Half-integer


In mathematics, a half-integer is a number of the form
where is an integer. For example,
are all half-integers. Half-integer is perhaps a misnomer, as the set may be misunderstood to include numbers such as 1. A name such as "integer-plus-half" may be more representative, but "half integer" is the traditional term. Half-integers occur frequently enough in mathematics 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 special case of the dyadic rationals.

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 cannot be itself a half-integer.

Uses

Sphere packing

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: 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 n-dimensional ball of radius R,
The values of the gamma function on half-integers are integer multiples of the square root of pi:
where n!! denotes the double factorial.