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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a half-integer is a number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
 of the form

,

where is an integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
. 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. The half-integers are precisely those numbers that are half of an odd integer.

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.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a half-integer is a number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
 of the form

,

where is an integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
. 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. The half-integers are precisely those numbers that are half of an odd integer.

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 sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
s 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 quaternion
Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
s whose real coefficients are either all integers or all half-integers.

Moreover, the Pauli exclusion principle
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
 results from definition of fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
s as particles which have spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
s that are half-integers. The energy level
Energy level

A Quantum mechanics system or particle that is Bound state, confined spatially, can only take on certain discrete values of energy, as opposed to Classical mechanics particles, which can have any energy....
s of the quantum harmonic oscillator
Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum mechanics analogue of the harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point....
 occur at half-integers and thus its lowest energy is not zero.