Half-integer

Last updated February 23, 2024

In mathematics, a half-integer is a number of the form

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 while not literally true, "half integer" is the conventional term.^{[ citation needed ]} 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).^[1]

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^[2]

{\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. $~{\tfrac {1}{2}}\times {\tfrac {1}{2}}~=~{\tfrac {1}{4}}~\notin ~{\tfrac {1}{2}}\mathbb {Z} ~.$ ^[3] The smallest ring containing them is $\mathbb {Z} \left[{\tfrac {1}{2}}\right]$ , 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 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.^[4]

Physics

In physics, the Pauli exclusion principle results from definition of fermions as particles which have spins that are half-integers.^[5]

The energy levels of the quantum harmonic oscillator occur at half-integers and thus its lowest energy is not zero.^[6]

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 $n$ -dimensional ball of radius $R$ ,^[7]

V_{n}(R)={\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}~.

The values of the gamma function on half-integers are integer 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.

Related Research Articles

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer $n$ ,

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter $ζ$ (zeta), is a mathematical function of a complex variable defined as

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

In mathematics, the floor function (or greatest integer function) is the function that takes as input a real number $x$ , and gives as output the greatest integer less than or equal to $x$ , denoted $⌊ x ⌋$ or $floor(x)$ . Similarly, the ceiling function maps $x$ to the smallest integer greater than or equal to $x$ , denoted $⌈ x ⌉$ or $ceil(x)$ .

<span class="mw-page-title-main">Fourier series</span> Decomposition of periodic functions into sums of simpler sinusoidal forms

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.

In mathematics, Stirling's approximation is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of $. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.$

In mechanics and geometry, the 3D rotation group, often denoted O(3), is the group of all rotations about the origin of three-dimensional Euclidean space $under the operation of composition.$

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

<span class="mw-page-title-main">Digamma function</span> Mathematical function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.

One half is the irreducible fraction resulting from dividing one (1) by two (2), or the fraction resulting from dividing any number by its double.

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function $Li s (z)$ of order $s$ and argument $z$ . Only for special values of $s$ does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

In mathematics, the double factorial of a number $n$ , denoted by $n ‼$ , is the product of all the positive integers up to $n$ that have the same parity as $n$ . That is,

In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.

The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x³ ≡ p is solvable if and only if x³ ≡ q is solvable.

In the 1760s, Johann Heinrich Lambert was the first to prove that the number $π$ is irrational, meaning it cannot be expressed as a fraction $, where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction.$

In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An $n$ -ball is a ball in an $n$ -dimensional Euclidean space. The volume of a $n$ -ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a $n$ -ball of radius $R$ is $where is the volume of the unit n -ball, the n -ball of radius 1 .$

In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.

References

↑ Sabin, Malcolm (2010). Analysis and Design of Univariate Subdivision Schemes. Geometry and Computing. Vol. 6. Springer. p. 51. ISBN 9783642136481.
↑ Turaev, Vladimir G. (2010). Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics. Vol. 18 (2nd ed.). Walter de Gruyter. p. 390. ISBN 9783110221848.
↑ Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge University Press. p. 105. ISBN 9780521007580.
↑ Baez, John C. (2005). "Review On Quaternions and Octonions: Their geometry, arithmetic, and symmetry by John H. Conway and Derek A. Smith". Bulletin of the American Mathematical Society (book review). 42: 229–243. doi: 10.1090/S0273-0979-05-01043-8 .
↑ Mészáros, Péter (2010). The High Energy Universe: Ultra-high energy events in astrophysics and cosmology. Cambridge University Press. p. 13. ISBN 9781139490726.
↑ Fox, Mark (2006). Quantum Optics: An introduction. Oxford Master Series in Physics. Vol. 6. Oxford University Press. p. 131. ISBN 9780191524257.
↑ "Equation 5.19.4". NIST Digital Library of Mathematical Functions. U.S. National Institute of Standards and Technology. 6 May 2013. Release 1.0.6.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Sabin, Malcolm (2010). Analysis and Design of Univariate Subdivision Schemes. Geometry and Computing. Vol. 6. Springer. p. 51. ISBN 9783642136481.

[2] Turaev, Vladimir G. (2010). Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics. Vol. 18 (2nd ed.). Walter de Gruyter. p. 390. ISBN 9783110221848.

[3] Boolos, George; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge University Press. p. 105. ISBN 9780521007580.

[4] Baez, John C. (2005). "Review On Quaternions and Octonions: Their geometry, arithmetic, and symmetry by John H. Conway and Derek A. Smith". Bulletin of the American Mathematical Society (book review). 42: 229–243. doi: 10.1090/S0273-0979-05-01043-8 .

[5] Mészáros, Péter (2010). The High Energy Universe: Ultra-high energy events in astrophysics and cosmology. Cambridge University Press. p. 13. ISBN 9781139490726.

[6] Fox, Mark (2006). Quantum Optics: An introduction. Oxford Master Series in Physics. Vol. 6. Oxford University Press. p. 131. ISBN 9780191524257.

[7] "Equation 5.19.4". NIST Digital Library of Mathematical Functions. U.S. National Institute of Standards and Technology. 6 May 2013. Release 1.0.6.

[1]

[2]

[3]

[4]

[5]

[6]

[7]