Half-integer
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In mathematics, a half-integer is a number of the form [n + 1/2], where [n] is an integer. For example, 3/2, 7/2, -13/2 are all half-integers. Note that a half of an integer is not always a half-integer. For example, half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers which are half of an odd integer.
The set of all half-integers is often denoted [\mathbb Z + ].
Half-integers occur frequently enough in mathematical contexts that a special name for them is convenient. For example, the densest lattice packing of unit spheres in four dimensions places a sphere at every point whose coordinates are either all integers or all half-integers; this packing is closely related to the Hurwitz integers, which are quaternions whose real coefficients are either all integers or all half-integers. Moreover, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers. The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.
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