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.

@ __History__: Menninger 69; Ifrah 85; Lines 86; Crossley 87; Rech(95)jul-aug;
Guedj
98; Gazalé 00 [I]; Hauser AS(00)mar [and animals].

> __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 *x* ∈ *S*, *y* ≤ *x* implies *y* ∈ *S*.

* __Addition__: *S* + *T*:= {*x*+*y* | *x* ∈ *S*,
*y* ∈ *Y*}; __Zero__:
0:= {*x* ∈ \(\mathbb Q\) | *x* < 0}; __One__:
1:= {*x* ∈ \(\mathbb Q\) | *x* < 1};
__Multiplication__: *S* · *T*:= {*x* · *y* | *x* ∈ *S*, *y* ∈ *T*}
(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*, |*x*–*p*/*q*| > *ε q*^{–λ};
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 [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 (*a*–*b*)/*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__: Number which are
roots of some algebraic equation with integer coefficients (opposed to transcendental
number).

* __Examples__: All rationals, as
well as many irrationals (e.g., 2^{1/2}),
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.

**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*};
*a*, *b* ∈ \(\mathbb R\)}
of *M*(2, \(\mathbb R\)).

* __Applications__: Important
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 2^{1/2}; 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 *γ*^{2} = *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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