The atomic number, Z, should not be confused with the mass number, A, which is the total number of protons and neutrons in the nucleus of an atom. The number of neutrons, N, is known as the neutron number of the atom; thus, A = Z + N. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes), the atomic mass of an atom is roughly equal to A. Note that the atomic mass number A of an atom, which is always an integer, is different than the atomic weight of a element, because in general an element consists of a mixture of atoms with the same Z but differing N (isotopes).
A numbering based on the periodic table was never entirely satisfactory. For one thing, the gradual identification of more and more lanthanoids over many decades led to long-term instability in the numbering of all elements from hafnium on up.
The situation improved dramatically after research by Henry Moseley in 1913. Moseley discovered a strict relationship between the x-ray diffraction spectra of elements and their most logical location in the periodic table. This led to the conclusion that the atomic number corresponds to the electric charge of the nucleus, i.e. the proton number Z. Among other things, Moseley demonstrated that the lanthanoid series (from lanthanum to lutetium inclusive) must have 15 members -- no less and no more -- which was far from obvious from the chemistry at that time.
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