# 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.

- 1, the multiplicative identity. Also the only natural number (not including 0) that isn't prime or composite.
- 2, the base of the binary number system, used in almost all modern computers and information systems.
- 3, 2
^{2}-1, the first Mersenne prime. It is the first odd prime, and it is also the 2 bit integer maximum value. - 4, the first composite number
- 6, the first of the series of perfect numbers, whose proper factors sum to the number itself.
- 9, the first odd number that is composite
- 11, the fifth prime and first palindromic multi-digit number in base 10.
- 12, the first sublime number.
- 17, the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.
- 24, all Dirichlet characters mod
*n*are real if and only if*n*is a divisor of 24. - 25, the first centered square number besides 1 that is also a square number.
- 27, the cube of 3, the value of 3
^{3}. - 28, the second perfect number.
- 30, the smallest sphenic number.
- 32, the smallest nontrivial fifth power.
- 36, the smallest number which is perfect power but not prime power.
- 72, the smallest Achilles number.
- 255, 2
^{8}− 1, the smallest perfect totient number that is neither a power of three nor thrice a prime; it is also the largest number that can be represented using an 8-bit unsigned integer - 341, the smallest base 2 Fermat pseudoprime.
- 496, the third perfect number.
- 1729, the Hardy–Ramanujan number, also known as the second taxicab number; that is, the smallest positive integer that can be written as the sum of two positive cubes in two different ways.[1]
- 8128, the fourth perfect number.
- 142857, the smallest base 10 cyclic number.
- 9814072356, the largest perfect power that contains no repeated digits in base ten.

#### 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.

- 3, significant in Christianity as the Trinity. Also considered significant in Hinduism (Trimurti, Tridevi). Holds significance in a number of ancient mythologies.
- 4, considered an "unlucky number" in modern China, Japan and Korea due to its audible similarity to the word "death."
- 7, considered a "lucky" number in Western cultures.
- 8, considered a "lucky" number in Chinese culture due to its aural similarity to the term for prosperity.
- 12, a common grouping known as a dozen and the number of months in a year.
- 13, considered an "unlucky" number in Western superstition. Also known as a "Baker's Dozen".
- 18, considered a "lucky" number due to it being the value for life in Jewish numerology.
- 42, the "answer to the ultimate question of life, the universe, and everything" in the popular 1979 science fiction work
*The Hitchhiker's Guide to the Galaxy*. - 69, used as slang to refer to a sexual act.
- 86, a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.[3]
- 108, considered sacred by the Dharmic Religions. Approximately equal to the ratio of the distance from Earth to Sun and diameter of the Sun.
- 420, a code-term that refers to the consumption of cannabis.
- 666, the Number of the Beast from the Book of Revelation.
- 786, regarded as sacred in the Muslim Abjad numerology.
- 5040, mentioned by Plato in the
*Laws*as one of the most important numbers for the city.

- 10, the number of digits in the decimal number system.
- 12, the number base for measuring time in many civilizations.
- 14, the number of days in a fortnight.
- 16, the number of digits in the hexadecimal number system.
- 24, number of hours in a day
- 31, the number of days most months of the year have.
- 60, the number base for some ancient counting systems, such as the Babylonians', and the basis for many modern measuring systems.
- 365, the number of days in the common year.

- 8, the number of bits in a byte
- 256, The number of possible combinations within 8 bits, or a byte.
- 1024, the number of bytes in a kibibyte. It's also the number of bits in a kibibit.
- 65535, 2
^{16}− 1, the maximum value of a 16-bit unsigned integer. - 65536, 2
^{16}, the number of possible 16-bit combinations. - 65537, 2
^{16}+ 1, the most popular RSA public key prime exponent in most SSL/TLS certificates on the Web/Internet. - 16777216, 2
^{24}, or 16^{6}; the hexadecimal "million" (0x1000000), and the total number of possible color combinations in 24/32-bit True Color computer graphics. - 2147483647, 2
^{31}− 1, the maximum value of a 32-bit signed integer using two's complement representation. - 9223372036854775807, 2
^{63}− 1, the maximum value of a 64-bit signed integer using two's complement representation.

## 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* × 10^{n}. The number 394,000 is written in this form as 3.94 × 10^{5}.

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.

Value | 1000^{m} |
Name |
---|---|---|

1000 | 1000^{1} | Kilo |

1000000 | 1000^{2} | Mega |

1000000000 | 1000^{3} | Giga |

1000000000000 | 1000^{4} | Tera |

1000000000000000 | 1000^{5} | Peta |

1000000000000000000 | 1000^{6} | Exa |

1000000000000000000000 | 1000^{7} | Zetta |

1000000000000000000000000 | 1000^{8} | Yotta |

## 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 , 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).

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 . It can be proven that this number exceeds . |

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 () | √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 | 1.324717957244746025960908854478 | The unique real root of the cubic equation x^{3} = 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 | 1.465571231876768026656731225220 | The only real solution of . 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 x^{2} = 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 | 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 (δ_{S}) |
√2 + 1 | 2.414213562373095048801688724210 | The larger of the two real roots of x^{2} = 2x + 1.Altitude of a regular octagon with side length 1. |

Bronze ratio (S_{3}) |
√13 + 3/2 | 3.302775637731994646559610633735 | The larger of the two real roots of x^{2} = 3x + 1. |

#### Transcendental numbers

Name | Symbol
or Formula |
Decimal expansion | Notes and notability |
---|---|---|---|

Gelfond's constant | e^{π} |
23.14069263277925... | |

Ramanujan's constant | e^{π√163} |
262537412640768743.99999999999925... | |

Gaussian integral | √π | 1.772453850905516... | |

Komornik–Loreti constant | q | 1.787231650... | |

Universal parabolic constant | P_{2} |
2.29558714939... | |

Gelfond–Schneider constant | 2^{√2} |
2.665144143... | |

Euler's number | e | 2.718281828459045235360287471352662497757247... | |

Pi | π | 3.141592653589793238462643383279502884197169399375... | |

Super square-root of 2 | √2_{s} |
1.559610469...[6] | |

Liouville constant | c | 0.110001000000000000000001000... | |

Champernowne constant | C_{10} |
0.12345678910111213141516... | |

Prouhet–Thue–Morse constant | τ | 0.412454033640... | |

Omega constant | Ω | 0.5671432904097838729999686622... | |

Cahen's constant | c | 0.64341054629... | |

Natural logarithm of 2 | ln 2 | 0.693147180559945309417232121458 | |

Gauss's constant | G | 0.8346268... | |

Tau | 2π: τ | 6.283185307179586476925286766559... | The ratio of the circumference to a radius, and the number of radians in a complete circle[7][8] |

#### 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.

Name | Decimal expansion | Proof of irrationality | Reference of unknown transcendentality |
---|---|---|---|

ζ(3), also known as Apéry's constant | 1.202056903159594285399738161511449990764986292 | [9] | [10] |

Erdős–Borwein constant, E | 1.606695152415291763... | [11][12] | ^{[citation needed]} |

Copeland–Erdős constant | 0.235711131719232931374143... | Can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most six primes. It also follows directly from its normality. | ^{[citation needed]} |

Prime constant, ρ | 0.414682509851111660248109622... | Proof of the number's irrationality is given at prime constant. | ^{[citation needed]} |

Reciprocal Fibonacci constant, ψ | 3.359885666243177553172011302918927179688905133731... | [13][14] | [15] |

## 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 and the Euler-Gompertz constant is transcendental.[17][18] It was also shown that all but at most one number in an infinite list containing 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 and the Euler-Gompertz constant is transcendental.[17][18] |

Catalan's constant, G | 0.915965594177219015054603514932384110774... | It is not known whether this number is irrational.[22] |

Khinchin's constant, K_{0} |
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, B_{2} |
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, B_{4} |
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, λ_{1} |
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, C_{2} |
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
*n*th 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: ℵ
_{0}: the smallest infinite cardinal, and the cardinality of ℕ, the set of natural numbers - Aleph-one: ℵ
_{1}: the cardinality of ω_{1}, the set of all countable ordinal numbers - Beth-one: ℶ
_{1}the cardinality of the continuum 2^{ℵ0} - ℭ or : 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}= 6.02214076×10^{23}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}m^{3}⋅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:
*ε*_{0}= 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, 10
^{100} - 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

- English-language numerals
- Floating point
- Fraction (mathematics)
- Integer sequence
- Interesting number paradox
- Large numbers
- List of mathematical constants
- List of numbers in various languages
- List of prime numbers
- List of types of numbers
- Mathematical constant
- Names of large numbers
- Names of small numbers
- Negative number
- Number prefix
- Numeral (linguistics)
- Orders of magnitude (numbers)
- Ordinal number
*The Penguin Dictionary of Curious and Interesting Numbers*- Power of two
- Powers of 10
- SI prefix
- Surreal number
- Table of prime factors

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## Further reading

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