# Half-integer

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{{Short description|Rational number equal to an integer plus 1/2}}
{{use dmy dates|date=February 2021}}
In [mathematics](/source/mathematics), a '''half-integer''' is a [number](/source/number) of the form 
<math display=block>n + \tfrac{1}{2},</math>
where <math>n</math> is an integer. For example, 
<math display=block>4\tfrac12,\quad 7/2,\quad -\tfrac{13}{2},\quad 8.5</math>
are all ''half-integers''. The name "half-integer" is perhaps misleading, as each integer <math>n</math> is itself half of the integer <math>2n</math>. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.{{citation needed|date=February 2020}}

Note that halving an integer does not always produce a half-integer; this is only true for [odd integer](/source/odd_integer)s. For this reason, half-integers are also sometimes called '''half-odd-integers'''. Half-integers are a subset of the [dyadic rational](/source/dyadic_rational)s (numbers produced by dividing an integer by a [power of two](/source/power_of_two)).<ref>{{cite book |first=Malcolm |last=Sabin |year=2010 |title=Analysis and Design of Univariate Subdivision Schemes |volume=6 |series=Geometry and Computing |publisher=Springer |isbn=9783642136481 |page=51 |url=https://books.google.com/books?id=18UC7d7h0LQC&pg=PA51}}</ref>

==Notation and algebraic structure==
The [set](/source/Set_(mathematics)) of all half-integers is often denoted 
<math display=block>\mathbb Z + \tfrac{1}{2} \quad = \quad \left( \tfrac{1}{2} \mathbb Z \right) \smallsetminus \mathbb Z ~.</math>
The integers and half-integers together form a [group](/source/group_(mathematics)) under the addition operation, which may be denoted<ref>{{cite book |first=Vladimir G. |last=Turaev |year=2010 |title=Quantum Invariants of Knots and 3-Manifolds |edition=2nd |series=De Gruyter Studies in Mathematics |volume=18 |publisher=Walter de Gruyter |isbn=9783110221848 |page=390}}</ref>
<math display=block>\tfrac{1}{2} \mathbb Z ~.</math>
However, these numbers do not form a [ring](/source/ring_(mathematics)) because the product of two half-integers is not a half-integer; e.g. <math>~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~.</math><ref>{{cite book |first1=George |last1=Boolos |first2=John P. |last2=Burgess |first3=Richard C. |last3=Jeffrey |year=2002 |title=Computability and Logic |page=105 |publisher=Cambridge University Press |isbn=9780521007580 |url=https://books.google.com/books?id=0LpsXQV2kXAC&pg=PA105}}</ref> The [smallest ring containing](/source/subring) them is <math>\Z\left[\tfrac12\right]</math>, the ring of [dyadic rational](/source/dyadic_rational)s.

==Properties==
*The sum of <math>n</math> half-integers is a half-integer if and only if <math>n</math> is odd. This includes <math>n=0</math> since the empty sum 0 is not half-integer.
*The negative of a half-integer is a half-integer.
*The [cardinality](/source/cardinality) of the set of half-integers is equal to that of the integers. This is due to the existence of a [bijection](/source/bijection) from the integers to the half-integers: <math>f:x\to x+0.5</math>, where <math>x</math> is an integer.

==Uses==
===Sphere packing===
The densest [lattice packing](/source/lattice_packing) of [unit sphere](/source/unit_sphere)s in four dimensions (called the [''D''<sub>4</sub> lattice](/source/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 integer](/source/Hurwitz_integer)s: [quaternion](/source/quaternion)s whose real coefficients are either all integers or all half-integers.<ref>{{cite journal |first=John C. |last=Baez |authorlink=John C. Baez |year=2005 |title=Review ''On Quaternions and Octonions: Their geometry, arithmetic, and symmetry'' by John H. Conway and Derek A. Smith |type=book review |journal=Bulletin of the American Mathematical Society |volume=42 |pages=229–243 |url=http://math.ucr.edu/home/baez/octonions/conway_smith/ |doi=10.1090/S0273-0979-05-01043-8 |doi-access=free}}</ref>

===Physics===
In physics, the [Pauli exclusion principle](/source/Pauli_exclusion_principle) results from definition of [fermion](/source/fermion)s as particles which have [spin](/source/Spin_(physics))s that are half-integers.<ref>{{cite book |first=Péter |last=Mészáros |year=2010 |title=The High Energy Universe: Ultra-high energy events in astrophysics and cosmology |page=13 |publisher=Cambridge University Press |isbn=9781139490726 |url=https://books.google.com/books?id=NXvE_zQX5kAC&pg=PA13}}</ref>

The [energy level](/source/energy_level)s of the [quantum harmonic oscillator](/source/quantum_harmonic_oscillator) occur at half-integers and thus its lowest energy is not zero.<ref>{{cite book |first=Mark |last=Fox |year=2006 |title=Quantum Optics: An introduction |page=131 |series=Oxford Master Series in Physics |volume=6 |publisher=Oxford University Press |isbn=9780191524257 |url=https://books.google.com/books?id=Q-4dIthPuL4C&pg=PA131}}</ref>

===Sphere volume===
Although the [factorial](/source/factorial) function is defined only for integer arguments, it can be extended to fractional arguments using the [gamma function](/source/gamma_function). The gamma function for half-integers is an important part of the formula for the [volume of an {{mvar|n}}-dimensional ball](/source/volume_of_an_n-ball) of radius <math>R</math>,<ref>{{cite web |title=Equation&nbsp;5.19.4 |website=NIST Digital Library of Mathematical Functions |url=http://dlmf.nist.gov/ |publisher=U.S. [National Institute of Standards and Technology](/source/National_Institute_of_Standards_and_Technology) |id=Release&nbsp;1.0.6 |date=2013-05-06}}</ref>
<math display=block>V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~.</math>
The values of the gamma function on half-integers are rational multiples of the square root of [pi](/source/pi):
<math display=block>\Gamma\left(\tfrac{1}{2} + n\right) ~=~ \frac{\,(2n-1)!!\,}{2^n}\, \sqrt{\pi\,} ~=~ \frac{(2n)!}{\,4^n \, n!\,} \sqrt{\pi\,} ~</math>
where <math>n!!</math> denotes the [double factorial](/source/double_factorial).

==References==
{{reflist}}

{{Rational numbers}}

Category:Rational numbers
Category:Elementary number theory
Category:Parity (mathematics)

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Adapted from the Wikipedia article [Half-integer](https://en.wikipedia.org/wiki/Half-integer) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Half-integer?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.