Demi-entier

In mathematics, a half-integer is a number of the form where is a whole number. For example, are all half-integers. The name "half-integer" is perhaps 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. 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 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). 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 However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. The smallest ring containing them is , the ring of dyadic rationals. The sum of half-integers is a half-integer if and only if is odd. This includes 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: , where is an integer The densest lattice packing of unit spheres 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 Hurwitz integers: quaternions whose real coefficients are either all integers or all half-integers. 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.