The problem with doing this is that you give up a lot of properties of functions that you might want, for example that continuous functions which cross the x-ax

The problem with doing this is that you give up a lot of properties of functions that you might want, for example that continuous functions which cross the x-axis somewhere actually intercept the x-axis. This is not true if your x-axis is made of rational numbers: consider the function x^2 - 2.