Types
of Numbers| Types
of Numbers |
In General > s.a. Complex
Numbers; Number [special ones].
* 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.
* Notation: We count in modulus 10, Babylonians used to count in modulus
60.
@ History: Menninger 69; Ifrah 85; Lines 86; Rech(95)jul-aug; Guedj
97; Gazalé 00 [I]; Hauser AS(00)mar
[and animals].
> Online Resources:
see Internet Encyclopedia of Science pages.
Integer Numbers > s.a. number
theory.
* Gaussian integers:
The numbers of the form a + i b, with a and b integers;
They form an integral domain.
@ Gaussian integers: Hardy & Wright 54.
Real Numbers > s.a. Continuum;
formalism of quantum mechanics [and quantum states].
$ Def: R is the complete ordered field, defined by the Dedekind
cuts S 
Q,
such that S 
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 Q | x < 0}; One:
1:= {x 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 
, 
such
that for all p and q, |x–p/q| > q^{–};
The larger is,
the closer to being rational.
@ Dedekind cut construction: E Landau 60.
@ Related topics: Benioff qp/05 [Fock-type representation].
Rational Numbers
$ Def: Q is
an ordered field defined by Q = N3/
,
with (a, b, c) (a', b', c')
iff 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.
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., 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.
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 R}
of M(2, R).
* Applications: Important
for the foundations of the conformal group.
@ References: Yaglom 68; Fjelstad AJP(86)
[and special relativity].
Transcendental Numbers
$ Def: A real number which is not an algebraic number.
* Examples: The numbers 
and e,
but not 21/2; There are uncountably
many (in terms of Lebesgue measure, most of the reals are transcendental).
@ References: Baker 90.
Surreal Numbers
* History: Discovered by J Conway, but named by D Knuth in his novel.
@ References: Knuth 74; Gonshor 86.
Two-Component Number (s.a. Perplex Numbers above).
* Idea: Objects of the form z = x + 
y,
with x, y R,
and satisfying 2 = 
+
, for fixed ,
R.
* Examples: =
–1, =
0: Complex numbers; =
+1, =
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].
@ 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)
[question], López AJP(95),
Phelps AJP(95),
Krantz AJP(98),
Milley AJP(98)
[answers]; Viswanath & Müller 94.
Other > s.a. functions.
@ Non-Archimedean: Rosinger m.HO/05 [use in physics]; > s.a. Adelic
Numbers; p-Adic
Numbers.
God made the integers; the rest is the work of Man – Leopold Kronecker
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008