Half-integer
In mathematics, a half-integer is a number of the form where formula_2 is an integer. For example, are all "half-integers". The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers such as 1 (being half the integer 2). A name such as "integer-plus-half" may be more accurate, but even though not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that a 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). Notation and algebraic structure. The set of all half-integers is often denoted The integers and half-integers together form a group under the addition operation, which may be denoted However, these numbers do not form a ring because the product of two half-integers is often not a half-integer; e.g. formula_5 Uses. Sphere packing. The densest lattice packing  of unit spheres in four dimensions (called the "D" 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 , The values of the gamma function on half-integers are integer multiples of the square root of pi: where !! denotes the double factorial.