where is an integer. For example,
- 4½, 7/2, −13/2, 8.5
are all half-integers.
Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer. The half-integers are precisely those numbers that are half of an odd integer, and for this reason are also called the half-odd-integers. Half-integers are a special case of the dyadic rationals, numbers that can be formed by dividing an integer by a power of two.
Notation and algebraic structure
The set of all half-integers is often denoted
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, which are quaternions whose real coefficients are either all integers or all half-integers.
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,
where n!! denotes the double factorial.
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|year= / |date= mismatch(help).
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