Half-integer explained

In mathematics, a half-integer is a number of the form $$n\; +\; \backslash tfrac,$$where

n is an integer. For example, $$4\backslash tfrac12,\backslash quad\; 7/2,\backslash quad\; -\backslash tfrac,\backslash quad\; 8.5$$are all half-integers. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers such as 1 (being half the integer 2). 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).^[1]

Notation and algebraic structure

The set of all half-integers is often denoted $$\backslash mathbb\; Z\; +\; \backslash tfrac\; \backslash quad\; =\; \backslash quad\; \backslash left(\backslash tfrac\; \backslash mathbb\; Z\; \backslash right)\; \backslash smallsetminus\; \backslash mathbb\; Z\; ~.$$The integers and half-integers together form a group under the addition operation, which may be denoted^[2] $$\backslash tfrac\; \backslash mathbb\; Z\; ~.$$However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g.

~\tfrac{ 1 }{ 2 } x \tfrac{ 1 }{ 2 } ~ = ~ \tfrac{ 1 }{ 4 } ~ \notin ~ \tfrac{ 1 }{ 2 } Z ~. ^[3] The smallest ring containing them is \Z\left[\tfrac 12 \right] , the ring of dyadic rationals.

Properties

• The sum of

n half-integers is a half-integer if and only if n is odd. This includes 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\to x +0 . 5 , where 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 -dimensional ball of radius

R ,^[7] $$V\_n(R)\; =\; \backslash fracR^n~.$$The values of the gamma function on half-integers are integer multiples of the square root of pi:$$\backslash Gamma\backslash left(\backslash tfrac\; +\; n\backslash right)\; ~=~\; \backslash frac\backslash ,\; \backslash sqrt\; ~=~\; \backslash frac\; \backslash sqrt\; ~$$where n!! denotes the double factorial.

Notes and References

1. Book: Sabin, Malcolm . 2010 . Analysis and Design of Univariate Subdivision Schemes . 6 . Geometry and Computing . Springer . 9783642136481 . 51 .
2. Book: Turaev, Vladimir G. . 2010 . Quantum Invariants of Knots and 3-Manifolds . 2nd . De Gruyter Studies in Mathematics . 18 . Walter de Gruyter . 9783110221848 . 390.
3. Book: George . Boolos . John P. . Burgess . Richard C. . Jeffrey . 2002 . Computability and Logic . 105 . Cambridge University Press . 9780521007580 .
4. John C. . Baez . John C. Baez . 2005 . Review On Quaternions and Octonions: Their geometry, arithmetic, and symmetry by John H. Conway and Derek A. Smith . book review . Bulletin of the American Mathematical Society . 42 . 229–243 . 10.1090/S0273-0979-05-01043-8 . free.
5. Book: Mészáros, Péter . 2010 . The High Energy Universe: Ultra-high energy events in astrophysics and cosmology . 13 . Cambridge University Press . 9781139490726 .
6. Book: Fox, Mark . 2006 . Quantum Optics: An introduction . 131 . Oxford Master Series in Physics . 6 . Oxford University Press . 9780191524257 .
7. Web site: Equation 5.19.4 . NIST Digital Library of Mathematical Functions . . Release 1.0.6 . 2013-05-06.