Английская Википедия:Half-integer
Шаблон:Short description Шаблон:Use dmy dates In mathematics, a half-integer is a number of the form <math display=block>n + \tfrac{1}{2},</math> where <math>n</math> is an integer. For example, <math display=block>4\tfrac12,\quad 7/2,\quad -\tfrac{13}{2},\quad 8.5</math> 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.Шаблон: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 <math display=block>\mathbb Z + \tfrac{1}{2} \quad = \quad \left( \tfrac{1}{2} \mathbb Z \right) \smallsetminus \mathbb Z ~.</math> The integers and half-integers together form a group under the addition operation, which may be denoted[2] <math display=block>\tfrac{1}{2} \mathbb Z ~.</math> However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. <math>~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~.</math>[3] The smallest ring containing them is <math>\Z\left[\tfrac12\right]</math>, the ring of dyadic rationals.
Properties
- The sum of <math>n</math> half-integers is a half-integer if and only if <math>n</math> is odd. This includes <math>n=0</math> 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: <math>f:x\to x+0.5</math>, where <math>x</math> 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-ball|volume of an Шаблон:Mvar-dimensional ball]] of radius <math>R</math>,[7] <math display=block>V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~.</math> The values of the gamma function on half-integers are integer multiples of the square root of pi: <math display=block>\Gamma\left(\tfrac{1}{2} + n\right) ~=~ \frac{\,(2n-1)!!\,}{2^n}\, \sqrt{\pi\,} ~=~ \frac{(2n)!}{\,4^n \, n!\,} \sqrt{\pi\,} ~</math> where <math>n!!</math> denotes the double factorial.
References
