Elementary arithmetic: Difference between revisions

Revision as of 05:06, 12 February 2014

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-{{unreferenced|date=August 2007}}
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-In [[mathematics]], a '''half-integer''' is a [[number]] of the form
-:<math>n + {1\over 2}</math>,
-where <math>n</math> is an [[integer]].  For example,
-:4&frac12;, 7/2, &minus;13/2, 8.5
-are all half-integers. 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]].
-The [[Set (mathematics)|set]] of all half-integers is often denoted
-:<math>\mathbb Z + {1\over 2}.</math>
-==Uses==
-Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. For example, the densest [[lattice packing]] of unit [[sphere]]s in four dimensions 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.
-Moreover, the [[Pauli exclusion principle]] results from definition of [[fermion]]s as particles which have [[spin (physics)|spin]]s that are half-integers. The [[energy level]]s of the [[quantum harmonic oscillator]] occur at half-integers and thus its lowest energy is not zero.
-Also, the [[factorial]] and [[gamma function]]s, while not defined for [[negative number|negative and non-positive integers]] respectively, are defined for all half-integers as rational multiples of the [[square root]] of [[pi]].
-[[Category:Rational numbers]]
-[[Category:Elementary number theory]]
-[[Category:Parity]]

Elementary arithmetic: Difference between revisions

Revision as of 05:06, 12 February 2014

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