# 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.

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.

The definition of what is classed as a number is not concrete and is 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).

## 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 270 280 290 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000 60000 70000 80000 90000 105 106 107 108 109 1010 10100 1010100 Larger numbers

### Natural numbers notable for their mathematical properties

Integers 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.

Subsets of the integers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members.

### 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.

### 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. There may be connections between the mathematical and practical properties of an integer.

#### Orders of magnitude and SI prefixes for powers of 10

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 SI prefix Name Binary prefix Approximate value as power of 2
1024m = 210m Value
1000 10001 k Kilo Ki 10241 1 024
1000000 10002 M Mega Mi 10242 1 048 576
1000000000 10003 G Giga Gi 10243 1 073 741 824
1000000000000 10004 T Tera Ti 10244 1 099 511 627 776
1000000000000000 10005 P Peta Pi 10245 1 125 899 906 842 624
1000000000000000000 10006 E Exa Ei 10246 1 152 921 504 606 846 976
1000000000000000000000 10007 Z Zetta Zi 10247 1 180 591 620 717 411 303 424
1000000000000000000000000 10008 Y Yotta Yi 10248 1 208 925 819 614 629 174 706 176

## 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".

Keep in mind that rational numbers like 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), twelve percent (12%), three twenty-fifths (3/25), nine seventy-fifths (9/75), six fiftieths (6/50), twelve hundredths (12/100), twenty-four two-hundredths (24/200), etc.

This section will focus on non-integral rational numbers.

A list of rational numbers is shown below. The names of fractions can be found at numeral (linguistics).

Table of notable rational numbers. Click to
Value Fraction
1 1/1
0.916666... 11/12
0.9 9/10
0.833333... 5/6
0.8 4/5
0.75 3/4
0.7 7/10
0.666666... 2/3
0.6 3/5
0.583333.. 7/12
0.5 1/2
0.416666... 5/12
0.4 2/5
0.333333... 1/3
0.3 3/10
0.25 1/4
0.2 1/5
0.166666... 1/6
0.142857142857... 1/7
0.125 1/8
0.111111... 1/9
0.1 1/10
0.090909... 1/11
0.083333... 1/12
0 0/1

### Algebraic numbers

Expression Approximate value Notes
3/4 0.433012701892219323381861585376 Area of an equilateral triangle with side length 1.
5 − 1/2 0.618033988749894848204586834366 Golden ratio conjugate Φ, reciprocal of and one less than the golden ratio.
3/2 0.866025403784438646763723170753 Height of an equilateral triangle with side length 1.
122 1.059463094359295264561825294946 Twelfth root of two.
Proportion between the frequencies of adjacent semitones in the equal temperament scale.
32/4 1.060660171779821286601266543157 The size of the cube that satisfies Prince Rupert's cube.
32 1.259921049894873164767210607278 Cube root of two.
Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
(cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots) 1.303577269034296391257099112153 Conway's constant, defined as the unique positive real root of a certain polynomial of degree 71.
${\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}$  1.324717957244746025960908854478 Plastic number, the unique real root of the cubic equation x3 = x + 1.
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).
${\displaystyle {\frac {1}{3}}+{\frac {2}{3{\sqrt[{3}]{116+12{\sqrt {93}}}}}}+{\frac {1}{6}}{\sqrt[{3}]{116+12{\sqrt {93}}}}}$  1.465571231876768026656731225220 The supergolden ratio, 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.
${\displaystyle {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}}$  1.538841768587626701285145288018 Altitude of a regular pentagon with side length 1.
17 − 1/2 1.561552812808830274910704927987 The Triangular root of 2.
5 + 1/2 1.618033988749894848204586834366 Golden ratio (φ), the larger of the two real roots of x2 = x + 1.
${\displaystyle {\frac {5}{4{\sqrt {5-2{\sqrt {5}}}}}}}$  1.720477400588966922759011977389 Area of a regular pentagon with side length 1.
3 1.732050807568877293527446341506 3 = 2 sin 60° = 2 cos 30°
Square root of three a.k.a. the measure of the fish.
Length of the space diagonal of a cube with edge length 1.
Length of the diagonal of a 1 × 2 rectangle.
Altitude of an equilateral triangle with side length 2.
Altitude of a regular hexagon with side length 1 and diagonal length 2.
${\displaystyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}}$  1.839286755214161132551852564653 The Tribonacci constant.
Appears in the volume and coordinates of the snub cube and some related polyhedra.
It satisfies the equation x + x−3 = 2.
5 2.236067977499789696409173668731 Square root of five.
Length of the diagonal of a 1 × 2 rectangle.
Length of the diagonal of a 2 × 3 rectangle.
Length of the space diagonal of a 1 × 2 × 2 rectangular box.
2 + 1 2.414213562373095048801688724210 Silver ratioS), the larger of the two real roots of x2 = 2x + 1.
Altitude of a regular octagon with side length 1.
6 2.449489742783178098197284074706 2 · 3 = area of a 2 × 3 rectangle.
Length of the space diagonal of a 1 × 1 × 2 rectangular box.
Length of the diagonal of a 1 × 5 rectangle.
Length of the diagonal of a 2 × 2 rectangle.
Length of the diagonal of a square with side length 3.
33/2 2.598076113533159402911695122588 Area of a regular hexagon with side length 1.
7 2.645751311064590590501615753639 Length of the space diagonal of a 1 × 2 × 2 rectangular box.
Length of the diagonal of a 1 × 6 rectangle.
Length of the diagonal of a 2 × 3 rectangle.
Length of the diagonal of a 2 × 5 rectangle.
8 2.828427124746190097603377448419 22
Volume of a cube with edge length 2.
Length of the diagonal of a square with side length 2.
Length of the diagonal of a 1 × 7 rectangle.
Length of the diagonal of a 2 × 6 rectangle.
Length of the diagonal of a 3 × 5 rectangle.
10 3.162277660168379331998893544433 2 · 5 = area of a 2 × 5 rectangle.
Length of the diagonal of a 1 × 3 rectangle.
Length of the diagonal of a 2 × 6 rectangle.
Length of the diagonal of a 3 × 7 rectangle.
Length of the diagonal of a square with side length 5.
13 + 3/2 3.302775637731994646559610633735 Bronze ratio (S3), the larger of the two real roots of x2 = 3x + 1.
11 3.316624790355399849114932736671 Length of the space diagonal of a 1 × 1 × 3 rectangular box.
Length of the diagonal of a 1 × 10 rectangle.
Length of the diagonal of a 2 × 7 rectangle.
Length of the diagonal of a 3 × 2 rectangle.
Length of the diagonal of a 3 × 8 rectangle.
Length of the diagonal of a 5 × 6 rectangle.
12 3.464101615137754587054892683012 23
Length of the space diagonal of a cube with edge length 2.
Length of the diagonal of a 1 × 11 rectangle.
Length of the diagonal of a 2 × 8 rectangle.
Length of the diagonal of a 3 × 3 rectangle.
Length of the diagonal of a 2 × 10 rectangle.
Length of the diagonal of a 5 × 7 rectangle.
Length of the diagonal of a square with side length 6.

### Transcendental numbers

Name or formula Decimal expansion Alternate name or formula
Gelfond's constant: eπ 23.14069263277925...
Ramanujan's constant: eπ163 262537412640768743.99999999999925...
ii 4.810477381... eπ
Gaussian integral: π 1.772453850905516...
Komornik–Loreti constant: q 1.787231650...
Universal parabolic constant: P2 2.29558714939...
Gelfond–Schneider constant: 22 2.665144143...
e 2.718281828459045235360287471352662497757247...
π 3.141592653589793238462643383279502884197169399375...
π2/6 1.644934066848226436472415... ζ(2)
ei + e−i 1.08060461... 2 cos 1
π4/90 1.082323...[5] ζ(4)
2s: 1.559610469...[6]
log2 3: 1.584962501... (the logarithm of any positive integer to any integer base greater than 1 is either rational or transcendental)
(−1)i 0.0432139183... eπ
ii 0.207879576... eπ
Liouville constant: c 0.110001000000000000000001000...
Champernowne constant: C10 0.12345678910111213141516...
1/π 0.318309886183790671537767526745028724068919291480...[7]
1/e 0.367879441171442321595523770161460867445811131031...[7]
Prouhet–Thue–Morse constant: τ 0.412454033640...
log10 e 0.434294481903251827651128918916605082294397005803...[7]
Omega constant: Ω 0.5671432904097838729999686622...
Cahen's constant: c 0.64341054629...
ln 2: 0.693147180559945309417232121458...
π/18 0.7404... the maximum density of sphere packing in three dimensional Euclidean space according to the Kepler conjecture[8]
Gauss's constant: G 0.8346268...
π/12 0.9068..., the fraction of the plane covered by the densest possible circle packing[9]
Tau, or 2π: τ 6.283185307179586476925286766559..., The ratio of the circumference to a radius, and the number of radians in a complete circle[10][11]

#### Suspected transcendentals

These are irrational numbers that are thought to be,[according to whom?] but have not yet been proved to be, transcendental.

## Hypercomplex numbers

Hypercomplex number is a traditional 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".[31] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".[32]

## Notes

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 1 April 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. ^ a b "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
9. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 30.
10. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
11. ^ Sequence .
12. ^
13. ^
14. ^
15. ^
16. ^
17. ^ [1]
18. ^
19. ^ Weisstein, Eric W. "Continued Fraction Constant". Wolfram Research, Inc. Archived from the original on 2011-10-24.
20. ^
21. ^
22. ^ [2]
23. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
24. ^
25. ^
26. ^
27. ^
28. ^ [3]
29. ^
30. ^
31. ^
32. ^ Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"