 Half-integer
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Half-integer

In mathematics, a half-integer is a number of the form

$n+{\tfrac {1}{2}}$ ,

where $n$ is an integer. For example,

4+1/2, 72, +13/2, 8.5

are all half-integers. The name "half-integer" is perhaps a 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

$\mathbb {Z} +{\tfrac {1}{2}}\quad =\quad \left({\tfrac {1}{2}}\mathbb {Z} \right)\smallsetminus \mathbb {Z} ~.$ 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 often not a half-integer; e.g. $~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.$ ## 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 n-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 integer 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.