In mathematics, a half-integer is a number of the form
is an integer.
The name "half-integer" is perhaps misleading, as each integer
is itself half of the integer
A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.
[citation needed] Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient.
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 integers and half-integers together form a group under the addition operation, which may be denoted[2]
However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g.
[3] The smallest ring containing them is
, the ring of dyadic rationals.
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
The values of the gamma function on half-integers are integer multiples of the square root of pi:
denotes the double factorial.