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 *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,
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 (*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__: 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., 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;
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 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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