Definitions

# List of numbers

## Rational numbers

### Notable 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
210 220 230 240 250 260 270 280 290
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-1000M Larger #s

### Integers

#### Notable integers

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

• -40 An equal point in the Fahrenheit and Celsius scales.
• -1
• 255
• 496 A perfect number.
• 666 The number of the beast, according to the Bible.
• 786 The number of God, according to the Koran.
• 911 The emergency phone number in the United States.
• 142857 A base 10 cyclic number.
• 9814072356

### Prime numbers

A prime number is a positive integer which has exactly two divisors: one and itself.

The first 100 prime numbers:

 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

### Perfect numbers

A perfect number is an integer that is the sum of its positive proper divisors (all divisors except itself).

The first ten 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 ten million -- names for which all variants of English agree.
Value Name Alternate names
0 Zero aught, cipher, cypher, goose egg, nada, naught, nil, none, nought, nowt, null, ought, oh, squat, zed, zilch, zip
1 One ace, single, singleton, unary, unit, unity
2 Two binary, brace, couple, couplet, distich, deuce, double, doubleton, duad, duality, duet, duo, dyad, pair, span, twain, twosome, yoke
3 Three deuce-ace, leash, set, tercet, ternary, ternion, terzetto, threesome, tierce, trey, triad, trine, trinity, trio, triplet, troika
5 Five cinque, fin, fivesome, pentad, quint, quintet, quintuplet
6 Six half dozen, hexad, sestet, sextet, sextuplet, sise
11 Eleven
12 Twelve dozen
13 Thirteen baker's dozen, long dozen
14 Fourteen
15 Fifteen
16 Sixteen
17 Seventeen
18 Eighteen
19 Nineteen
20 Twenty score
21 Twenty-one
22 Twenty-two
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
40 Forty
50 Fifty Half - century
60 Sixty shock
70 Seventy three-score and ten
80 Eighty four-score
87 Eighty-seven four-score and seven
90 Ninety
100 One hundred centred, century, ton, short hundred
101 One hundred [and] one
110 One hundred [and] ten
111 One hundred [and] eleven,
120 One hundred [and] twenty long hundred, 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
666 Six Hundred [and] sixty-six Number of the Beast
1 000 One thousand chiliad, grand (or G), thou, yard, kilo (often shortened to k)
1 001 One thousand [and] one
1 010 One thousand [and] ten
1 011 One thousand [and] eleven
1 024 One thousand [and] twenty-four kilo (in computing, see binary prefix) (often shortened to K)
1 100 One thousand one hundred
1 101 One thousand one hundred [and] one
1 728 One thousand seven hundred [and] twenty-eight great gross, long gross, dozen gross
2 000 Two thousand
100 000 One hundred thousand lakh
500 000 Five hundred thousand crore (Iranian)
1 000 000 One million meg, mil, (often shortened to M)
1 048 576 One million forty-eight thousand five hundred [and] seventy-six meg (in computing, see binary prefix) (often shortened to M)
10 000 000 Ten million crore (Indian)

#### 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 names of numbers in English or English-language numerals for more information on naming numbers.
Short scale Long scale Power
Value American & Modern British Traditional 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
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 ... ... 100024+1 100000012.5
... ... ... ... ...
1093 Trigintillion Thousand quindecillion Quindecilliard 100030+1 100000015.5
... ... ... ... ...
10120 Novemtrigintillion Vigintillion 100039+1 100000020
10123 Quadragintillion Thousand vigintillion Vigintilliard 100040+1 100000020.5
... ... ... ... ...
10153 Quinquagintillion Thousand duovigintillion Duovigintilliard 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
... ... ... ... ...
... ... ... ... ...
... ... ... ... ...
10300 Novemnonagintillion Quinquagintillion 100099+1 100000050
10303 Centillion Thousand quinquagintillion Quinquagintilliard 1000100+1 100000050.5
... ... ... ...
10360 Sexagintillion 1000119+1 100000060
10420 Septuagintillion 1000139+1 100000070
10480 Octogintillion 1000159+1 100000080
10540 Nonagintillion 1000179+1 100000090
10600 Centillion 1000199+1 1000000100
10603 ducentillion Thousand Centillion Centilliard 1000200+1 1000000100.5

#### Proposed systematic names for powers of 10

##### Gillion system
As proposed by Russ Rowlett, based on Greek-derived numerical prefixes:
Value Name
103 Thousand
106 Million
109 Gillion
1012 Tetrillion
1015 Pentillion
1018 Hexillion
1021 Heptillion
1024 Oktillion
1027 Ennillion
1030 Dekillion
Value Name
1033 Hendekillion
1036 Dodekillion
1039 Trisdekillion
1060 Icosillion
Value Name
1063 Icosihenillion
1066 Icosidillion
1069 Icositrillion
1072 Icositetrillion
1075 Icosipentillion
1078 Icosihexillion
1081 Icosiheptillion
1084 Icosioktillion
1087 Icosiennillion
1090 Triacontillion
Proposed by Donald E. Knuth:
Value Name Notation
100 One 1
101 Ten 10
102 Hundred 100
103 Ten hundred 1000
108 Myllion 1,0000,0000
1016 Byllion 1,0000,0000,0000,0000
1024 Myllion byllion 1,0000,0000:0000,0000;0000,0000
1032 Tryllion 1 0000,0000;0000,0000:0000,0000;0000,0000
10128 Quintyllion 1'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000
10256 Sextyllion 1'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,00000000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000
10512 Septyllion
101024 Octyllion
102048 Nonyllion
104096 Decyllion
108192 Undecyllion
1016,384 Duodecyllion
1032,768 Tredecyllion
1065,536 Quattuordecyllion
10131,072 Quindecyllion
10262,144 Sexdecyllion
10524,288 Septendecyllion
101,048,576 Octodecyllion
102,097,152 Novemdecyllion
104,194,304 Vigintyllion
108,388,608 Cafaolion
1016,777,216 Saralion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{40\right\}\right\}$ Quadragintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{50\right\}\right\}$ Quinquagintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{60\right\}\right\}$ Sexagintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{70\right\}\right\}$ Septuagintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{80\right\}\right\}$ Octogintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{90\right\}\right\}$ Nonagintyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{100\right\}\right\}$ Centyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{1000\right\}\right\}$ Millyllion
$\left\{10\right\}^\left\{,! 4 * 2^\left\{10000\right\}\right\}$ Myryllion
##### Googol and others
10100 Googol
$\left\{10\right\}^\left\{,!10^\left\{100\right\}\right\}$ Googolplex
10-N N-minex
10N N-plex
##### SI-derived
Value SI prefix Name Binary prefix Value
1 000 k Kilo (k) Ki 1024
1 000 000 M Meg (M) Mi 1 048 576
1 000 000 000 G Gig Gi 1 073 741 824

### 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 $left\left(\left\{3 over 25\right\}right\right)$, nine seventy-fifths $left\left(\left\{9 over 75\right\} right\right)$, six fiftieths $left\left(\left\{6 over 50\right\}right\right)$, twelve hundredths $left\left(\left\{12 over 100\right\}right\right)$, twenty-four two-hundredths $left\left(\left\{24 over 200\right\}right\right)$, etc.

Value Fraction Common names Alternative names
1 $1 over 1$ One 0.999...
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 one One 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 sixtieth One minute
0.012345679 (012345679) $1 over 81$ One eighty-first
0.01 $1 over 100$ One hundredth, [zero] point zero one One percent
0.001 $1 over 1000$ One thousandth, [zero] point zero zero one One permille
0.000 27 (777 777)... $1 over 3600$ One thirty-six hundredth One second
0.000 1 $1 over 10000$ One ten-thousandth, [zero] point zero zero zero one One myriadth, one permyria, one permyriad, one basis point
0.000 01 $1 over 10^5$ One hundred-thousandth One lakhth, one perlakh
0.000 001 $1 over 10^6$ One millionth One perion, one ppm
0.000 000 1 $1 over 10^7$ One ten-millionth One crorth, one percrore
0.000 000 01 $1 over 10^8$ One hundred-millionth One awkth, one perawk
0.000 000 001 $1 over 10^9$ One billionth (in some dialects) One ppb
0 $0 over 1$ Zero

## Irrational numbers

### Algebraic numbers

Expression Value Notes
$\left\{sqrt\left\{5\right\} - 1\right\} over 2$ 0.618 033 988 749 894 848 204 586 834 366... Golden ratio conjugate $Phi,$, reciprocal of and one less than the golden ratio.
$sqrt\left[12\right]\left\{2\right\}$ 1.059 463 094 359 295 264 561 825 294 946... Twelfth root of two.
Proportion between the frequencies of adjacent semitones in the equal temperament scale.
$sqrt\left[3\right]\left\{2\right\}$ 1.259 921 049 894 873 164 767 210 607 278... Cube root of two.
Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
$sqrt\left[3\right]\left\{frac\left\{1\right\}\left\{2\right\}+frac\left\{1\right\}\left\{6\right\}sqrt\left\{frac\left\{23\right\}\left\{3\right\}\right\}\right\}+$
$sqrt\left[3\right]\left\{frac\left\{1\right\}\left\{2\right\}-frac\left\{1\right\}\left\{6\right\}sqrt\left\{frac\left\{23\right\}\left\{3\right\}\right\}\right\}$
1.324 717 957 244 746 025 960 908 854 478... Plastic number.
$sqrt\left\{2\right\}$ 1.414 213 562 373 095 048 801 688 724 210... $sqrt\left\{2\right\} = 2 sin 45^circ = 2 cos 45^circ$
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).
$\left\{sqrt\left\{5\right\} + 1\right\} over 2$ 1.618 033 988 749 894 848 204 586 834 366... Golden ratio $left\left(phiright\right)$.
$sqrt\left\{3\right\}$ 1.732 050 807 568 877 193 176 604 123 437... $sqrt\left\{3\right\} = 2 sin 60^circ = 2 cos 30^circ$
Square root of three a.k.a. the measure of the fish.
Length of the diagonal of a cube with edge length 1.
Length of the diagonal of a $1 times sqrt\left\{2\right\}$ 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.
$sqrt\left\{5\right\}$ 2.236 067 977 499 789 805 051 477 742 381... Square root of five.
Length of the diagonal of a $1 times 2$ rectangle.
Length of the diagonal of a $sqrt\left\{2\right\} times sqrt\left\{3\right\}$ rectangle.
Length of the diagonal of a $1 times sqrt\left\{2\right\} times sqrt\left\{2\right\}$ rectangular box.
$sqrt\left\{2\right\} + 1$ 2.414 213 562 373 095 048 801 688 724 210... Silver ratio $left\left(delta_Sright\right)$.
$sqrt\left\{6\right\}$ 2.449 489 742 783 177 881 335 632 264 381... $sqrt\left\{2\right\} cdot sqrt\left\{3\right\}$ = area of a $sqrt\left\{2\right\} times sqrt\left\{3\right\}$ rectangle.
Length of the diagonal of a $1 times 1 times 2$ rectangular box.
Length of the diagonal of a $1 times sqrt\left\{5\right\}$ rectangle.
Length of the diagonal of a $2 times sqrt\left\{2\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{3\right\} times sqrt\left\{3\right\}$ rectangle.
$sqrt\left\{7\right\}$ 2.645 751 311 064 590 716 171 096 573 817... Length of the diagonal of a $1 times 2 times sqrt\left\{2\right\}$ rectangular box.
Length of the diagonal of a $1 times sqrt\left\{6\right\}$ rectangle.
Length of the diagonal of a $2 times sqrt\left\{3\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{2\right\} times sqrt\left\{5\right\}$ rectangle.
$sqrt\left\{8\right\}$ 2.828 427 124 746 190 290 949 243 717 478... $2 sqrt\left\{2\right\}$
Volume of a cube with edge length $sqrt\left\{2\right\}$.
Length of the diagonal of a $2 times 2$ rectangle.
Length of the diagonal of a $1 times sqrt\left\{7\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{2\right\} times sqrt\left\{6\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{3\right\} times sqrt\left\{5\right\}$ rectangle.
$sqrt\left\{10\right\}$ 3.162 277 660 168 379 522 787 063 251 599... $sqrt\left\{2\right\} cdot sqrt\left\{5\right\}$ = area of a $sqrt\left\{2\right\} times sqrt\left\{5\right\}$ rectangle.
Length of the diagonal of a $1 times 3$ rectangle.
Length of the diagonal of a $2 times sqrt\left\{6\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{3\right\} times sqrt\left\{7\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{5\right\} times sqrt\left\{5\right\}$ rectangle.
$sqrt\left\{11\right\}$ 3.316 624 790 355 399 849 114 932 736 671 Length of the diagonal of a $1 times 1 times 3$ rectangular box.
Length of the diagonal of a $1 times sqrt\left\{10\right\}$ rectangle.
Length of the diagonal of a $2 times sqrt\left\{7\right\}$ rectangle.
Length of the diagonal of a $3 times sqrt\left\{2\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{3\right\} times sqrt\left\{8\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{5\right\} times sqrt\left\{6\right\}$ rectangle.
$sqrt\left\{12\right\}$ 3.464 101 615 137 754 587 054 892 683 012... $2 sqrt\left\{3\right\}$
Length of the diagonal of a cube with edge length 2.
Length of the diagonal of a $1 times sqrt\left\{11\right\}$ rectangle.
Length of the diagonal of a $2 times sqrt\left\{8\right\}$ rectangle.
Length of the diagonal of a $3 times sqrt\left\{3\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{2\right\} times sqrt\left\{10\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{5\right\} times sqrt\left\{7\right\}$ rectangle.
Length of the diagonal of a $sqrt\left\{6\right\} times sqrt\left\{6\right\}$ rectangle.

### Numbers not known with high precision

Grothendieck constant: between 1.67 and 1.79

## Hypercomplex numbers

### Algebraic complex numbers

• Imaginary unit: $i = sqrt\left\{-1\right\}$

## Transfinite numbers

• Infinity in general: $infty$
• Aleph-null: $aleph_0$
• Aleph-one: $aleph_1$
• Beth-one: ($beth_1$) is the cardinality of the continuum: $2^\left\{aleph_0\right\}$

## Bases

See also positional systems of numeral system for bases which might not be listed here.