Half-integer

In mathematics, a half-integer is a number of the form $${\displaystyle n+{\tfrac {1}{2}},}$${\displaystyle n+{\tfrac {1}{2}},} where ${\displaystyle n}${\displaystyle n} is an integer. For example, $${\displaystyle 4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5}$${\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 each integer ${\displaystyle n}${\displaystyle n} is itself half of the integer ${\displaystyle 2n}${\displaystyle 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 ]} 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 $${\displaystyle \mathbb {Z} +{\tfrac {1}{2}}\quad =\quad \left({\tfrac {1}{2}}\mathbb {Z} \right)\smallsetminus \mathbb {Z} ~.}$${\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] $${\displaystyle {\tfrac {1}{2}}\mathbb {Z} ~.}$${\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. ${\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.}${\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.}^[3] The smallest ring containing them is ${\displaystyle \mathbb {Z} \left[{\tfrac {1}{2}}\right]}${\displaystyle \mathbb {Z} \left[{\tfrac {1}{2}}\right]}, the ring of dyadic rationals.

• The sum of ${\displaystyle n}${\displaystyle n} half-integers is a half-integer if and only if ${\displaystyle n}${\displaystyle n} is odd. This includes ${\displaystyle n=0}${\displaystyle 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: ${\displaystyle f:x\to x+0.5}${\displaystyle f:x\to x+0.5}, where ${\displaystyle x}${\displaystyle x} is an integer.

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 ${\displaystyle R}${\displaystyle R},^[7] $${\displaystyle V_{n}(R)={\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}~.}$${\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: $${\displaystyle \Gamma \left({\tfrac {1}{2}}+n\right)~=~{\frac {,円(2n-1)!!,円}{2^{n}}},円{\sqrt {\pi ,円}}~=~{\frac {(2n)!}{,4円^{n},円n!,円}}{\sqrt {\pi ,円}}~}$${\displaystyle \Gamma \left({\tfrac {1}{2}}+n\right)~=~{\frac {,円(2n-1)!!,円}{2^{n}}},円{\sqrt {\pi ,円}}~=~{\frac {(2n)!}{,4円^{n},円n!,円}}{\sqrt {\pi ,円}}~} where ${\displaystyle n!!}${\displaystyle n!!} denotes the double factorial.

• Rational numbers
• Elementary number theory
• Parity (mathematics)

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