kb/data/en.wikipedia.org/wiki/List_of_numbers-1.md

title	chunk	source	category	tags	date_saved	instance
List of numbers	2/2	https://en.wikipedia.org/wiki/List_of_numbers	reference	science, encyclopedia	2026-05-05T08:19:08.279774+00:00	kb-cron

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is trivially a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold

      Q
    
  

{\displaystyle \mathbb {Q} }

, Unicode U+211A ℚ DOUBLE-STRUCK CAPITAL Q); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient". Rational numbers such as 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), three twenty-fifths (⁠3/25⁠), nine seventy-fifths (⁠9/75⁠), etc. This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction. A list of rational numbers is shown below. The names of fractions can be found at numeral (linguistics).

== Real numbers == Real numbers are least upper bounds of sets of rational numbers that are bounded above, or greatest lower bounds of sets of rational numbers that are bounded below, or limits of convergent sequences of rational numbers. Real numbers that are not rational numbers are called irrational numbers. The real numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not; all rational numbers are algebraic.

=== Algebraic numbers ===

=== Transcendental numbers ===

=== Irrational but not known to be transcendental === Some numbers are known to be irrational numbers, but have not been proven to be transcendental. This differs from the algebraic numbers, which are known not to be transcendental.

=== Real but not known to be irrational, nor transcendental === For some numbers, it is not known whether they are algebraic or transcendental. The following list includes real numbers that have not been proved to be irrational, nor transcendental.

=== Numbers not known with high precision ===

Some real numbers, including transcendental numbers, are not known with high precision.

The constant in the Berry–Esseen Theorem: 0.4097 < C < 0.4748 De Bruijn–Newman constant: 0 ≤ Λ ≤ 0.2 Chaitin's constants Ω, which are transcendental and provably impossible to compute. Bloch's constant (also 2nd Landau's constant): 0.4332 < B < 0.4719 1st Landau's constant: 0.5 < L < 0.5433 3rd Landau's constant: 0.5 < A ≤ 0.7853 Grothendieck constant: 1.67 < k < 1.79 Romanov's constant in Romanov's theorem: 0.107648 < d < 0.49094093, Romanov conjectured that it is 0.434

== Hypercomplex numbers ==

Hypercomplex number is a term for an element of a unital algebra over the field of real numbers. The complex numbers are often symbolised by a boldface C (or blackboard bold

        C
      
    
  

{\displaystyle \mathbb {\mathbb {C} } }

, Unicode U+2102 ℂ DOUBLE-STRUCK CAPITAL C), while the set of quaternions is denoted by a boldface H (or blackboard bold

      H
    
  

{\displaystyle \mathbb {H} }

, Unicode U+210D ℍ DOUBLE-STRUCK CAPITAL H).

=== Algebraic complex numbers === Imaginary unit:

    i
    =
    
      
        −
        1
      
    
  

{\textstyle i={\sqrt {-1}}}

nth roots of unity:

      ξ
      
        n
      
      
        k
      
    
    =
    cos
    ⁡
    
      
        (
      
    
    2
    π
    
      
        k
        n
      
    
    
      
        )
      
    
    +
    i
    sin
    ⁡
    
      
        (
      
    
    2
    π
    
      
        k
        n
      
    
    
      
        )
      
    
  

{\textstyle \xi _{n}^{k}=\cos {\bigl (}2\pi {\frac {k}{n}}{\bigr )}+i\sin {\bigl (}2\pi {\frac {k}{n}}{\bigr )}}

, while

    0
    ≤
    k
    ≤
    n
    −
    10
  

{\textstyle 0\leq k\leq n-10}

, GCD(k, n) = 1

=== Other hypercomplex numbers === The quaternions The octonions The sedenions The trigintaduonions The dual numbers (with an infinitesimal)

== Transfinite numbers ==

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.

Aleph-null: ℵ0, the smallest infinite cardinal, and the cardinality of

      N
    
  

{\displaystyle \mathbb {N} }

, the set of natural numbers Aleph-one: ℵ1, the cardinality of ω1, the set of all countable ordinal numbers Beth-one:

      ℶ
      
        1
      
    
  

{\displaystyle \beth _{1}}

or

        c
      
    
  

{\displaystyle {\mathfrak {c}}}

, the cardinality of the continuum 2ℵ0 Omega: ω, the smallest infinite ordinal

== Numbers representing physical quantities ==

Physical quantities that appear in the universe are often described using physical constants.

Avogadro constant: NA = 6.02214076×1023 mol−1‍ Electron mass: me = 9.1093837139(28)×10−31 kg‍ Fine-structure constant: α = 0.0072973525643(11)‍ Gravitational constant: G = 6.67430(15)×10−11 m3⋅kg−1⋅s−2‍ Molar mass constant: Mu = 1.00000000105(31)×10−3 kg⋅mol−1‍ Planck constant: h = 6.62607015×10−34 J⋅Hz−1‍ Rydberg constant: R∞ = 10973731.568157(12) m−1‍ Speed of light in vacuum: c = 299792458 m⋅s−1‍ Vacuum permittivity: ε0 = 8.8541878188(14)×10−12 F⋅m−1‍

== Numbers representing geographical and astronomical distances == 6378.137, the average equatorial radius of Earth in kilometers (following GRS 80 and WGS 84 standards). 40075.0167, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards). 384399, the semi-major axis of the orbit of the Moon, in kilometers, roughly the distance between the center of Earth and that of the Moon. 149597870700, the average distance between the Earth and the Sun or Astronomical Unit (AU), in meters. 9460730472580800, one light-year, the distance travelled by light in one Julian year, in meters. 30856775814913673, the distance of one parsec, another astronomical unit, in whole meters.

== Numbers without specific values ==

Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier". Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".

== Named numbers == Hardy–Ramanujan number, 1729 Kaprekar's constant, 6174 Eddington number, ~1080 Googol, 10100 Shannon number Centillion, 10303 Skewes's number Googolplex, 10(10100) Mega/Circle(2) Moser's number Graham's number TREE(3) SSCG(3) Rayo's number

== See also ==

== References ==

Finch, Steven R. (2003), "Anmol Kumar Singh", Mathematical Constants (Encyclopedia of Mathematics and its Applications, Series Number 94), Cambridge University Press, pp. 130–133, ISBN 0521818052 Apéry, Roger (1979), "Irrationalité de

    ζ
    (
    2
    )
  

{\displaystyle \zeta (2)}

et

    ζ
    (
    3
    )
  

{\displaystyle \zeta (3)}

", Astérisque, 61: 11–13.

== Further reading == Kingdom of Infinite Number: A Field Guide by Bryan Bunch, W.H. Freeman & Company, 2001. ISBN 0-7167-4447-3

== External links == What's Special About This Number? A Zoology of Numbers: from 0 to 500 Name of a Number See how to write big numbers About big numbers at the Wayback Machine (archived 27 November 2010) Robert P. Munafo's Large Numbers page Different notations for big numbers – by Susan Stepney Names for Large Numbers, in How Many? A Dictionary of Units of Measurement by Russ Rowlett What's Special About This Number? (from 0 to 9999)