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A quantum number is a number used in quantum mechanics that represents an amount of some quality that is preserved through interactions between particles, e.g., if two particles were to simply combine, then the resulting particle has the sum of their individual quantum numbers representing that quality. An atom or subatomic particle has some of these quantum numbers, of various types. Through any interaction among the particles, including splitting or combining, the sum of the quantum numbers of each type remains unchanged. (This is analogous to the physics-concept of the conservation of energy: that total energy remains the same through an interaction.)
There are several well-known quantum numbers associated with typical matter (i.e., matter consisting of electrons, protons, and neutrons), which are used in quantum-mechanical descriptions of their interactions of interest.
Examples of interactions in which quantum number conservation is of interest include emission of a photon by an atom whose electrons have a more-than-minimal state of excitation, the analogous absorption, photoionization, fission, fusion, and beta decay.
Example types of quantum numbers are spin and electric charge. Quantum numbers are generally specified so as to always be integral, though spin is defined such that the smallest unit is termed 1/2. (See "electron orbital" for more regarding quantum numbers associated with electron orbits.)