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In mathematics, a **half-integer** is a number of the form
where is an integer. For example,
are all *half-integers*. The name "half-integer" is perhaps misleading, as each integer is itself half of the integer . A name such as "integer-plus-half" may be more accurate, but while 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]}

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 is not a half-integer; e.g. ^{[3]} The smallest ring containing them is , the ring of dyadic rationals.

- The sum of half-integers is a half-integer if and only if is odd. This includes 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: , where is an integer

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: quaternions whose real coefficients are either all integers or all half-integers.^{[4]}

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]}

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 ,^{[7]}
The values of the gamma function on half-integers are integer multiples of the square root of pi:
where denotes the double factorial.

**^**Sabin, Malcolm (2010).*Analysis and Design of Univariate Subdivision Schemes*. Geometry and Computing. Vol. 6. Springer. p. 51. ISBN 9783642136481.**^**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.**^**Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002).*Computability and Logic*. Cambridge University Press. p. 105. ISBN 9780521007580.**^**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.**^**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. Vol. 6. Oxford University Press. p. 131. ISBN 9780191524257.**^**"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.