Having a set of ordered numbers $ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \} $ If you put them in a matrix so the index is at the same time the content
Matrix dimension $=$ row x col $= 3\times 4$ $ \begin{bmatrix} 1 &2 &3 & 4\\ 5 &6 & 7 & 8\\ 9 &10 &11 &12 \end{bmatrix} $ examples of getting the coordinates from $n=$index
$f_{row}(n)$ get the row, $f_{col}(n)$ get the column
$f_{row}(2) = 1, f_{col}(2) = 2$
$f_{row}(10) = 3, f_{col}(10) = 2$
$f_{row}(12) = 3, f_{col}(12) = 4$
They seems to be complex functions, i.e. (non-linear, bounded, etc..)
How are $f_{row}$ and $f_{col}$ defined?
I think a software equation, but it has functions like
$\operatorname{int}(\cdot), \operatorname{floor}(\cdot),\operatorname{mod}(\cdot)\ldots$ or worst a sequenced algorithm..
I don't know if there is some "math way" to do this, I mean with "basic operations" an "arithmetic" solution. Of course you could say "Floor and ceiling functions are math and even algorithms!" but simply put: Is there any series expansions/ other path for doing this? I would like to "relate index with dimentions", in some equation.