List of Numbers
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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 least "interesting" 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.

 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 270 280 290 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000 60000 70000 80000 90000 105 106 107 108 109 Larger numbers, including 10100 and 1010100

### 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 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:

 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541

### 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:

1.   6
2.   28
3.   496
4.   8 128
5.   33 550 336
6.   8 589 869 056
7.   137 438 691 328
8.   2 305 843 008 139 952 128
9.   2 658 455 991 569 831 744 654 692 615 953 842 176
10.   191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216

## 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 (numbers). A power of ten is a number 10k, 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.

Value 1000m Name
1000 10001 Kilo
1000000 10002 Mega
1000000000 10003 Giga
1000000000000 10004 Tera
1000000000000000 10005 Peta
1000000000000000000 10006 Exa
1000000000000000000000 10007 Zetta
1000000000000000000000000 10008 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.[3] 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);[4] 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 , nine seventy-fifths , 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 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 counter-intuitively ascribed to the series 1+2+3....
0.5 One half occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle: × 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 (${\displaystyle \Phi }$) Reciprocal of (and one less than) the golden ratio.
Twelfth root of two Proportion between the frequencies of adjacent semitones in the equal temperament scale.
Cube root of two 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) Defined as the unique positive real root of a certain polynomial of degree 71.
Plastic number ${\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}$ The unique real root of the cubic equation x3 = x + 1.
Square root of two = 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 ${\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {93}}}{2}}}}{3}}}$ 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 .
Triangular root of 2.
Golden ratio (?) The larger of the two real roots of x2 = x + 1.
Square root of three = 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. ${\displaystyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}}$ 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. Length of the diagonal of a 1 × 2 rectangle.
Silver ratio (?S) + 1 The larger of the two real roots of x2 = 2x + 1.
Altitude of a regular octagon with side length 1.
Square root of 6 · = area of a × rectangle. Length of the space diagonal of a 1 × 1 × 2 rectangular box.
Square root of 7
Square root of 8 2
Square root of 10 · . Length of the diagonal of a 1 × 3 rectangle.
Bronze ratio (S3) The larger of the two real roots of x2 = 3x + 1.
Square root of 11 Length of the space diagonal of a 1 × 1 × 3 rectangular box.
Square root of 12 2 . Length of the space diagonal of a cube with edge length 2.

### Transcendental numbers

Name Symbol

or

Formula

Decimal expansion Notes and notability
Gelfond's constant e? ...
Ramanujan's constant e? ...
Gaussian integral ...
Komornik-Loreti constant q ...
Universal parabolic constant P2 ...
Gelfond-Schneider constant 2 ...
Euler's number e ...
Pi ? ...
Riemann zeta function at s=2 ... Also represented as ?(2)
Riemann zeta function at s=4 ...[5] Also represented as ?(4)
Super square-root of 2 s ...[6]
Liouville constant c ...
Champernowne constant C10 ...
Reciprocal of pi ...[7]
Reciprocal of Euler's number ...[7]
Prouhet-Thue-Morse constant ? ...
Base ten logarithm of Euler's number log10e ...[7]
Omega constant ? ...
Cahen's constant c ...
Natural logarithm of 2 ln 2
Gauss's constant G ...
Tau 2?: ? ... The ratio of the circumference to a radius, and the number of radians in a complete circle[8][9]

### 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
?(3), also known as Apéry's constant [10] [11]
Erd?s-Borwein constant, E ... [12][13] []
Copeland-Erd?s constant ... 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. []
Prime constant, ? ... Proof of the number's irrationality is given at prime constant. []
Reciprocal Fibonacci constant, ? ... [14][15] [16]

## 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
1st Feigenbaum constant, ? 4.6692... Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[17]
2nd Feigenbaum constant, ? 2.5029... Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[17]
Barban's constant ...[18]
Backhouse's constant ...
Fransén-Robinson constant, F ...
Glaisher-Kinkelin constant, A ...
Khinchin's constant, K0 ...[19] It is not known whether this number is irrational.[20]
Lévy's constant, ? ...
Mills' constant, A ... It is not known whether this number is irrational.(Finch 2003)
Murata's constant ...[21]
Ramanujan-Soldner constant, ? ...
Sierpi?ski's constant, K ...
Totient summatory constant ...[22]
Van der Pauw's constant, ...[23]
Vardi's constant, E ...
Favard constant, K1 ...
Somos' quadratic recurrence constant, ? ...
Niven's constant, c ...
Brun's constant, B2 ... The irrationality of this number would be a consequence of the truth of the infinitude of twin primes.
Landau's totient constant ...[24]
Brun's constant for prime quadruplets, B4 ...
Catalan's constant, G ... It is not known whether this number is irrational.[26]
Viswanath's constant, ?(1) ...
Khinchin-Lévy constant ...[27] This number represents the probability that three random numbers have no common factor greater than 1.[28]
Sarnak's constant ...[29]
Landau-Ramanujan constant ...
C(1) ...
Z(1) ...
Heath-Brown-Moroz constant, C ...
Kepler-Bouwkamp constant ...
MRB constant ... It is not known whether this number is irrational.
Meissel-Mertens constant, M ...
Bernstein's constant, ? ...
Strongly carefree constant ...[30]
Gauss-Kuzmin-Wirsing constant, ?1 ...[31]
Hafner-Sarnak-McCurley constant ...
Artin's constant ...
Carefree constant ...[32]
S(1) ...
F(1) ...
Stephens' constant ...[33]
Euler-Mascheroni constant, ? ... It is not known whether this number is irrational.
Golomb-Dickman constant, ? ...
Twin prime constant, C2 ...
Feller-Tornier constant ...[34]
Laplace limit, ? ...[35]
Taniguchi's constant ...[36]
Continued Fraction Constant, C ...[37]
Embree-Trefethen constant ...

### Numbers not known with high precision

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

## Hypercomplex numbers

Hypercomplex number is a term for an element of a unital algebra over the field of real numbers.

## 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.

## Numbers representing physical quantities

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

## 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".[38] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".[39]

## References

1. ^ Weisstein, Eric W. "Hardy-Ramanujan Number". Archived from the original on 2004-04-08.
2. ^ "Eighty-six - Definition of eighty-six by Merriam-Webster". merriam-webster.com. Archived from the original on 2013-04-08.
3. ^ Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158-160. ISBN 978-0-07-288008-3.
4. ^ Rouse, Margaret. "Mathematical Symbols". Retrieved 2015.
5. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33.
6. ^ "Nick's Mathematical Puzzles: Solution 29". Archived from the original on 2011-10-18.
7. ^ a b c "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 27.
8. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
9. ^ Sequence .
10. ^ See Apéry 1979.
11. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
12. ^ Erd?s, P. (1948), "On arithmetical properties of Lambert series" (PDF), J. Indian Math. Soc. (N.S.), 12: 63-66, MR 0029405
13. ^ Borwein, Peter B. (1992), "On the irrationality of certain series", Mathematical Proceedings of the Cambridge Philosophical Society, 112 (1): 141-146, doi:10.1017/S030500410007081X, MR 1162938
14. ^ André-Jeannin, Richard; 'Irrationalité de la somme des inverses de certaines suites récurrentes.'; Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 308, issue 19 (1989), pp. 539-541.
15. ^ S. Kato, 'Irrationality of reciprocal sums of Fibonacci numbers', Master's thesis, Keio Univ. 1996
16. ^ Duverney, Daniel, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa; 'Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers';
17. ^ a b Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
18. ^
19. ^ [1]
20. ^
21. ^
22. ^
23. ^
24. ^
25. ^
26. ^ Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153-170, doi:10.1134/s0081543816010107
27. ^ [2]
28. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
29. ^
30. ^
31. ^
32. ^
33. ^
34. ^
35. ^ [3]
36. ^
37. ^ Weisstein, Eric W. "Continued Fraction Constant". Wolfram Research, Inc. Archived from the original on 2011-10-24.
38. ^
39. ^ Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"
• Finch, Steven R. (2003), "Mills' Constant", Mathematical Constants, Cambridge University Press, pp. 130-133, ISBN 0-521-81805-2.
• Apéry, Roger (1979), "Irrationalité de ${\displaystyle \zeta (2)}$ et ${\displaystyle \zeta (3)}$", Astérisque, 61: 11-13.