Incommensurables

HAVING grown up with such quantities as π, ∊, √2, we have become callous to their unreasonableness. Nevertheless, from time to time we are forced to ask ourselves what is wrong with our system of numeration that these quantities should refuse to be expressible either as fractions or as decimal quantities using a finite number of digits. Surely our system permits steps as fine as we please. What change of system would meet the needs, for instance, change from base 10? Or again, we ask, how many of these rebels are there? Is there a mere sprinkling? Is there a generic incommensurable unit — call it I — whereby all incommensurable quantities can be expressed as so many ordinary units plus so many I units, p+qI, a method we are used to for complex quantities?