The question is a bit misleading because natural Ag is a mix of
107Ag and
109Ag whose
mean value is ~107.87, so 108 shouldn't be understood as an integer number.
http://www.webelements.com/silver/isotopes.htmlBecause the neutron happens to weigh about as much as a proton plus an electron (this lead to wrong theories in the past), and because a neutral atom has as many electrons as protons, one can multiply the number of nucleons (107, 109...) by the mass of one neutron to approximate the mass of the neutral atom.
Though, elements show deviations to that, which are commonly observed. Beyond neutrons' mass differing from protons plus electrons, this is because energy has a mass, and the interaction between the nucleons is so strong that the mass difference is seen on lab scales.
https://en.wikipedia.org/wiki/Nuclear_binding_energySo even for monoisotopic matter, the number of amu isn't an integer. Compare
4He (4.0026) with
59Co (54.942).
Because these deviations have practical implications, some convention is needed for the amu. Carbon is in the midrange between a small (He) and a big (Co) mass defect, and many atoms in organic chemistry have a mass defect similar to carbon: useful. On the other hand, natural carbon comprises 1%
13C too, so the standard specifies
12C.