{
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  "slug": "half-integer",
  "title": "Half-integer",
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  "lead": "In mathematics, a half-integer is a number of the form n + \\tfrac{1}{2}, where n is an integer. For example, 4\\tfrac12,\\quad 7/2,\\quad -\\tfrac{13}{2},\\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. Note that 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).",
  "body": "]] In mathematics, a half-integer is a number of the form n + \\tfrac{1}{2}, where n is an integer. For example, 4\\tfrac12,\\quad 7/2,\\quad -\\tfrac{13}{2},\\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. Note that 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). ==Notation and algebraic structure== The set of all half-integers is often denoted \\mathbb Z + \\tfrac{1}{2} \\quad = \\quad \\left( \\tfrac{1}{2} \\mathbb Z \\right) \\smallsetminus \\mathbb Z ~. The integers and half-integers together form a group under the addition operation, which may be denoted \\tfrac{1}{2} \\mathbb Z ~. However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. ~\\tfrac{1}{2} \\times \\tfrac{1}{2} ~=~ \\tfrac{1}{4} ~ \\notin ~ \\tfrac{1}{2} \\mathbb Z ~. The smallest ring containing them is \\Z\\left[\\tfrac12\\right] , the ring of dyadic rationals. ==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\\to x+0.5 , where x is an integer. ==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: quaternions whose real coefficients are either all integers or all half-integers. ===Physics=== In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers. The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero. ===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) = \\frac{\\pi^{n/2}}{\\Gamma(\\frac{n}{2} + 1)}R^n~. The values of the gamma function on half-integers are rational multiples of the square root of pi: \\Gamma\\left(\\tfrac{1}{2} + n\\right) ~=~ \\frac{\\,(2n-1)!!\\,}{2^n}\\, \\sqrt{\\pi\\,} ~=~ \\frac{(2n)!}{\\,4^n \\, n!\\,} \\sqrt{\\pi\\,} ~ where n!! denotes the double factorial. ==References==",
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    "title": "Half-integer",
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    "publisher-place": "Dubai, United Arab Emirates",
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