Half-integer

 ( Rational Numbers ) 

In mathematics, a half-integer is a number of the form $$n\; +\; \backslash tfrac\{1\}\{2\},$$ where $n$ is an integer. For example, $$4\backslash tfrac12,\backslash quad\; 7/2,\backslash quad\; -\backslash tfrac\{13\}\{2\},\backslash quad\; 8.5$$ 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. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient.

Note that halving an integer does not always produce a half-integer; this is only true for . For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the (numbers produced by dividing an integer by a power of two).

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Notation and algebraic structure The set of all half-integers is often denoted $$\backslash mathbb\; Z\; +\; \backslash tfrac\{1\}\{2\}\; \backslash quad\; =\; \backslash quad\; \backslash left(\; \backslash tfrac\{1\}\{2\}\; \backslash mathbb\; Z\; \backslash right)\; \backslash smallsetminus\; \backslash mathbb\; Z\; ~.$$ The integers and half-integers together form a group under the addition operation, which may be denoted

Vladimir G. Turaev (2025). 9783110221848, Walter de Gruyter. ISBN 9783110221848

$$\backslash tfrac\{1\}\{2\}\; \backslash mathbb\; Z\; ~.$$ However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. $~\backslash tfrac\{1\}\{2\}\; \backslash times\; \backslash tfrac\{1\}\{2\}\; ~=~\; \backslash tfrac\{1\}\{4\}\; ~\; \backslash notin\; ~\; \backslash tfrac\{1\}\{2\}\; \backslash mathbb\; Z\; ~.$

(2025). 9780521007580, Cambridge University Press. . ISBN 9780521007580

The subring them is $\backslash Z\backslash left\backslash tfrac12\backslash right$, the ring of .

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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: $f:x\backslash to\; x+0.5$, where $x$ is an integer.

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Uses

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Sphere packing The densest lattice packing of 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 : whose real coefficients are either all integers or all half-integers.

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Physics In physics, the Pauli exclusion principle results from definition of as particles which have spins that are half-integers.

Péter Mészáros (2025). 9781139490726, Cambridge University Press. . ISBN 9781139490726

The of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.

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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 $R$, $$V\_n(R)\; =\; \backslash frac\{\backslash pi^\{n/2\}\}\{\backslash Gamma(\backslash frac\{n\}\{2\}\; +\; 1)\}R^n~.$$ The values of the gamma function on half-integers are integer multiples of the square root of pi: $$\backslash Gamma\backslash left(\backslash tfrac\{1\}\{2\}\; +\; n\backslash right)\; ~=~\; \backslash frac\{\backslash ,(2n-1)!!\backslash ,\}\{2^n\}\backslash ,\; \backslash sqrt\{\backslash pi\backslash ,\}\; ~=~\; \backslash frac\{(2n)!\}\{\backslash ,4^n\; \backslash ,\; n!\backslash ,\}\; \backslash sqrt\{\backslash pi\backslash ,\}\; ~$$ where $n!!$ denotes the double factorial.

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