# Half-integer

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]}

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

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.^{[4]}

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]}

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]}

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