\left( A_n\right)\left( B_m\right)}\approx 1$$
and the second order approximation to include the next two potentially irrational terms
$$\frac{A \times B}{\left( A_n\right)\left( B_m\right)} \approx 1+\frac{\left( A(R)_n\right)}{\left( A_n\right)}+\frac{\left( B(R)_m\right)}{\left( B_m\right)}\tag{3}$$
Therefore it is only necessary to define what we mean by multiplying two potentially irrational numbers if we are interested in terms of third order smallness.
However how do we reliably determine the size of $\frac{\left( A(R)_n\right)}{\left( A_n\right)}$ and $\frac{\left( B(R)_m\right)}{\left( B_m\right)}$ compared to $1$ in terms of $n$ and $m$ given two convergence rates $C_n$ and $C_m$ (i.e. a generalised calculation of the maximum error in our rational first order approximation).
These convergence rates could involve a linear number of extra terms per decimal significant figure, or a quadratic number, or exponentially increasing number of extra terms required to converge on each succeeding decimal digit.
It would also be helpful to know if there is standard notation for this type of generalised arithmetic.