Half-integer

In mathematics, a half-integer is a number of the form n + \tfrac{1}{2}, where n is an integer. For example, 4\tfrac12,\quad 7/2,\quad -\tfrac{13}{2},\quad 8.5 are all half-integers. The name "half-integer" is perhaps misleading, as each integer n is itself half of the integer 2n . A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term. 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).

]] In mathematics, a half-integer is a number of the form n + \tfrac{1}{2}, where n is an integer. For example, 4\tfrac12,\quad 7/2,\quad -\tfrac{13}{2},\quad 8.5 are all half-integers. The name "half-integer" is perhaps misleading, as each integer n is itself half of the integer 2n . A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term. 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). ==Notation and algebraic structure== The set of all half-integers is often denoted \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 \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. ~\tfrac{1}{2} \times \tfrac{1}{2} ~=~ \tfrac{1}{4} ~ \notin ~ \tfrac{1}{2} \mathbb Z ~. The smallest ring containing them is \Z\left[\tfrac12\right] , the ring of dyadic rationals. ==Properties== The sum of n half-integers is a half-integer if and only if n is odd. This includes 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: f:x\to x+0.5 , where x is an integer. ==Uses== ===Sphere packing=== The densest lattice packing of unit spheres in four dimensions (called the D 4 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. ===Physics=== In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers. The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero. ===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 -dimensional ball of radius R , V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~. The values of the gamma function on half-integers are rational multiples of the square root of pi: \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==

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