Suppose a,b are real numbers constructed in some well-defined logical way, for example having in their decimal expansion the sequence of all odd & all even numbers respectively:
a = 0.135791113151719.... and b = 0.2468101214161820....
I think that logically these numbers are well-defined, but i have no clue as to how to proceed to get the decimal expansion of their sum, product or ratio. So my question how can we compute a+b, a*b, a/b ?
The usual algorithms can give you the result to as many decimal places as you want. It is not guaranteed that you can express the full result with any neat pattern like you described $a,b$ with. The usual propagation of errors will tell you how many extra places to keep. If you want $ab$ to $20$ places, you have $\Delta (ab) \approx a\Delta(b)+b\Delta(a)$. Since both $a$ and $b$ are less than $1$, if you keep an extra place on each when multiplying you will be fine.