In mathematics, a half-integer is a number of the form
Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two).[1]
The set of all half-integers is often denoted
The densest lattice packing of unit spheres in four dimensions (called the D4 lattice) 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: quaternions whose real coefficients are either all integers or all half-integers.[4]
In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.[5]
The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.[6]
Although the factorial function is defined only for integer arguments, it can be extended to fractional arguments using the gamma function. The gamma function for half-integers is an important part of the formula for the volume of an n-dimensional ball of radius ,[7]