Half-integer

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

n + {1\over 2},

where n is an integer. For example,

4½, 7/2, −13/2, 8.5

are all half-integers.

Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers that are half of an odd integer, and for this reason are also called the half-odd-integers. Half-integers are a special case of the dyadic rationals, numbers that can be formed by dividing an integer by a power of two.[1]

Notation and algebraic structure[edit]

The set of all half-integers is often denoted

\mathbb Z + {1\over 2}.

The integers and half-integers together form a group under the addition operation, which may be denoted[2]

\frac{1}{2} \mathbb Z.

However, these numbers do not form a ring because the product of two half-integers is generally not itself a half-integer.[3]

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, which are 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 R,[7]

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:[8]

\Gamma\left(\frac{1}{2}+n\right) = \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = {(2n)! \over 4^n n!} \sqrt{\pi}

where n!! denotes the double factorial.

References[edit]

  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 (August 12, 2004), "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.
  8. ^ Bonnar, James (2013), The Gamma Function, Applied Research Press, p. 43, ISBN 9781493775439 .