Half-integer

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]} 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 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

• 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.