Half-integer

In mathematics, a half-integer is a number of the form

,

where is an integer. For example,

4½, 7/2, −13/2, 8.5

are all half-integers. Half-integer is perhaps a misnomer, 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 representative, but "half integer" is the traditional term.[citation needed] Half-integers occur frequently enough in mathematics that a distinct term is convenient.

Note that a 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 special case of the dyadic rationals (numbers produced by dividing an integer by a power of two).[1]

Notation and algebraic structureEdit

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 cannot be itself a half-integer.[3]

UsesEdit

Sphere packingEdit

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]

PhysicsEdit

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 volumeEdit

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,[7]

 

The values of the gamma function on half-integers are integer multiples of the square root of pi:

 

where n!! denotes the double factorial.

ReferencesEdit

  1. ^ Sabin, Malcolm (2010), Analysis and Design of Univariate Subdivision Schemes, Geometry and Computing, 6, Springer, p. 51, ISBN 9783642136481.
  2. ^ Turaev, Vladimir G. (2010), Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics, 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. ^ John, Baez (2005), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith", Bulletin of the American Mathematical Society, 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, 6, Oxford University Press, p. 131, ISBN 9780191524257.
  7. ^ Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.