List of numbers

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This is a list of articles about numbers (not about numerals).

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.[1] Since q may be equal to 1, every integer is 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);[2] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".

Natural 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". There are infinitely many natural 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

(Note that the status of 0 is ambiguous. In set theory and computer science, 0 is considered a natural number. In number theory, it usually is not.)

Powers of ten (scientific notation)

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.

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

Notable integers

Integers that are notable for their mathematical properties or cultural meanings include:

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

Cardinal numbers

In the following tables, [and] indicates that the word and is used in some dialects (such as British English), and omitted in other dialects (such as American English).

Small numbers

This table demonstrates the standard English construction of small cardinal numbers up to one hundred million—names for which all variants of English agree.

Value Name Alternate names, and names for sets of the given size
0 Zero aught, cipher, cypher, donut, dot, duck, goose egg, love, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip, zippo
1 One ace, individual, single, singleton, unary, unit, unity
2 Two binary, brace, couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twin, twosome, yoke
3 Three deuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika, hat-trick
4 Four foursome, quadruplet, quatern, quaternary, quaternity, quartet, tetrad
5 Five cinque, fin, fivesome, pentad, quint, quintet, quintuplet
6 Six half dozen, hexad, sestet, sextet, sextuplet, sise
7 Seven heptad, septet, septuple, walking stick
8 Eight octad, octave, octet, octonary, octuplet, ogdoad
10 Ten deca, decade
11 Eleven onze, ounze, ounce, banker's dozen
12 Twelve dozen
13 Thirteen baker's dozen, long dozen[5]
14 Fourteen
15 Fifteen
16 Sixteen
17 Seventeen
18 Eighteen
19 Nineteen
20 Twenty score
21 Twenty-one long score[5]
22 Twenty-two Deuce-deuce
23 Twenty-three
24 Twenty-four two dozen
25 Twenty-five
26 Twenty-six
27 Twenty-seven
28 Twenty-eight
29 Twenty-nine
30 Thirty
31 Thirty-one
32 Thirty-two
40 Forty two-score
50 Fifty half-century
60 Sixty three-score
70 Seventy three-score and ten
80 Eighty four-score
87 Eighty-seven four-score and seven
90 Ninety four-score and ten
100 One hundred centred, century, ton, short hundred
101 One hundred [and] one
110 One hundred [and] ten
111 One hundred [and] eleven eleventy-one[6]
120 One hundred [and] twenty long hundred,[5] great hundred, (obsolete) hundred
121 One hundred [and] twenty-one
144 One hundred [and] forty-four gross, dozen dozen, small gross
200 Two hundred
300 Three hundred
400 Four hundred
500 Five hundred
600 Six hundred
666 Six hundred [and] sixty-six
700 Seven hundred
777 Seven hundred [and] seventy-seven
800 Eight hundred
900 Nine hundred
1000 One thousand chiliad, grand, G, thou, yard, kilo, k, millennium
1001 One thousand [and] one
1010 One thousand [and] ten
1011 One thousand [and] eleven
1024 One thousand [and] twenty-four kibi or kilo in computing, see binary prefix (kilo is shortened to K, Kibi to Ki)
1100 One thousand one hundred Eleven hundred
1101 One thousand one hundred [and] one
1728 One thousand seven hundred [and] twenty-eight great gross, long gross, dozen gross
2000 Two thousand
3000 Three thousand
10000 Ten thousand myriad, wan (China)
100000 One hundred thousand lakh
500000 Five hundred thousand crore (Iranian)
1000000 One million Mega, meg, mil, (often shortened to M)
1048576 One million forty-eight thousand five hundred [and] seventy-six Mibi or Mega in computing, see binary prefix (Mega is shortened to M, Mibi to Mi)
10000000 Ten million crore (Indian)(Pakistan)
100000000 One hundred million yi (China)

English names for powers of 10

This table compares the English names of cardinal numbers according to various American, British, and Continental European conventions. See English numerals or names of large numbers for more information on naming numbers.

Short scale Long scale Power
Value American British
(Nicolas Chuquet)
Continental European
(Jacques Peletier du Mans)
of a thousand of a million
100 One 1000−1+1 10000000
101 Ten
102 Hundred
103 Thousand 10000+1 10000000.5
106 Million 10001+1 10000001
109 Billion Thousand million Milliard 10002+1 10000001.5
1012 Trillion Billion 10003+1 10000002
1015 Quadrillion Thousand billion Billiard 10004+1 10000002.5
1018 Quintillion Trillion 10005+1 10000003
1021 Sextillion Thousand trillion Trilliard 10006+1 10000003.5
1024 Septillion Quadrillion 10007+1 10000004
1027 Octillion Thousand quadrillion Quadrilliard 10008+1 10000004.5
1030 Nonillion Quintillion 10009+1 10000005
1033 Decillion Thousand quintillion Quintilliard 100010+1 10000005.5
1036 Undecillion Sextillion 100011+1 10000006
1039 Duodecillion Thousand sextillion Sextilliard 100012+1 10000006.5
1042 Tredecillion Septillion 100013+1 10000007
1045 Quattuordecillion Thousand septillion Septilliard 100014+1 10000007.5
1048 Quindecillion Octillion 100015+1 10000008
1051 Sexdecillion Thousand octillion Octilliard 100016+1 10000008.5
1054 Septendecillion Nonillion 100017+1 10000009
1057 Octodecillion Thousand nonillion Nonilliard 100018+1 10000009.5
1060 Novemdecillion Decillion 100019+1 100000010
1063 Vigintillion Thousand decillion Decilliard 100020+1 100000010.5
1066 Unvigintillion Undecillion 100021+1 100000011
1069 Duovigintillion Thousand undecillion Undecilliard 100022+1 100000011.5
1072 Trevigintillion Duodecillion 100023+1 100000012
1075 Quattuorvigintillion Thousand duodecillion Duodecilliard 100024+1 100000012.5
1078 Quinvigintillion Tredecillion 100025+1 100000013
1081 Sexvigintillion Thousand tredecillion Tredecilliard 100026+1 100000013.5
1084 Septenvigintillion Quattuordecillion 100027+1 100000014
1087 Octovigintillion Thousand quattuordecillion Quattuordecilliard 100028+1 100000014.5
1090 Novemvigintillion Quindecillion 100029+1 100000015
1093 Trigintillion Thousand quindecillion Quindecilliard 100030+1 100000015.5
1096 Untrigintillion Sexdecillion 100031+1 100000016
1099 Duotrigintillion Thousand sexdecillion Sexdecilliard 100032+1 100000016.5
... ... ... ... ...
10120 Novemtrigintillion Vigintillion 100039+1 100000020
10123 Quadragintillion Thousand vigintillion Vigintilliard 100040+1 100000020.5
... ... ... ... ...
10153 Quinquagintillion Thousand quinvigintillion Quinvigintilliard 100050+1 100000025.5
... ... ... ... ...
10180 Novemquinquagintillion Trigintillion 100059+1 100000030
10183 Sexagintillion Thousand trigintillion Trigintilliard 100060+1 100000030.5
... ... ... ... ...
10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard 100070+1 100000035.5
... ... ... ... ...
10240 Novemseptuagintillion Quadragintillion 100079+1 100000040
10243 Octogintillion Thousand quadragintillion Quadragintilliard 100080+1 100000040.5
... ... ... ... ...
10273 Nonagintillion Thousand quinquadragintillion Quinquadragintilliard 100090+1 100000045.5
... ... ... ... ...
10300 Novemnonagintillion Quinquagintillion 100099+1 100000050
10303 Centillion Thousand quinquagintillion Quinquagintilliard 1000100+1 100000050.5
... ... ... ... ...
10360 Cennovemdecillion Sexagintillion 1000119+1 100000060
10420 Cennovemtrigintillion Septuagintillion 1000139+1 100000070
10480 Cennovemquinquagintillion Octogintillion 1000159+1 100000080
10540 Cennovemseptuagintillion Nonagintillion 1000179+1 100000090
10600 Cennovemnonagintillion Centillion 1000199+1 1000000100
10603 Ducentillion Thousand centillion Centilliard 1000200+1 1000000100.5

There is no consistent and widely accepted way to extend cardinals beyond centillion (centilliard).

SI prefixes for powers of 10

Value 1000m SI prefix Name Binary prefix 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

Fractional numbers

This is a table of English names for non-negative rational numbers less than or equal to 1. It also lists alternative names, but there is no widespread convention for the names of extremely small positive numbers.

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.

Value Fraction Common names Alternative names
1 1/1 One 0.999..., Unity
0.9 9/10 Nine tenths, [zero] point nine
0.8 4/5 Four fifths, eight tenths, [zero] point eight
0.7 7/10 Seven tenths, [zero] point seven
0.6 3/5 Three fifths, six tenths, [zero] point six
0.5 1/2 One half, five tenths, [zero] point five
0.4 2/5 Two fifths, four tenths, [zero] point four
0.333333... 1/3 One third
0.3 3/10 Three tenths, [zero] point three
0.25 1/4 One quarter, one fourth, twenty-five hundredths, [zero] point two five
0.2 1/5 One fifth, two tenths, [zero] point two
0.166666... 1/6 One sixth
0.142857142857... 1/7 One seventh
0.125 1/8 One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
0.111111... 1/9 One ninth
0.1 1/10 One tenth, [zero] point one One perdecime, one perdime
0.090909... 1/11 One eleventh
0.09 9/100 Nine hundredths, [zero] point zero nine
0.083333... 1/12 One twelfth
0.08 2/25 Two twenty-fifths, eight hundredths, [zero] point zero eight
0.0625 1/16 One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
0.05 1/20 One twentieth, [zero] point zero five
0.047619047619... 1/21 One twenty-first
0.045454545... 1/22 One twenty-second
0.043478260869565217391304347... 1/23 One twenty-third
0.041666... 1/24 One twenty-fourth
0.033333... 1/30 One thirtieth
0.016666... 1/60 One sixtieth
0.012345679012345679... 1/81 One eighty-first
0.01 1/100 One hundredth, [zero] point zero one One percent
0.001 1/1000 One thousandth, [zero] point zero zero one One permille
0.000277777... 1/3600 One thirty-six hundredth
0.0001 1/10000 One ten-thousandth, [zero] point zero zero zero one One myriadth, one permyria, one permyriad, one basis point
0.00001 1/100000 One hundred-thousandth One lakhth, one perlakh
0.000001 1/1000000 One millionth One perion, one ppm
0.0000001 1/10000000 One ten-millionth One crorth, one percrore
0.00000001 1/100000000 One hundred-millionth One awkth, one perawk
0.000000001 1/1000000000 One billionth (in some dialects) One ppb
0 0/1 Zero Nil

Irrational and suspected irrational numbers

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.
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 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.
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

Suspected transcendentals

These are irrational numbers that are thought to be, 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 measured quantities

Various terms have arisen to describe commonly used measured quantities.

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".[30] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".[31]

Notes

1. ^ 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.
2. ^ Rouse, Margaret. "Mathematical Symbols". Retrieved 1 April 2015.
3. ^ "Eighty-six – Definition of eighty-six by Merriam-Webster". merriam-webster.com. Archived from the original on 2013-04-08.
4. ^ Weisstein, Eric W. "Hardy–Ramanujan Number". Archived from the original on 2004-04-08.
5. ^ a b c Blunt, Joseph (1 January 1837). "The Shipmaster's Assistant, and Commercial Digest: Containing Information Useful to Merchants, Owners, and Masters of Ships". E. & G.W. Blunt – via Google Books.
6. ^ Ezard, John (2 Jan 2003). "Tolkien catches up with his hobbit". The Guardian. Retrieved 6 Apr 2018.
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 33.
11. ^ "Nick's Mathematical Puzzles: Solution 29". Archived from the original on 2011-10-18.
12. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
13. ^ Sequence .
14. ^
15. ^
16. ^
17. ^
18. ^
19. ^
20. ^ Weisstein, Eric W. "Continued Fraction Constant". Wolfram Research, Inc. Archived from the original on 2011-10-24.
21. ^
22. ^
23. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
24. ^
25. ^
26. ^
27. ^
28. ^
29. ^
30. ^
31. ^ Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"