# Half-integer

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

- ,

where is an integer. For example,

- 4½, 7/2, −13/2, 8.5

are all half-integers.

Half-integers occur frequently enough in mathematical contexts in which 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

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

- .

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 *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, 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]

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

where *n*!! denotes the double factorial.

## References

- Sabin, Malcolm (2010),
*Analysis and Design of Univariate Subdivision Schemes*, Geometry and Computing,**6**, Springer, p. 51, ISBN 9783642136481. - 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. - Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002),
*Computability and Logic*, Cambridge University Press, p. 105, ISBN 9780521007580. - 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. - Mészáros, Péter (2010),
*The High Energy Universe: Ultra-High Energy Events in Astrophysics and Cosmology*, Cambridge University Press, p. 13, ISBN 9781139490726. - Fox, Mark (2006),
*Quantum Optics : An Introduction*, Oxford Master Series in Physics,**6**, Oxford University Press, p. 131, ISBN 9780191524257. - Equation 5.19.4,
*NIST Digital Library of Mathematical Functions.*http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.