Half-integer

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In mathematics, a half-integer is a number of the form ${\displaystyle n+{\tfrac {1}{2,}$ where ${\displaystyle n}$ is an integer. 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 each integer ${\displaystyle n}$ is itself half of the integer ${\displaystyle 2n}$ . A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.^{[citation needed]}

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#edit

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 not a half-integer; e.g. ${\displaystyle ~{\tfrac {1}{2\times {\tfrac {1}{2~=~{\tfrac {1}{4~\notin ~{\tfrac {1}{2\mathbb {Z} ~.}$ ^[3] The smallest ring containing them is ${\displaystyle \mathbb {Z} \left[{\tfrac {1}{2\right]}$ , the ring of dyadic rationals.

Properties#edit

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#edit

Sphere packing#edit

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#edit

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#edit

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 rational 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#edit

^ 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.

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