In mathematics, a half-integer is a number are all half-integers. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.

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).

Notation and algebraic structure

The set 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 The smallest ring containing, the ring of dyadic rationals.

Properties

• 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 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 spinss 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, The values of the gamma function on half-integers are rational multiples of the square root of pi denotes the double factorial.

References