List of numbers

This is a list of articles about numbers. Due to the infinitude of many sets of numbers, this list will invariably be incomplete. Hence, only particularly notable numbers will be included. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities which could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as the interesting number paradox.

The definition of what is classed as a number is rather diffuse and based on historical distinctions. For example, the pair of numbers (3,4) is commonly regarded as a number when it is in the form of a complex number (3+4i), but not when it is in the form of a vector (3,4). This list will also be categorised with the standard convention of types of numbers.

This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five (the noun referring to the number).

Natural numbers[]

The natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used for counting and often have ethno-cultural significance (see below). Beyond this, natural numbers are widely used as a building block for other number systems including the integers, rational numbers and real numbers. Natural numbers are those used for counting (as in "there are six (6) coins on the table") and ordering (as in "this is the third (3rd) largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Defined by the Peano axioms, the natural numbers form an infinitely large set.

The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions. In set theory and computer science, 0 is typically considered a natural number. In number theory, it usually is not. The ambiguity can be solved with the terms "non-negative integers", which includes 0, and "positive integers", which does not.

Natural numbers may be used as cardinal numbers, which may go by various names. Natural numbers may also be used as ordinal numbers.

Mathematical significance[]

Natural numbers may have properties specific to the individual number or may be part of a set (such as prime numbers) of numbers with a particular property.

Cultural or practical significance[]

Along with their mathematical properties, many integers have cultural significance^[2] or are also notable for their use in computing and measurement. As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.

Classes of natural numbers[]

Subsets of the natural numbers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members. Infinitely many such sets are possible. A list of notable classes of natural numbers may be found at classes of natural numbers.

Prime numbers[]

A prime number is a positive integer which has exactly two divisors: 1 and itself.

The first 100 prime numbers are:

Highly composite numbers[]

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement.

The first 20 highly composite numbers are:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560

Perfect numbers[]

A perfect number is an integer that is the sum of its positive proper divisors (all divisors except itself).

The first 10 perfect numbers:

Integers[]

The integers are a set of numbers commonly encountered in arithmetic and number theory. There are many subsets of the integers, including the natural numbers, prime numbers, perfect numbers, etc. Many integers are notable for their mathematical properties.

Notable integers include −1, the additive inverse of unity, and 0, the additive identity.

As with the natural numbers, the integers may also have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.

SI prefixes[]

One important use of integers is in orders of magnitude. A power of 10 is a number 10^k, where k is an integer. For instance, with k = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, k = -3 gives 1/1000, or 0.001. This is used in scientific notation, real numbers are written in the form m × 10ⁿ. The number 394,000 is written in this form as 3.94 × 10⁵.

Integers are used as prefixes in the SI system. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.

Rational numbers[]

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.^[4] 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 ${\displaystyle \mathbb {Q} }$, Unicode ℚ);^[5] 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).

Table of notable rational numbers hide

Decimal expansion	Fraction	Notability
1	1/1	One is the multiplicative identity. One is trivially a rational number, as it is equal to 1/1.
-0.083 333...	−+1/12	The value assigned to the series 1+2+3... by zeta function regularization and Ramanujan summation.
0.5	1/2	One half occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle: 1/2 × base × perpendicular height and in the formulae for figurate numbers, such as triangular numbers and pentagonal numbers.
3.142 857...	22/7	A widely used approximation for the number ${\displaystyle \pi }$. It can be proven that this number exceeds ${\displaystyle \pi }$.
0.166 666...	1/6	One sixth. Often appears in mathematical equations, such as in the sum of squares of the integers and in the solution to the Basel problem.

Irrational numbers[]

The irrational numbers are a set of numbers that includes all real numbers that are not rational numbers. The irrational numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not.

Algebraic numbers[]

Name	Expression	Decimal expansion	Notability
Golden ratio conjugate (${\displaystyle \Phi }$)	√5 − 1/2	0.618033988749894848204586834366	Reciprocal of (and one less than) the golden ratio.
Twelfth root of two	12√2	1.059463094359295264561825294946	Proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale.
Cube root of two	3√2	1.259921049894873164767210607278	Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
Conway's constant	(cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots)	1.303577269034296391257099112153	Defined as the unique positive real root of a certain polynomial of degree 71.
Plastic number	${\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}$	1.324717957244746025960908854478	The unique real root of the cubic equation x3 = x + 1.
Square root of two	√2	1.414213562373095048801688724210	√2 = 2 sin 45° = 2 cos 45° Square root of two a.k.a. Pythagoras' constant. Ratio of diagonal to side length in a square. Proportion between the sides of paper sizes in the ISO 216 series (originally DIN 476 series).
Supergolden ratio	${\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {93}}}{2}}}}{3}}}$	1.465571231876768026656731225220	The only real solution of ${\displaystyle x^{3}=x^{2}+1}$. Also the limit to the ratio between subsequent numbers in the binary Look-and-say sequence and the Narayana's cows sequence (OEIS: A000930).
Triangular root of 2	√17 − 1/2	1.561552812808830274910704927987
Golden ratio (φ)	√5 + 1/2	1.618033988749894848204586834366	The larger of the two real roots of x2 = x + 1.
Square root of three	√3	1.732050807568877293527446341506	√3 = 2 sin 60° = 2 cos 30° . A.k.a. the measure of the fish. Length of the space diagonal of a cube with edge length 1. Altitude of an equilateral triangle with side length 2. Altitude of a regular hexagon with side length 1 and diagonal length 2.
Tribonacci constant	${\displaystyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}}$	1.839286755214161132551852564653	Appears in the volume and coordinates of the snub cube and some related polyhedra. It satisfies the equation x + x−3 = 2.
Square root of five	√5	2.236067977499789696409173668731	Length of the diagonal of a 1 × 2 rectangle.
Silver ratio (δS)	√2 + 1	2.414213562373095048801688724210	The larger of the two real roots of x2 = 2x + 1. Altitude of a regular octagon with side length 1.
Bronze ratio (S3)	√13 + 3/2	3.302775637731994646559610633735	The larger of the two real roots of x2 = 3x + 1.

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 numbers[]

The real numbers are a superset containing the algebraic and the transcendental numbers. 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.

Real but not known to be irrational, nor transcendental[]

Name and symbol	Decimal expansion	Notes
Euler–Mascheroni constant, γ	0.577215664901532860606512090082...[16]	Believed to be transcendental but not proven to be so. However, it was shown that at least one of ${\displaystyle \gamma }$ and the Euler-Gompertz constant ${\displaystyle \delta }$ is transcendental.[17][18] It was also shown that all but at most one number in an infinite list containing ${\displaystyle {\frac {\gamma }{4}}}$ have to be transcendental.[19][20]
Euler–Gompertz constant, δ	0.596 347 362 323 194 074 341 078 499 369...[21]	It was shown that at least one of the Euler-Mascheroni constant ${\displaystyle \gamma }$ and the Euler-Gompertz constant ${\displaystyle \delta }$ is transcendental.[17][18]
Catalan's constant, G	0.915965594177219015054603514932384110774...	It is not known whether this number is irrational.[22]
Khinchin's constant, K0	2.685452001...[23]	It is not known whether this number is irrational.[24]
1st Feigenbaum constant, δ	4.6692...	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[25]
2nd Feigenbaum constant, α	2.5029...	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[25]
Glaisher–Kinkelin constant, A	1.28242712...
Backhouse's constant	1.456074948...
Fransén–Robinson constant, F	2.8077702420...
Lévy's constant, γ	3.275822918721811159787681882...
Mills' constant, A	1.30637788386308069046...	It is not known whether this number is irrational.(Finch 2003)
Ramanujan–Soldner constant, μ	1.451369234883381050283968485892027449493...
Sierpiński's constant, K	2.5849817595792532170658936...
Totient summatory constant	1.339784...[26]
Vardi's constant, E	1.264084735305...
Somos' quadratic recurrence constant, σ	1.661687949633594121296...
Niven's constant, c	1.705211...
Brun's constant, B2	1.902160583104...	The irrationality of this number would be a consequence of the truth of the infinitude of twin primes.
Landau's totient constant	1.943596...[27]
Brun's constant for prime quadruplets, B4	0.8705883800...
Viswanath's constant, σ(1)	1.1319882487943...
Khinchin–Lévy constant	1.1865691104...[28]	This number represents the probability that three random numbers have no common factor greater than 1.[29]
Landau–Ramanujan constant	0.76422365358922066299069873125...
C(1)	0.77989340037682282947420641365...
Z(1)	−0.736305462867317734677899828925614672...
Heath-Brown–Moroz constant, C	0.001317641...
Kepler–Bouwkamp constant	0.1149420448...
MRB constant	0.187859...	It is not known whether this number is irrational.
Meissel–Mertens constant, M	0.2614972128476427837554268386086958590516...
Bernstein's constant, β	0.2801694990...
Gauss–Kuzmin–Wirsing constant, λ1	0.3036630029...[30]
Hafner–Sarnak–McCurley constant	0.3532363719...
Artin's constant	0.3739558136...
S(1)	0.438259147390354766076756696625152...
F(1)	0.538079506912768419136387420407556...
Stephens' constant	0.575959...[31]
Golomb–Dickman constant, λ	0.62432998854355087099293638310083724...
Twin prime constant, C2	0.660161815846869573927812110014...
Feller–Tornier constant	0.661317...[32]
Laplace limit, ε	0.6627434193...[33]
Embree–Trefethen constant	0.70258...

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 ≤ 0.22
• 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.

Algebraic complex numbers[]

• Imaginary unit: i = √−1
• nth roots of unity: (ξ_n)^k = cos (2π k/n) + i sin (2π k/n), while 0 ≤ k ≤ n−1, GCD(k, n) = 1

Other hypercomplex numbers[]

• The quaternions
• The octonions
• The sedenions
• 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, the set of natural numbers
• Aleph-one: ℵ₁: the cardinality of ω₁, the set of all countable ordinal numbers
• Beth-one: ℶ₁ the cardinality of the continuum 2^ℵ₀
• ℭ or ${\displaystyle {\mathfrak {c}}}$: the cardinality of the continuum 2^ℵ₀
• 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: N_A = 6.02214076×10²³ mol^−1[34]
• Electron mass: m_e = 9.1093837015(28)×10⁻³¹ kg^[35]
• Fine-structure constant: α = 7.2973525693(11)×10^−3[36]
• Gravitational constant: G = 6.67430(15)×10⁻¹¹ m³⋅kg⁻¹⋅s^−2[37]
• Molar mass constant: M_u = 0.99999999965(30)×10⁻³ kg⋅mol^−1[38]
• Planck constant: h = 6.62607015×10⁻³⁴ J⋅Hz^−1[39]
• Rydberg constant: R_∞ = 10973731.568160(21) m^−1[40]
• Speed of light in vacuum: c = 299792458 m⋅s^−1[41]
• Vacuum electric permittivity: ε₀ = 8.8541878128(13)×10⁻¹² F⋅m^−1[42]

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".^[43] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".^[44]

Named numbers[]

• Eddington number
• Googol, 10¹⁰⁰
• Googolplex, 10^(10100)
• Graham's number
• Hardy–Ramanujan number, 1729
• Kaprekar's constant, 6174
• Moser's number
• Rayo's number
• Shannon number
• Skewes's number

See also[]

References[]

• Finch, Steven R. (2003), "Mills' Constant", Mathematical Constants, Cambridge University Press, pp. 130–133, ISBN 0-521-81805-2^{[permanent dead link]}.
• Apéry, Roger (1979), "Irrationalité de ${\displaystyle \zeta (2)}$ et ${\displaystyle \zeta (3)}$", 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[]

• The Database of Number Correlations: 1 to 2000+
• 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? Archived 2018-02-23 at the Wayback Machine (from 0 to 9999)