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Half-integer
Half-integer
Half-integer
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In mathematics, a half-integer is a number of the form where is an integer. For example, are all half-integers. The name "half-integer" is perhaps misleading, as each integer is itself half of the integer . 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

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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[2] However, these numbers do not form a ring because the product of two half-integers is not a half-integer; e.g. [3] The smallest ring containing them is , the ring of dyadic rationals.

Properties

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  • 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.

Uses

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Sphere packing

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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

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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

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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 ,[7] The values of the gamma function on half-integers are integer multiples of the square root of pi: where denotes the double factorial.

References

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Half-integer

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A half-integer is a number of the form n+12n + \frac{1}{2}, where nn is an , equivalent to 2k+12\frac{2k + 1}{2} for some kk. Examples include …,−32,−12,12,32,52,…\dots, -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \dots. The collection of all half-integers forms a set that lies midway between consecutive and plays roles in various mathematical contexts, such as modular forms and like the , where values at half-integer arguments yield explicit expressions involving square roots of π. In physics, particularly , half-integers are significant for the ss, which characterizes the intrinsic of elementary particles. Particles with half-integer spin values (such as s=12,32s = \frac{1}{2}, \frac{3}{2}) are classified as fermions, including electrons, protons, and neutrons, and they obey the , which prohibits identical fermions from occupying the same . This fermionic nature arises from the antisymmetric wavefunctions required for half-integer spin systems under particle exchange. In contrast, particles with spin are bosons, which can occupy the same state, leading to phenomena like Bose-Einstein condensation. The concept of half-integer spin originates from the relativistic quantum theory developed by in 1928, where the naturally predicts for electrons, resolving inconsistencies in the non-relativistic . Half-integer spins also manifest in higher representations, such as spin-3/2 for certain exotic particles, described by multi-component wavefunctions. This dichotomy between integer and half-integer spins underpins the spin-statistics theorem in , dictating the statistical behavior of particles in multi-particle systems.

Fundamentals

Definition

A half-integer is a number expressible as n+12n + \frac{1}{2}, where nn is an integer (positive, negative, or zero). This yields the sequence …,−32,−12,12,32,…\dots, -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \dots. Integers form the foundational set of whole numbers, and half-integers extend this by incorporating the fractional offset of one-half, positioning them midway between consecutive integers on the number line. Examples include 12\frac{1}{2} (between 0 and 1), 52\frac{5}{2} (between 2 and 3), and −32-\frac{3}{2} (between -2 and -1). Half-integers constitute a proper of the rational numbers, specifically those expressible as an odd divided by 2 in lowest terms, such as 12=12\frac{1}{2} = \frac{1}{2}, 52\frac{5}{2}, and −32=−32-\frac{3}{2} = \frac{-3}{2}. These are dyadic rationals distinguished by their odd numerator over a denominator of 2, and the set of half-integers is not closed under multiplication.

Notation

Half-integers are primarily denoted in fractional form as 2k+12\frac{2k+1}{2}, where kk is an , emphasizing their structure as odd multiples of one-half. For instance, when k=1k=1, this yields 32\frac{3}{2}, a common example in analytic contexts such as the values. In decimal notation, half-integers are expressed as terminating s, such as 0.5 for 12\frac{1}{2}, due to their dyadic rational nature, which ensures finite decimal expansions without repetition. The set of all half-integers is conventionally denoted using Z+12\mathbb{Z} + \frac{1}{2} or equivalently 12+Z\frac{1}{2} + \mathbb{Z}, where Z\mathbb{Z} represents the . An explicit is H={n+12∣n∈Z}H = \{ n + \frac{1}{2} \mid n \in \mathbb{Z} \}, capturing all elements like …,−1.5,−0.5,0.5,1.5,…\dots, -1.5, -0.5, 0.5, 1.5, \dots. The notation for half-integers draws from broader conventions in involving fractional indices. In contextual usage, half-integers frequently serve as indices or labels in mathematical sequences, such as in approximations for partial sums of the harmonic series using half-integer variables or in evaluations of numbers at half-integer points.

Mathematical Properties

Algebraic Structure

The set of half-integers HH, when considered together with the Z\mathbb{Z}, forms the set 12Z\frac{1}{2}\mathbb{Z}, which is an under addition isomorphic to the additive group Z\mathbb{Z}. The of this group is 0, an rather than a half-integer, and the inverse of any half-integer h∈Hh \in H is −h-h, which is also in HH. The is given by the map ϕ:12Z→Z\phi: \frac{1}{2}\mathbb{Z} \to \mathbb{Z} defined by ϕ(x)=2x\phi(x) = 2x, which preserves the group operation since ϕ(x+y)=2(x+y)=2x+2y=ϕ(x)+ϕ(y)\phi(x + y) = 2(x + y) = 2x + 2y = \phi(x) + \phi(y). The set HH itself is not closed under addition; the sum of two half-integers is always an integer. For instance, 12+12=1\frac{1}{2} + \frac{1}{2} = 1. Every half-integer lies precisely halfway between two consecutive integers, such as 32\frac{3}{2} between 1 and 2. The set HH is in bijective correspondence with Z\mathbb{Z} via the map ϕ:H→Z\phi: H \to \mathbb{Z} given by ϕ(n+12)=2n+1\phi\left(n + \frac{1}{2}\right) = 2n + 1 for n∈Zn \in \mathbb{Z}, which sends half-integers to the odd integers (and vice versa). Under multiplication, the set HH is not closed; the product of two half-integers is generally a dyadic rational with denominator at least 4. For example, 12×12=14\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}, a quarter-integer. More precisely, if h1=2m+12h_1 = \frac{2m+1}{2} and h2=2n+12h_2 = \frac{2n+1}{2} with m,n∈Zm, n \in \mathbb{Z}, then h1h2=(2m+1)(2n+1)4h_1 h_2 = \frac{(2m+1)(2n+1)}{4}, where the numerator is odd and the denominator is 4. The half-integers do not form a ring, primarily due to the lack of closure under multiplication (and also under addition). They are, however, a subset of the rational numbers Q\mathbb{Q}, and adjoining HH to Z\mathbb{Z} generates the ring of dyadic rationals Z[12]\mathbb{Z}\left[\frac{1}{2}\right], consisting of all rationals whose denominators are powers of 2.

Analytic Properties

Half-integers play a significant role in the analytic extension of the function through the , Γ(z)\Gamma(z), which provides values at non-integer points. For positive integers nn, the evaluates to Γ(n+12)=(2n)!4nn!π\Gamma\left(n + \frac{1}{2}\right) = \frac{(2n)!}{4^n n!} \sqrt{\pi}
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