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In [[mathematics]], a '''half-integer''' is a [[number]] of the form |
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#REDIRECT [[Mig enter]] |
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:<math>n + {1\over 2}</math>, |
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where <math>n</math> is an [[integer]]. For example, |
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:4½, 7/2, −13/2, 8.5 |
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are all half-integers. |
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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 rational]]s, numbers that can be formed by dividing an integer by a [[power of two]].<ref>{{citar ref|títol=Analysis and Design of Univariate Subdivision Schemes|volum=6|series=Geometry and Computing|nom=Malcolm|cognom=Sabin|editorial=Springer|any=2010|isbn=9783642136481|pàgina=51|url=https://books.google.com/books?id=18UC7d7h0LQC&pg=PA51}}</ref> |
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==Notation and algebraic structure== |
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The [[Set (mathematics)|set]] of all half-integers is often denoted |
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:<math>\mathbb Z + {1\over 2}.</math> |
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The integers and half-integers together form a [[group (mathematics)|group]] under the addition operation, which may be denoted<ref>{{citar ref|títol=Quantum Invariants of Knots and 3-Manifolds|volum=18|series=De Gruyter Studies in Mathematics|nom=Vladimir G.|cognom=Turaev|edició=2nd|editorial=Walter de Gruyter|any=2010|isbn=9783110221848|pàgina=390}}</ref> |
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:<math>\frac{1}{2} \mathbb Z</math>. |
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However, these numbers do not form a [[ring (mathematics)|ring]] because the product of two half-integers cannot be itself a half-integer.<ref>{{citar ref|títol=Computability and Logic|nom1=George|cognom1=Boolos|nom2=John P.|cognom2=Burgess|nom3=Richard C.|cognom3=Jeffrey|editorial=Cambridge University Press|any=2002|isbn=9780521007580|pàgina=105|url=https://books.google.com/books?id=0LpsXQV2kXAC&pg=PA105}}</ref> |
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==Uses== |
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===Sphere packing=== |
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The densest [[lattice packing]] of unit [[sphere]]s in four dimensions, called the [[D4 lattice|''D''<sub>4</sub> 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 integer]]s, which are [[quaternion]]s whose real coefficients are either all integers or all half-integers.<ref>{{citar ref|nom=Baez|enllaçautor=John C. Baez|cognom=John|títol=''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry'' by John H. Conway and Derek A. Smith|data= 12 agost 2004|publicació=Bulletin of the American Mathematical Society|volum=42|any=2005|pàgines=229–243|url=http://math.ucr.edu/home/baez/octonions/conway_smith/|doi=10.1090/S0273-0979-05-01043-8}}</ref> |
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===Physics=== |
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In physics, the [[Pauli exclusion principle]] results from definition of [[fermion]]s as particles which have [[spin (physics)|spin]]s that are half-integers.<ref>{{citar ref|títol=The High Energy Universe: Ultra-High Energy Events in Astrophysics and Cosmology|nom=Péter|cognom=Mészáros|editorial=Cambridge University Press|any=2010|isbn=9781139490726|pàgina=13|url=https://books.google.com/books?id=NXvE_zQX5kAC&pg=PA13}}</ref> |
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The [[energy level]]s of the [[quantum harmonic oscillator]] occur at half-integers and thus its lowest energy is not zero.<ref>{{citar ref|títol=Quantum Optics : An Introduction|volum=6|series=Oxford Master Series in Physics|nom=Mark|cognom=Fox|editorial=Oxford University Press|any=2006|isbn=9780191524257|pàgina=131|url=https://books.google.com/books?id=Q-4dIthPuL4C&pg=PA131}}</ref> |
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===Sphere volume=== |
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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|volume of an ''n''-dimensional ball]] of radius ''R'',<ref>Equation 5.19.4, ''NIST Digital Library of Mathematical Functions.'' http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.</ref> |
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:<math>V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n.</math> |
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The values of the gamma function on half-integers are integer multiples of the square root of [[pi]]: |
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:<math>\Gamma\left(\frac{1}{2}+n\right) = \frac{(2n-1)!!}{2^n}\, \sqrt{\pi} = {(2n)! \over 4^n n!} \sqrt{\pi} </math> |
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where ''n''!! denotes the [[double factorial]]. |
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==Referències== |
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{{referències}} |
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{{autoritat}} |
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[[Categoria:Nombres enters]] |
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Revisió del 23:28, 4 oct 2018
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-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
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]
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, which are 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 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.
Referències
- ↑ Sabin, Malcolm. Analysis and Design of Univariate Subdivision Schemes. 6. Springer, 2010, p. 51. ISBN 9783642136481.
- ↑ Turaev, Vladimir G. Quantum Invariants of Knots and 3-Manifolds. 18. 2nd. Walter de Gruyter, 2010, p. 390. ISBN 9783110221848.
- ↑ Boolos, George; Burgess, John P.; Jeffrey, Richard C. Computability and Logic. Cambridge University Press, 2002, p. 105. ISBN 9780521007580.
- ↑ John, Baez «On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith». Bulletin of the American Mathematical Society, 42, 12-08-2004, p. 229–243. DOI: 10.1090/S0273-0979-05-01043-8.
- ↑ Mészáros, Péter. The High Energy Universe: Ultra-High Energy Events in Astrophysics and Cosmology. Cambridge University Press, 2010, p. 13. ISBN 9781139490726.
- ↑ Fox, Mark. Quantum Optics : An Introduction. 6. Oxford University Press, 2006, p. 131. ISBN 9780191524257.
- ↑ Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.