Why are atomic masses not whole numbers?

This is because, the atom mass of an atom is the average of the atomic masses of all its isotopes multiplied by their abundance in nature. Therefore, is it highly unlikely to get a whole number as the atomic mass.
Another reason is that, since carbon-12 is the reference atom, the atomic mass is calculated relative to it.

10.063g / 10.00 ml = 1.0063 g/ml which rounds off to 4 sig figs as: 1.006 g/ml