Half-integer

{{Short description|Rational number equal to an integer plus 1/2}}

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

$n + \tfrac{1}{2},$

where $n$ is an integer. For example,

$4\tfrac12,\quad 7/2,\quad -\tfrac{13}{2},\quad 8.5$

are all half-integers. The name "half-integer" is perhaps misleading, as each integer $n$ is itself half of the integer $2n$ . A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.{{citation needed|date=February 2020}} 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).{{cite book |first=Malcolm |last=Sabin |year=2010 |title=Analysis and Design of Univariate Subdivision Schemes |volume=6 |series=Geometry and Computing |publisher=Springer |isbn=9783642136481 |page=51 |url=https://books.google.com/books?id=18UC7d7h0LQC&pg=PA51}}

Notation and algebraic structure

The set of all half-integers is often denoted

$\mathbb Z + \tfrac{1}{2} \quad = \quad \left( \tfrac{1}{2} \mathbb Z \right) \smallsetminus \mathbb Z ~.$

The integers and half-integers together form a group under the addition operation, which may be denoted{{cite book |first=Vladimir G. |last=Turaev |year=2010 |title=Quantum Invariants of Knots and 3-Manifolds |edition=2nd |series=De Gruyter Studies in Mathematics |volume=18 |publisher=Walter de Gruyter |isbn=9783110221848 |page=390}}

$\tfrac{1}{2} \mathbb Z ~.$

However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. $~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~.$ {{cite book |first1=George |last1=Boolos |first2=John P. |last2=Burgess |first3=Richard C. |last3=Jeffrey |year=2002 |title=Computability and Logic |page=105 |publisher=Cambridge University Press |isbn=9780521007580 |url=https://books.google.com/books?id=0LpsXQV2kXAC&pg=PA105}} The smallest ring containing them is $\Z\left[\tfrac12\right]$ , 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.{{cite journal |first=John C. |last=Baez |authorlink=John C. Baez |year=2005 |title=Review On Quaternions and Octonions: Their geometry, arithmetic, and symmetry by John H. Conway and Derek A. Smith |type=book review |journal=Bulletin of the American Mathematical Society |volume=42 |pages=229–243 |url=http://math.ucr.edu/home/baez/octonions/conway_smith/ |doi=10.1090/S0273-0979-05-01043-8 |doi-access=free}}

=Physics=

In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.{{cite book |first=Péter |last=Mészáros |year=2010 |title=The High Energy Universe: Ultra-high energy events in astrophysics and cosmology |page=13 |publisher=Cambridge University Press |isbn=9781139490726 |url=https://books.google.com/books?id=NXvE_zQX5kAC&pg=PA13}}

The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.{{cite book |first=Mark |last=Fox |year=2006 |title=Quantum Optics: An introduction |page=131 |series=Oxford Master Series in Physics |volume=6 |publisher=Oxford University Press |isbn=9780191524257 |url=https://books.google.com/books?id=Q-4dIthPuL4C&pg=PA131}}

=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-ball of radius $R$ ,{{cite web |title=Equation 5.19.4 |website=NIST Digital Library of Mathematical Functions |url=http://dlmf.nist.gov/ |publisher=U.S. National Institute of Standards and Technology |id=Release 1.0.6 |date=2013-05-06}}

$V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~.$

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

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

where $n!!$ denotes the double factorial.

References

Category:Rational numbers

Category:Elementary number theory

Category:Parity (mathematics)