# Half-integer

In mathematics, a **half-integer** is a number of the form

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.^{[1]}

## Contents

## Notation and algebraic structure

The set of all half-integers is often denoted

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

However, these numbers do not form a ring because the product of two half-integers is generally not itself a half-integer.^{[3]}

## 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, which are 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 *R*,^{[7]}

The values of the gamma function on half-integers are integer multiples of the square root of pi:^{[8]}

where *n*!! denotes the double factorial.

## References

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