In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a half-integer is a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
of the form
:
,
where
is an whole number. For example,
:, , , 8.5
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
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negat ...
).
Notation and algebraic structure
The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 often not a half-integer; e.g.
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from the integers to the half-integers: ''f'': ''x → 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:
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s whose real coefficients are either all integers or all half-integers.
Physics
In physics, the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
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
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
. The gamma function for half-integers is an important part of the formula for the
volume of an -dimensional ball of radius ,
:
The values of the gamma function on half-integers are integer multiples of the square root of
pi:
:
where !! denotes the
double factorial.
References
{{Rational numbers
Rational numbers
Elementary number theory
Parity (mathematics)