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Half-integer

## Summary

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

${\displaystyle n+{\tfrac {1}{2}},}$
where ${\displaystyle n}$ is a whole number. For example,
${\displaystyle 4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5}$
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.[citation needed] 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).[1]

## Notation and algebraic structure

The set of all half-integers is often denoted

${\displaystyle \mathbb {Z} +{\tfrac {1}{2}}\quad =\quad \left({\tfrac {1}{2}}\mathbb {Z} \right)\smallsetminus \mathbb {Z} ~.}$

The integers and half-integers together form a group under the addition operation, which may be denoted[2]
${\displaystyle {\tfrac {1}{2}}\mathbb {Z} ~.}$

However, these numbers do not form a ring because the product of two half-integers is often not a half-integer; e.g. ${\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.}$ [3]

## Properties

• The sum of ${\displaystyle n}$  half-integers is a half-integer if and only if ${\displaystyle n}$  is odd. This includes ${\displaystyle n=0}$  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: ${\displaystyle f:x\to x+0.5}$ , where ${\displaystyle x}$  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.[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 ${\displaystyle R}$ ,[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:
${\displaystyle \Gamma \left({\tfrac {1}{2}}+n\right)~=~{\frac {\,(2n-1)!!\,}{2^{n}}}\,{\sqrt {\pi \,}}~=~{\frac {(2n)!}{\,4^{n}\,n!\,}}{\sqrt {\pi \,}}~}$

where ${\displaystyle n!!}$  denotes the double factorial.

## References

1. ^ Sabin, Malcolm (2010). Analysis and Design of Univariate Subdivision Schemes. Geometry and Computing. Vol. 6. Springer. p. 51. ISBN 9783642136481.
2. ^ Turaev, Vladimir G. (2010). Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics. Vol. 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. ^ Baez, John C. (2005). "Review On Quaternions and Octonions: Their geometry, arithmetic, and symmetry by John H. Conway and Derek A. Smith". Bulletin of the American Mathematical Society (book review). 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. Vol. 6. Oxford University Press. p. 131. ISBN 9780191524257.
7. ^ "Equation 5.19.4". NIST Digital Library of Mathematical Functions. U.S. National Institute of Standards and Technology. 6 May 2013. Release 1.0.6.