Half-integer

The set of all half-integers is often denoted \mathbb Z + \tfrac{1}{2} \quad = \quad \left( \tfrac{1}{2} \mathbb Z \right) \smallsetminus \mathbb Z ~. The integers and half-integers together form a group under the addition operation, which may be denoted \tfrac{1}{2} \mathbb Z ~. However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. ~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~. The smallest ring containing them is \Z\left[\tfrac12\right], the ring of dyadic rationals. ==Properties==

• The sum of n half-integers is a half-integer if and only if n is odd. This includes n=0 since the empty sum 0 is not half-integer. • The negative of a half-integer is a half-integer. • The cardinality of the set of half-integers is equal to that of the integers. This is due to the existence of a bijection from the integers to the half-integers: f:x\to x+0.5, where x is an integer. ==Uses==

Sphere packing 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. Physics In physics, 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. Sphere volume 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 -dimensional ball of radius R, V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~. The values of the gamma function on half-integers are rational multiples of the square root of pi: \Gamma\left(\tfrac{1}{2} + n\right) ~=~ \frac{\,(2n-1)!!\,}{2^n}\, \sqrt{\pi\,} ~=~ \frac{(2n)!}{\,4^n \, n!\,} \sqrt{\pi\,} ~ where n!! denotes the double factorial. ==References==