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., 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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