Inmathematics, ahalf-integer is anumber of the form$${\displaystyle n+\frac{1}{2},}$$where${\displaystyle n}$ is an integer. For example,$${\displaystyle 4\frac{1}{2},7/2,-\frac{13}{2},8.5}$$are allhalf-integers. The name "half-integer" is perhaps misleading, as each integer${\displaystyle n}$ is itself half of the integer${\displaystyle 2n}$. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term.^{[citation needed]} Half-integers occur frequently enough in mathematics and inquantum mechanics that a distinct term is convenient.

Note that halving an integer does not always produce a half-integer; this is only true forodd integers. For this reason, half-integers are also sometimes calledhalf-odd-integers. Half-integers are a subset of thedyadic rationals (numbers produced by dividing an integer by apower of two).^[1]

Theset of all half-integers is often denoted$${\displaystyle \mathbb{Z}+\frac{1}{2}=\left(\frac{1}{2}\mathbb{Z}\right)\setminus \mathbb{Z}.}$$The integers and half-integers together form agroup under the addition operation, which may be denoted^[2]$${\displaystyle \frac{1}{2}\mathbb{Z}.}$$However, these numbers do not form aring because the product of two half-integers is not a half-integer; e.g.${\displaystyle \frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\notin \frac{1}{2}\mathbb{Z}.}$^[3] Thesmallest ring containing them is${\displaystyle \mathbb{Z}\left[\frac{1}{2}\right]}$, the ring ofdyadic rationals.

• The sum of${\displaystyle n}$ half-integers is a half-integer if and only if${\displaystyle n}$ is odd. This includes${\displaystyle n=0}$ since the empty sum 0 is not half-integer.
• The negative of a half-integer is a half-integer.
• Thecardinality of the set of half-integers is equal to that of the integers. This is due to the existence of abijection from the integers to the half-integers:${\displaystyle f:x\to x+0.5}$, where${\displaystyle x}$ is an integer.

The densestlattice packing ofunit spheres in four dimensions (called theD₄ lattice) places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to theHurwitz integers:quaternions whose real coefficients are either all integers or all half-integers.^[4]

In physics, thePauli exclusion principle results from definition offermions as particles which havespins that are half-integers.^[5]

Theenergy levels of thequantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.^[6]

Although thefactorial function is defined only for integer arguments, it can be extended to fractional arguments using thegamma function. The gamma function for half-integers is an important part of the formula for thevolume of ann-dimensional ball of radius${\displaystyle R}$,^[7]$${\displaystyle V_n(R)=\frac{{\pi }^{n/2}}{\mathrm{\Gamma }(\frac{n}{2}+1)}{R}^n.}$$The values of the gamma function on half-integers are integer multiples of the square root ofpi:$${\displaystyle \mathrm{\Gamma }\left(\frac{1}{2}+n\right)=\frac{(2n-1)!!}{2^n}\sqrt{\pi }=\frac{(2n)!}{4^{n}n!}\sqrt{\pi }}$$where${\displaystyle n!!}$ denotes thedouble factorial.

• Rational numbers
• Elementary number theory
• Parity (mathematics)

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