Half-integer
# Half-integer

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

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

where $n$ is a whole number. For example,

$4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5$

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 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 not a half-integer; e.g. $~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.$ The smallest ring containing them is $\mathbb {Z} \left[{\tfrac {1}{2}}\right]$, the ring of dyadic rationals.

## Properties

* The sum of $n$ half-integers is a half-integer if and only if $n$ is odd. This includes $n=0$ 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 from the integers to the half-integers: $f:x\to x+0.5$, where $x$ is an integer

## Uses

### Sphere packing

The densest lattice packing of unit spheres in four dimensions (called the $D_4$ 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.

