In mathematics, a half-integer is a number of the form
- n + 1 / 2,
where n is an integer. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French For example,
- 4½, 7/2, −13/2, 8. 5
are all half-integers. 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. In Mathematics, the parity of an object states whether it is even or odd The half-integers are precisely those numbers that are half of an odd integer. In Mathematics, the parity of an object states whether it is even or odd
The set of all half-integers is often denoted
Uses
Half-integers occur frequently enough in mathematical contexts that a special term for them is convenient. For example, the densest lattice packing of unit spheres in four dimensions 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. In Mathematics, sphere packing problems are problems concerning arrangements of non-overlapping identical Spheres which fill a space "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, a Hurwitz quaternion (or Hurwitz integer) is a Quaternion whose components are either all Integers or all Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician
Moreover, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero. A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator.
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