# Half-integer

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

${\displaystyle n+{1 \over 2}}$,

where ${\displaystyle n}$ is an integer. For example,

4½, 7/2, −13/2, 8.5

are all half-integers.

Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers that are half of an odd integer, and for this reason are also called the half-odd-integers. Half-integers are a special case of the dyadic rationals, numbers that can be formed by dividing an integer by a power of two.[1]

## Notation and algebraic structure

The set of all half-integers is often denoted

${\displaystyle \mathbb {Z} +{1 \over 2}.}$

The integers and half-integers together form a group under the addition operation, which may be denoted[2]

${\displaystyle {\frac {1}{2}}\mathbb {Z} }$.

However, these numbers do not form a ring because the product of two half-integers cannot be itself a half-integer.[3]

## 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, which are quaternions whose real coefficients are either all integers or all half-integers.[4]

### Physics

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]

### 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,[7]

${\displaystyle V_{n}(R)={\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}.}$

The values of the gamma function on half-integers are integer multiples of the square root of pi:

${\displaystyle \Gamma \left({\frac {1}{2}}+n\right)={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={(2n)! \over 4^{n}n!}{\sqrt {\pi }}}$

where n!! denotes the double factorial.

## References

1. ^ Sabin, Malcolm (2010), Analysis and Design of Univariate Subdivision Schemes, Geometry and Computing, 6, Springer, p. 51, ISBN 9783642136481.
2. ^ Turaev, Vladimir G. (2010), Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics, 18 (2nd ed.), Walter de Gruyter, p. 390, ISBN 9783110221848.
3. ^ Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002), Computability and Logic, Cambridge University Press, p. 105, ISBN 9780521007580.
4. ^ John, Baez (2005), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith", Bulletin of the American Mathematical Society, 42: 229–243, doi:10.1090/S0273-0979-05-01043-8.
5. ^ Mészáros, Péter (2010), The High Energy Universe: Ultra-High Energy Events in Astrophysics and Cosmology, Cambridge University Press, p. 13, ISBN 9781139490726.
6. ^ Fox, Mark (2006), Quantum Optics : An Introduction, Oxford Master Series in Physics, 6, Oxford University Press, p. 131, ISBN 9780191524257.
7. ^ Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.