# List of numbers

## Rational numbers

### 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 230 240 250 260 270 280 290 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000 60000 70000 80000 90000 100k–1M 1M–10M 10M–100M 100M–1G 1G–10G Larger numbers

### Integers

#### Notable integers

Other numbers 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: one and itself.

 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 (the seven values with more divisors than any lesser number than twice itself are in bold):

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1 260, 1 680, 2 520, 5 040, 7 560

### 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:

16
228
3496
48 128
533 550 336
68 589 869 056
7137 438 691 328
82 305 843 008 139 952 128
92 658 455 991 569 831 744 654 692 615 953 842 176
10   191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216

### 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.

ValueNameAlternate names, and names for sets of the given size
0Zeroaught, cipher, cypher, donut, goose egg, love, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip
1Oneace, individual, single, singleton, unary, unit, unity
2Twobinary, brace, couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twin, twosome, yoke
3Threedeuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika, hat-trick
5Fivecinque, fin, fivesome, pentad, quint, quintet, quintuplet
6Sixhalf dozen, hexad, sestet, sextet, sextuplet, sise
11Elevenonze, ounze, ounce
12Twelvedozen
13Thirteenbaker's dozen, long dozen[2]
14Fourteen
15Fifteen
16Sixteen
17Seventeen
18Eighteen
19Nineteen
20Twentyscore
21Twenty-onelong score[2]
22Twenty-twoDeuce-deuce
23Twenty-three
24Twenty-fourtwo dozen
25Twenty-five
26Twenty-six
27Twenty-seven
28Twenty-eight
29Twenty-nine
30Thirty
31Thirty-one
40Fortytwo-score
50Fiftyhalf-century
60Sixtythree-score
70Seventythree-score and ten
80Eightyfour-score
87Eighty-sevenfour-score and seven
90Ninety
100One hundredcentred, century, ton, short hundred
101One hundred [and] one
110One hundred [and] ten
111One hundred [and] eleven
120One hundred [and] twentylong hundred,[2] great hundred, (obsolete) hundred
121One hundred [and] twenty-one
144One hundred [and] forty-fourgross, dozen dozen, small gross
169One hundred [and] sixty-ninebaker's gross[citation needed]
200Two hundred
300Three hundred
400Four hundred
500Five hundred
600Six hundred
666Six hundred [and] sixty-sixNumber of the Beast
700Seven hundred
777Seven hundred [and] seventy-sevenNumber of Luck
800Eight hundred
900Nine hundred
1 000One thousandchiliad, grand, G, thou, yard, kilo, k, millennium
1 001One thousand [and] one
1 010One thousand [and] ten
1 011One thousand [and] eleven
1 024One thousand [and] twenty-fourkibi or kilo in computing, see binary prefix (kilo is shortened to K, Kibi to Ki)
1 100One thousand one hundredEleven hundred
1 101One thousand one hundred [and] one
1 728One thousand seven hundred [and] twenty-eightgreat gross, long gross, dozen gross
2 000Two thousand
3 000Three thousand
100 000One hundred thousandlakh
500 000Five hundred thousandcrore (Iranian)
1 000 000One millionMega, meg, mil, (often shortened to M)
1 048 576One million forty-eight thousand five hundred [and] seventy-sixMibi or Mega in computing, see binary prefix (Mega is shortened to M, Mibi to Mi)
10 000 000Ten millioncrore (Bhartia)
100 000 000One hundred millionyi (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 scaleLong scalePower
ValueAmericanBritish
(Nicolas Chuquet)
Continental European
(Jacques Peletier du Mans)
of a thousandof a million
100One1000−1+110000000
101Ten
102Hundred
103Thousand10000+110000000.5
106Million10001+110000001
109BillionThousand millionMilliard10002+110000001.5
1012TrillionBillion10003+110000002
1018QuintillionTrillion10005+110000003
1021SextillionThousand trillionTrilliard10006+110000003.5
1030NonillionQuintillion10009+110000005
1033DecillionThousand quintillionQuintilliard100010+110000005.5
1036UndecillionSextillion100011+110000006
1039DuodecillionThousand sextillionSextilliard100012+110000006.5
1042TredecillionSeptillion100013+110000007
1045QuattuordecillionThousand septillionSeptilliard100014+110000007.5
1048QuindecillionOctillion100015+110000008
1051SexdecillionThousand octillionOctilliard100016+110000008.5
1054SeptendecillionNonillion100017+110000009
1057OctodecillionThousand nonillionNonilliard100018+110000009.5
1060NovemdecillionDecillion100019+1100000010
1063VigintillionThousand decillionDecilliard100020+1100000010.5
1066UnvigintillionUndecillion100021+1100000011
1069DuovigintillionThousand undecillionUndecilliard100022+1100000011.5
1072TrevigintillionDuodecillion100023+1100000012
1075QuattuorvigintillionThousand duodecillionDuodecilliard100024+1100000012.5
1078QuinvigintillionTredecillion100025+1100000013
...............
1093TrigintillionThousand quindecillionQuindecilliard100030+1100000015.5
...............
10120NovemtrigintillionVigintillion100039+1100000020
...............
10153QuinquagintillionThousand quinvigintillionQuinvigintilliard100050+1100000025.5
...............
10180NovemquinquagintillionTrigintillion100059+1100000030
10183SexagintillionThousand trigintillionTrigintilliard100060+1100000030.5
...............
10213SeptuagintillionThousand quintrigintillionQuintrigintilliard100070+1100000035.5
...............
...............
...............
10300NovemnonagintillionQuinquagintillion100099+1100000050
10303CentillionThousand quinquagintillionQuinquagintilliard1000100+1100000050.5
............
10360Sexagintillion1000119+1100000060
10420Septuagintillion1000139+1100000070
10480Octogintillion1000159+1100000080
10540Nonagintillion1000179+1100000090
10600Centillion1000199+11000000100
10603DucentillionThousand centillionCentilliard1000200+11000000100.5

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

#### Proposed systematic names for powers of 10

ValueNameNotation
100One1
101Ten10
102Hundred100
103Ten hundred1000
108Myllion1;0000,0000
1016Byllion1:0000,0000;0000,0000
1024Myllion byllion1;0000,0000:0000,0000;0000,0000
1032Tryllion1'0000,0000;0000,0000:0000,0000;0000,0000
10128Quintyllion
10256Sextyllion
10512Septyllion
101024Octyllion
102048Nonyllion
104096Decyllion
108192Undecyllion
1016,384Duodecyllion
1032,768Tredecyllion
1065,536Quattuordecyllion
10131,072Quindecyllion
10262,144Sexdecyllion
10524,288Septendecyllion
101,048,576Octodecyllion
102,097,152Novemdecyllion
${10}^{\,\! 4\cdot 2^{20}}$Vigintyllion
${10}^{\,\! 4\cdot 2^{30}}$Trigintyllion
${10}^{\,\! 4 \cdot 2^{40}}$Quadragintyllion
${10}^{\,\! 4 \cdot 2^{50}}$Quinquagintyllion
${10}^{\,\! 4 \cdot 2^{60}}$Sexagintyllion
${10}^{\,\! 4 \cdot 2^{70}}$Septuagintyllion
${10}^{\,\! 4 \cdot 2^{80}}$Octogintyllion
${10}^{\,\! 4 \cdot 2^{90}}$Nonagintyllion
${10}^{\,\! 4 \cdot 2^{100}}$Centyllion
${10}^{\,\! 4 \cdot 2^{1000}}$Millyllion
${10}^{\,\! 4 \cdot 2^{10,000}}$Myryllion
##### SI-derived
Value1000mSI prefixNameBinary prefix1024m=210mValue
1 00010001kKiloKi102411 024
1 000 00010002MMegaMi102421 048 576
1 000 000 00010003GGigaGi102431 073 741 824
1 000 000 000 00010004TTeraTi102441 099 511 627 776
1 000 000 000 000 00010005PPetaPi102451 125 899 906 842 624
1 000 000 000 000 000 00010006EExaEi102461 152 921 504 606 846 976
1 000 000 000 000 000 000 00010007ZZettaZi102471 180 591 620 717 411 303 424
1 000 000 000 000 000 000 000 00010008YYottaYi102481 208 925 819 614 629 174 706 176

### Fractional numbers

This is a table of English names for positive 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 $\left({3 \over 25}\right)$, nine seventy-fifths $\left({9 \over 75} \right)$, six fiftieths $\left({6 \over 50}\right)$, twelve hundredths $\left({12 \over 100}\right)$, twenty-four two-hundredths $\left({24 \over 200}\right)$, etc.

ValueFractionCommon namesAlternative names
1$1 \over 1$One0.999..., Unity
0.9$9 \over 10$Nine tenths, [zero] point nine
0.8$4 \over 5$Four fifths, eight tenths, [zero] point eight
0.7$7 \over 10$Seven tenths, [zero] point seven
0.6$3 \over 5$Three fifths, six tenths, [zero] point six
0.5$1 \over 2$One half, five tenths, [zero] point five
0.4$2 \over 5$Two fifths, four tenths, [zero] point four
0.3 (333 333)...$1 \over 3$One third
0.3$3 \over 10$Three tenths, [zero] point three
0.25$1 \over 4$One quarter, one fourth, twenty-five hundredths, [zero] point two five
0.2$1 \over 5$One fifth, two tenths, [zero] point two
0.16 (666 666)...$1 \over 6$One sixth
0.142 857 (142 857)...$1 \over 7$One seventh
0.125$1 \over 8$One eighth, one-hundred-[and-]twenty-five thousandths, [zero] point one two five
0.1 (111 111)...$1 \over 9$One ninth
0.1$1 \over 10$One tenth, [zero] point oneOne perdecime, one perdime
0.090 (909 090)...$1 \over 11$One eleventh
0.09$9 \over 100$Nine hundredths, [zero] point zero nine
0.083 (333 333)...$1 \over 12$One twelfth
0.08$2 \over 25$Two twenty-fifths, eight hundredths, [zero] point zero eight
0.0625$1 \over 16$One sixteenth, six-hundred-[and-]twenty-five ten-thousandths, [zero] point zero six two five
0.05$1 \over 20$One twentieth, [zero] point zero five
0.047 619 (047 619)...$1 \over 21$One twenty-first
0.045 (454 545)...$1 \over 22$One twenty-second
0.043 478 260 869 565 217 3913 (043 478)...$1 \over 23$One twenty-third
0.03 (333 333)...$1 \over 30$One thirtieth
0.016 (666 666)...$1 \over 60$One sixtiethOne minute
0.012345679 (012345679)...$1 \over 81$One eighty-first
0.01$1 \over 100$One hundredth, [zero] point zero oneOne percent
0.001$1 \over 1000$One thousandth, [zero] point zero zero oneOne permille
0.000 27 (777 777)...$1 \over 3600$One thirty-six hundredthOne second
0.000 1$1 \over 10000$One ten-thousandth, [zero] point zero zero zero oneOne myriadth, one permyria, one permyriad, one basis point
0.000 01$1 \over 10^5$One hundred-thousandthOne lakhth, one perlakh
0.000 001$1 \over 10^6$One millionthOne perion, one ppm
0.000 000 1$1 \over 10^7$One ten-millionthOne crorth, one percrore
0.000 000 01$1 \over 10^8$One hundred-millionthOne awkth, one perawk
0.000 000 001$1 \over 10^9$One billionth (in some dialects)One ppb
0$0 \over 1$ZeroNil

## Irrational and suspected irrational numbers

### Algebraic numbers

ExpressionApproximate valueNotes
$\frac{\sqrt{3}}{4}$0.433 012 701 892 219 323 381 861 585 376Area of a triangle with sides of length one and half its height.
${\sqrt{5} - 1} \over 2$0.618 033 988 749 894 848 204 586 834 366Golden ratio conjugate $\Phi\,$, reciprocal of and one less than the golden ratio.
$\sqrt[12]{2}$1.059 463 094 359 295 264 561 825 294 946Twelfth root of two.
Proportion between the frequencies of adjacent semitones in the equal temperament scale.
$\frac{3 \sqrt{2}}{4}$1.060 660 171 779 821 286 601 266 543 157The size of the cube that satisfies Prince Rupert's cube.
$\sqrt[3]{2}$1.259 921 049 894 873 164 767 210 607 278Cube root of two.
Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
n/a1.303 577 269 034 296 391 257 099 112 153Conway's constant, defined as the unique positive real root of a certain polynomial of degree 71.
$\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+$
$\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}$
1.324 717 957 244 746 025 960 908 854 478Plastic number, the unique real root of the cubic equation $x^3=x+1\, .$
$\sqrt{2}$1.414 213 562 373 095 048 801 688 724 210$\sqrt{2} = 2 \sin 45^\circ = 2 \cos 45^\circ$
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).
$\frac{\sqrt{17}-1}{2}$1.561 552 812 808 830 274 910 704 927 987The Triangular root of 2.
${\sqrt{5} + 1} \over 2$1.618 033 988 749 894 848 204 586 834 366Golden ratio $\left(\phi\right)$, the larger of the two real roots of $x^2=x+1\, .$
$\sqrt{3}$1.732 050 807 568 877 293 527 446 341 506$\sqrt{3} = 2 \sin 60^\circ = 2 \cos 30^\circ$
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 $1 \times \sqrt{2}$ rectangle.
Altitude of an equilateral triangle with side length 2.
Twice the altitude of an equilateral triangle with side length 1.
Altitude of a regular hexagon with side length 1 and diagonal length 2.
$\frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}$1.839 286 755 214 161 132 551 852 564 653The Tribonacci constant.
Used in the formula for the volume of the snub cube and properties of some of its dual polyhedrons.
It satisfies the equation x + x−3 = 2.
$\sqrt{5}$2.236 067 977 499 789 696 409 173 668 731Square root of five.
Length of the diagonal of a $1 \times 2$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{3}$ rectangle.
Length of the space diagonal of a $1 \times \sqrt{2} \times \sqrt{2}$ rectangular box.
$\sqrt{2} + 1$2.414 213 562 373 095 048 801 688 724 210Silver ratio $\left(\delta_S\right)$, the larger of the two real roots of $x^2=2x+1\, .$
$\sqrt{6}$2.449 489 742 783 178 098 197 284 074 706$\sqrt{2} \cdot \sqrt{3}$ = area of a $\sqrt{2} \times \sqrt{3}$ rectangle.
Length of the space diagonal of a $1 \times 1 \times 2$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{5}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{2}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{3}$.
$\sqrt{7}$2.645 751 311 064 590 590 501 615 753 639Length of the space diagonal of a $1 \times 2 \times \sqrt{2}$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{6}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{3}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{5}$ rectangle.
$\sqrt{8}$2.828 427 124 746 190 097 603 377 448 419$2 \sqrt{2}$
Volume of a cube with edge length $\sqrt{2}$.
Length of the diagonal of a square with side length 2.
Length of the diagonal of a $1 \times \sqrt{7}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{6}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{5}$ rectangle.
$\sqrt{10}$3.162 277 660 168 379 331 998 893 544 433$\sqrt{2} \cdot \sqrt{5}$ = area of a $\sqrt{2} \times \sqrt{5}$ rectangle.
Length of the diagonal of a $1 \times 3$ rectangle.
Length of the diagonal of a $2 \times \sqrt{6}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{7}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{5}$.
$\sqrt{11}$3.316 624 790 355 399 849 114 932 736 671Length of the space diagonal of a $1 \times 1 \times 3$ rectangular box.
Length of the diagonal of a $1 \times \sqrt{10}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{7}$ rectangle.
Length of the diagonal of a $3 \times \sqrt{2}$ rectangle.
Length of the diagonal of a $\sqrt{3} \times \sqrt{8}$ rectangle.
Length of the diagonal of a $\sqrt{5} \times \sqrt{6}$ rectangle.
$\sqrt{12}$3.464 101 615 137 754 587 054 892 683 012$2 \sqrt{3}$
Length of the space diagonal of a cube with edge length 2.
Length of the diagonal of a $1 \times \sqrt{11}$ rectangle.
Length of the diagonal of a $2 \times \sqrt{8}$ rectangle.
Length of the diagonal of a $3 \times \sqrt{3}$ rectangle.
Length of the diagonal of a $\sqrt{2} \times \sqrt{10}$ rectangle.
Length of the diagonal of a $\sqrt{5} \times \sqrt{7}$ rectangle.
Length of the diagonal of a square with side length $\sqrt{6}$.

## Hypercomplex numbers

### Algebraic complex numbers

• Imaginary unit: $i = \sqrt{-1}$
• nth roots of unity: $\xi^k_n = \cos\left(2\pi \tfrac{k}{n}\right)+i\sin\left(2\pi \tfrac{k}{n}\right)$

## Transfinite numbers

• Infinity in general: $\infty$
• Aleph-null: $\aleph_0$
• Aleph-one: $\aleph_1$
• Beth-one: ($\beth_1$) is the cardinality of the continuum $(2^{\aleph_0})$
• ℭ or $\mathfrak c$: the cardinality of the continuum $(2^{\aleph_0})$
• omega: ω, the smallest infinite ordinal

## Notes

1. ^ http://mathworld.wolfram.com/Hardy-RamanujanNumber.html
2. ^ a b c The shipmaster's assistant, and commercial digest
3. ^ a b c "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 27.
4. ^ a b "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
5. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 30.
6. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33.
7. ^ http://www.qbyte.org/puzzles/p029s.html
8. ^
9. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
10. ^ Sequence .
11. ^
12. ^
13. ^
14. ^
15. ^
16. ^
17. ^ http://mathworld.wolfram.com/ContinuedFractionConstant.html
18. ^
19. ^
20. ^ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
21. ^
22. ^
23. ^
24. ^
25. ^