Half-integer

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

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

where ${\displaystyle n}$ is an integer. For example,

4+1/2, 7⁄2, −+13/2, 8.5

are all half-integers. The name "half-integer" is perhaps a 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 even though 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 a 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]

Notation and algebraic structure[]

The set of all half-integers is often denoted

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

The integers and half-integers together form a group under the addition operation, which may be denoted^[2]

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

However, these numbers do not form a ring because the product of two half-integers is often not a half-integer; e.g. ${\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.}$^[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: 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]

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

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

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

where n!! denotes the double factorial.

References[]