Ooh, good point. but that still only gets us to fractions, not irrational quantities*.
So let us define a unit of mass - the Andrew - to be exactly one integral-Avogardo's-number of C12 atoms - we'll pick a value for Advogadro's Number and round down. Doesn't matter which, really.
For any given object, there must be a rational relationship between the mass of that object expressed in kilograms and expressed in Andrews.
*although pi is *defined* as a ratio... I don't see how anything like that applies here.That's not quite proof by contradiction, but it's the schematic of one...
I cheated and looked - Avogadro's Number is good to 6 decimal places. :) We are so deep in the noise... still, the processes that produce Advogadro's number are inherently about counting, so I maintain that it is at least rational.
-- Les Cargill