Types of Numbers  

In General > s.a. Complex Numbers; Mathematical Constants and Numbers [notation, special numbers].
* Idea: A number is an element of a field, or of an algebra; Physically, a quantity we use as a scalar; The most commonly used ones by far are real and complex numbers.
@ General references: Pereyra 20 [for physicists].
@ History: Menninger 69; Ifrah 85; Lines 86; Crossley 87; Rech(95)jul-aug; Guedj 98; Gazalé 00 [I]; Hauser AS(00)mar [and animals]; Körner 20.
> Online Resources: see Internet Encyclopedia of Science pages.

Integer Numbers > s.a. number theory [including prime numbers].
* Gaussian integers: The numbers of the form a + i b, with a and b integers; They form an integral domain.
@ General references: Budinich AACA(16)-a1605 [binary representation, and Clifford algebras].
@ Gaussian integers: Hardy & Wright 54.
@ Other numbers: Abramowicz & Cetera a1208 [Hypatia and negative numbers]; Kenney & Bezuszka 15 [families of positive integers].

Real Numbers > s.a. Continuum; formalism of quantum mechanics [and quantum states].
$ Def: \(\mathbb R\) is the complete ordered field, defined by the Dedekind cuts S ⊂ \(\mathbb Q\), such that S ≠ \(\mathbb Q\), S has no greatest element, and xS, yx implies yS.
* Addition: S + T:= {x+y | xS, yY}; Zero: 0:= {x ∈ \(\mathbb Q\) | x < 0}; One: 1:= {x ∈ \(\mathbb Q\) | x < 1}; Multiplication: S · T:= {x · y | xS, yT} (leads to superable difficulties–exercise).
* Remark: How far a number is from being rational can be measured by numbers ε, λ such that for all p and q, |xp/q| > ε qλ; The larger λ is, the closer to being rational.
* Extended real numbers: The extended real number line is the real number line with the addition of infinity, \(\mathbb R\) ∪ {–∞, +∞}.
@ General references: Landau 60 [Dedekind cut construction]; Borwein & Borwein 90 ["dictionary of real numbers"]; Bukovský 11 [structure of the real line].
@ Extensions, deformations: Sergeyev CSF(09)-a1203 [infinite and infinitesimal numbers, and areas/volumes of fractals]; Corbett a0905 [quantum real numbers]; Giordano RJMP(10)-a0909 [extension containing nilpotent infinitesimals]; Kalogeropoulos PhyA(12)-a1104 [deformation, from Tsallis entropy]; > s.a. Hypercomplex Numbers; Hypernumbers; non-standard analysis.
@ Related topics: Benioff IJPAM(07)qp/05 [Fock-type representation]; Garrity a0811 [thermodynamic classification]; Visser a1212-FQXi [alternatives to the real number system for modelling empirical reality]; Gisin a1803-conf, a1909-conf [not physically real, and non-determinism].

Rational Numbers
$ Def: \(\mathbb Q\) is an ordered field defined by \(\mathbb Q\) = \(\mathbb N^3\)/~, with (a, b, c) ~ (a', b', c'), ac' + b'c = a'c + bc'.
* Remark: Represent (a, b, c) as (ab)/c; Then the operations and 0, 1, are defined as expected.
@ Irrational numbers: Pešić Isis(10)#3 [and music]; Havil 12 [I].

Algebraic Numbers
$ Def: Numbers which are roots of algebraic equations with integer coefficients (as opposed to transcendental numbers).
* Examples: All rationals, as well as many irrationals (e.g., 21/2), and there are countably many.
* History: Contributors to the theory were Abel, Galois, E I Zolotarev, G F Voronoi, N G Chebotarev.
@ References: Niven 61 [I]; Lang 64; Artin 67; Cassels & Fröhlich ed-67; news cosmos(18)sep [patterns in the complex plane].

Perplex Numbers > s.a. fractals.
* Idea: Numbers of the form z = x + h y, where the "hallucinatory'' h is such that ||h|| = −1.
$ Def: The subring {\(a\ \&\ b\) \cr \(b\ \&\ a\)}, with a, \(b \in \mathbb R\) of M(2, \(\mathbb R\)).
* Applications: It is mportant for the foundations of the conformal group.
@ References: Yaglom 68; Fjelstad AJP(86)may [and special relativity].

Transcendental Numbers
$ Def: A real number which is not an algebraic number.
* Examples: The numbers π and e, but not 21/2; There are uncountably many (in terms of Lebesgue measure, most of the reals are transcendental).
@ References: Baker 90; Murty & Rath 14.

Surreal Numbers
* History: Discovered by J Conway, but named by D Knuth in his novel.
@ References: Knuth 74; Gonshor 86.
> Online Resources: see MathWorld page; Wikipedia page.

Two-Component Number (s.a. Perplex Numbers above).
* Idea: Objects of the form z = x + γ y, with x, y ∈ \(\mathbb R\), and γ satisfying γ2 = α + γβ, for fixed α, β ∈ \(\mathbb R\).
* Examples: α = −1, β = 0: Complex numbers; α = +1, β = 0: Perplex numbers [= hyperbolic numbers? > see trigonometry].

Continued Fractions > s.a. stochastic processes [birth-and-death].
* Applications: Used to solve algebraic and diophantine equations.
@ References: Olds 63; Flajolet DM(80) [combinatorial aspects]; Bender & Milton JMP(94) [relationship with Taylor series]; Hensley 06; Lorentzen & Waadeland 08 [convergence theory].
@ And correlation functions: Mori PTP(65); Dupuis PTP(67); Karasudani et al PTP(79); Nagano et al PTP(80); Fujisaka & Inoue PTP(87).
@ Other physics: Lee PRL(82), PRB(82) [Langevin equation and recurrence relations]; Neuenschwander AJP(94)oct [question], López AJP(95)jul, Phelps AJP(95)jul, Krantz AJP(98)apr, Milley AJP(98)aug [answers]; Viswanath & Müller 94.

Other > s.a. Adelic, p-Adic and general Non-Archimedean Numbers; functions; quaternions.

God made the integers; the rest is the work of Man – Leopold Kronecker

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