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In mathematics, a half-integer is a number of the form n + 1 2 , {\displaystyle n+{\tfrac {1}{2}},} where n {\displaystyle n} is an integer. For example, 4 1 2 , 7 / 2 , − − 13 2 , 8.5 {\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 each integer n {\displaystyle n} is itself half of the integer 2 n {\displaystyle 2n} . A name such as "integer-plus-half" may be more accurate, but while 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).cite-ref-1[1]

• Notation and algebraic structure

• Properties

• Uses

• Sphere packing

• Physics

• Sphere volume

• References

The set of all half-integers is often denoted Z + 1 2 = ( 1 2 Z ) ∖ ∖ Z . {\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 denotedcite-ref-2[2] 1 2 Z . {\displaystyle {\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. 1 2 × × 1 2 = 1 4 ∉ ∉ 1 2 Z . {\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.} cite-ref-3[3] The smallest ring containing them is Z [ 1 2 ] {\displaystyle \mathbb {Z} \left[{\tfrac {1}{2}}\right]} , the ring of dyadic rationals.

• The sum of n {\displaystyle n} half-integers is a half-integer if and only if n {\displaystyle n} is odd. This includes n = 0 {\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: f : x → → x + 0.5 {\displaystyle f:x\to x+0.5} , where x {\displaystyle x} is an integer.

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.cite-ref-4[4]

In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.cite-ref-5[5]

The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.cite-ref-6[6]

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 {\displaystyle R} ,cite-ref-7[7] V n ( R ) = π π n / 2 Γ Γ ( n 2 + 1 ) R n . {\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: Γ Γ ( 1 2 + n ) = ( 2 n − − 1 ) ! ! 2 n π π = ( 2 n ) ! 4 n n ! π π {\displaystyle \Gamma \left({\tfrac {1}{2}}+n\right)~=~{\frac {\,(2n-1)!!\,}{2^{n}}}\,{\sqrt {\pi \,}}~=~{\frac {(2n)!}{\,4^{n}\,n!\,}}{\sqrt {\pi \,}}~} where n ! ! {\displaystyle n!!} denotes the double factorial.

cite-note-11. ↑ citerefsabin2010Sabin, Malcolm (2010). Analysis and Design of Univariate Subdivision Schemes. Geometry and Computing. Vol. 6. Springer. p. 51. ISBN 9783642136481.

cite-note-22. ↑ citerefturaev2010Turaev, 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.

cite-note-33. ↑ citerefboolosburgessjeffrey2002Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge University Press. p. 105. ISBN 9780521007580.

cite-note-44. ↑ citerefbaez2005Baez, 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.

cite-note-55. ↑ citerefm-sz-ros2010Mészáros, Péter (2010). The High Energy Universe: Ultra-high energy events in astrophysics and cosmology. Cambridge University Press. p. 13. ISBN 9781139490726.

cite-note-66. ↑ citereffox2006Fox, Mark (2006). Quantum Optics: An introduction. Oxford Master Series in Physics. Vol. 6. Oxford University Press. p. 131. ISBN 9780191524257.

cite-note-77. ↑ "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.