Mate, I mean this with the best of intentions: please reflect on your requirement for exactness and consider whether it serves you well.

I did so with mine and I concluded it did not. Perhaps you will conclude differently. Perhaps the same. In either case, I think you will find it worthwhile.

The claim is only missing a qualification that the impossibility exists within the framework of ieee 754. It can be easily inferred from context and it is important to note because thats what the major CPU architectures use to store native floating point numbers. Therefore, the point is, we need a non-native (wrt the cpu) way to store and calculate with irrationals and some rationals if mathematical precision is required.