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, $${\displaystyle 4{\tfrac {1}{2}},\quad 7/2,\quad -{\tfrac {13}{2}},\quad 8.5}$$ are all half-integers. The name "half-integer" is perhaps misleading, as each integer ${\displaystyle n}$ is itself half of the integer ${\displaystyle 2n}$. 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]

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 not a half-integer; e.g. ${\displaystyle ~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.}$^[3] The smallest ring containing them is ${\displaystyle \mathbb {Z} \left[{\tfrac {1}{2}}\right]}$, the ring of dyadic rationals.

Properties

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

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 ${\displaystyle R}$,^[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 ${\displaystyle n!!}$ denotes the double factorial.