In mathematics, a half-integer is a number of the form n + 1 2 , {\displaystyle n+{\tfrac {1}{2}},} where n {\displaystyle n} is an integer. For example, 4 1 2 , 7 / 2 , − − --> 13 2 , 8.5 {\displaystyle 4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5} are all half-integers. The name "half-integer" is perhaps misleading, 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 accurate, but while not literally true, "half integer" is the conventional term.[citation needed] 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).[1]
The set of all half-integers is often denoted Z + 1 2 = ( 1 2 Z ) ∖ ∖ --> Z . {\displaystyle \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[2] 1 2 Z . {\displaystyle {\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. 1 2 × × --> 1 2 = 1 4 ∉ ∉ --> 1 2 Z . {\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.} [3] The smallest ring containing them is Z [ 1 2 ] {\displaystyle \mathbb {Z} \left[{\tfrac {1}{2}}\right]} , the ring of dyadic rationals.
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]
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]
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 {\displaystyle R} ,[7] V n ( R ) = π π --> n / 2 Γ Γ --> ( n 2 + 1 ) R n . {\displaystyle 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: Γ Γ --> ( 1 2 + n ) = ( 2 n − − --> 1 ) ! ! 2 n π π --> = ( 2 n ) ! 4 n n ! π π --> {\displaystyle \Gamma \left({\tfrac {1}{2}}+n\right)~=~{\frac {\,(2n-1)!!\,}{2^{n}}}\,{\sqrt {\pi \,}}~=~{\frac {(2n)!}{\,4^{n}\,n!\,}}{\sqrt {\pi \,}}~} where n ! ! {\displaystyle n!!} denotes the double factorial.
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