List of numbers

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 01673261270 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 $\mathbb {Q}$ , 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	10^10¹⁰⁰	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 10^k, 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:

−40, the equal point in the Fahrenheit and Celsius scales.
−1, the additive inverse of unity.
0, the additive identity.
1, the multiplicative identity. Also the only natural number (not including 0) that isn't prime or composite.
2, the base of the binary number system, used in almost all modern computers and information systems. Also notable as the only even prime number.
3, significant in Christianity as the Trinity. Also considered significant in Hinduism (Trimurti, Tridevi).
4, the first composite number, also considered an "unlucky number" in modern China due to its audible similarity to the word "Death."
6, the first of the series of perfect numbers, whose proper factors sum to the number itself.
7, considered a "lucky" number in Western cultures.
8, considered a "lucky" number in Chinese culture.
9, the first odd number that is composite.
10, the number base for most modern counting systems.
12, the number base for some ancient counting systems and the basis for some modern measuring systems. Known as a dozen.
13, considered an "unlucky" number in Western superstition. Also known as a "Baker's Dozen".
20, known as a score.
28, the second perfect number.
42, the "answer to the ultimate question of life, the universe, and everything" in the popular science fiction work The Hitchhiker's Guide to the Galaxy.
60, the number base for some ancient counting systems, such as the Babylonians', and the basis for many modern measuring systems.
86, a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.^[3]
108, considered sacred by the Dharmic Religions. Approximately equal to the ratio of the distance from Earth to Sun and diameter of the Sun.
144, a dozen times dozen, known as a gross.
255, 2⁸ − 1, a Mersenne number and the smallest perfect totient number that is neither a power of three nor thrice a prime; it is also the largest number that can be represented using an 8-bit unsigned integer.
420, a code-term that refers to the consumption of cannabis.
496, the third perfect number.
666, the Number of the Beast from the Book of Revelation.
786, regarded as sacred in the Muslim Abjad numerology.
1729, the Hardy–Ramanujan number, also known as the second taxicab number; that is, the smallest positive integer that can be written as the sum of two positive cubes in two different ways.^[4]
5040, mentioned by Plato in the Laws as one of the most important numbers for the city. It is also the largest factorial (7! = 5040) that is also a highly composite number.
8128, the fourth perfect number.
65535, 2¹⁶ − 1, the maximum value of a 16-bit unsigned integer.
65537, 2¹⁶ + 1, the most popular RSA public key prime exponent in most SSL/TLS certificates on the Web/Internet.
142857, the smallest base 10 cyclic number.
2147483647, 2³¹ − 1, the maximum value of a 32-bit signed integer using two's complement representation.
9814072356, the largest perfect power that contains no repeated digits in base ten.
9223372036854775807, 2⁶³ − 1, the maximum value of a 64-bit signed integer using two's complement representation.

Named numbers

Googol (10¹⁰⁰) and googolplex (10^(10¹⁰⁰)) and googolplexian (10^{(10^(10¹⁰⁰))}) or 1 followed by a googolplex of zeros.
Graham's number
Moser's number
Shannon number
Hardy–Ramanujan number (1729)
Skewes' number
Kaprekar's constant (6174)

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

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
9	Nine	ennead
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	1000000^0.5
10⁶	Million			1000¹⁺¹	1000000¹
10⁹	Billion	Thousand million	Milliard	1000²⁺¹	1000000^1.5
10¹²	Trillion	Billion		1000³⁺¹	1000000²
10¹⁵	Quadrillion	Thousand billion	Billiard	1000⁴⁺¹	1000000^2.5
10¹⁸	Quintillion	Trillion		1000⁵⁺¹	1000000³
10²¹	Sextillion	Thousand trillion	Trilliard	1000⁶⁺¹	1000000^3.5
10²⁴	Septillion	Quadrillion		1000⁷⁺¹	1000000⁴
10²⁷	Octillion	Thousand quadrillion	Quadrilliard	1000⁸⁺¹	1000000^4.5
10³⁰	Nonillion	Quintillion		1000⁹⁺¹	1000000⁵
10³³	Decillion	Thousand quintillion	Quintilliard	1000¹⁰⁺¹	1000000^5.5
10³⁶	Undecillion	Sextillion		1000¹¹⁺¹	1000000⁶
10³⁹	Duodecillion	Thousand sextillion	Sextilliard	1000¹²⁺¹	1000000^6.5
10⁴²	Tredecillion	Septillion		1000¹³⁺¹	1000000⁷
10⁴⁵	Quattuordecillion	Thousand septillion	Septilliard	1000¹⁴⁺¹	1000000^7.5
10⁴⁸	Quindecillion	Octillion		1000¹⁵⁺¹	1000000⁸
10⁵¹	Sexdecillion	Thousand octillion	Octilliard	1000¹⁶⁺¹	1000000^8.5
10⁵⁴	Septendecillion	Nonillion		1000¹⁷⁺¹	1000000⁹
10⁵⁷	Octodecillion	Thousand nonillion	Nonilliard	1000¹⁸⁺¹	1000000^9.5
10⁶⁰	Novemdecillion	Decillion		1000¹⁹⁺¹	1000000¹⁰
10⁶³	Vigintillion	Thousand decillion	Decilliard	1000²⁰⁺¹	1000000^10.5
10⁶⁶	Unvigintillion	Undecillion		1000²¹⁺¹	1000000¹¹
10⁶⁹	Duovigintillion	Thousand undecillion	Undecilliard	1000²²⁺¹	1000000^11.5
10⁷²	Trevigintillion	Duodecillion		1000²³⁺¹	1000000¹²
10⁷⁵	Quattuorvigintillion	Thousand duodecillion	Duodecilliard	1000²⁴⁺¹	1000000^12.5
10⁷⁸	Quinvigintillion	Tredecillion		1000²⁵⁺¹	1000000¹³
10⁸¹	Sexvigintillion	Thousand tredecillion	Tredecilliard	1000²⁶⁺¹	1000000^13.5
10⁸⁴	Septenvigintillion	Quattuordecillion		1000²⁷⁺¹	1000000¹⁴
10⁸⁷	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard	1000²⁸⁺¹	1000000^14.5
10⁹⁰	Novemvigintillion	Quindecillion		1000²⁹⁺¹	1000000¹⁵
10⁹³	Trigintillion	Thousand quindecillion	Quindecilliard	1000³⁰⁺¹	1000000^15.5
10⁹⁶	Untrigintillion	Sexdecillion		1000³¹⁺¹	1000000¹⁶
10⁹⁹	Duotrigintillion	Thousand sexdecillion	Sexdecilliard	1000³²⁺¹	1000000^16.5
...	...	...		...	...
10¹²⁰	Novemtrigintillion	Vigintillion		1000³⁹⁺¹	1000000²⁰
10¹²³	Quadragintillion	Thousand vigintillion	Vigintilliard	1000⁴⁰⁺¹	1000000^20.5
...	...	...		...	...
10¹⁵³	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard	1000⁵⁰⁺¹	1000000^25.5
...	...	...		...	...
10¹⁸⁰	Novemquinquagintillion	Trigintillion		1000⁵⁹⁺¹	1000000³⁰
10¹⁸³	Sexagintillion	Thousand trigintillion	Trigintilliard	1000⁶⁰⁺¹	1000000^30.5
...	...	...		...	...
10²¹³	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard	1000⁷⁰⁺¹	1000000^35.5
...	...	...		...	...
10²⁴⁰	Novemseptuagintillion	Quadragintillion		1000⁷⁹⁺¹	1000000⁴⁰
10²⁴³	Octogintillion	Thousand quadragintillion	Quadragintilliard	1000⁸⁰⁺¹	1000000^40.5
...	...	...		...	...
10²⁷³	Nonagintillion	Thousand quinquadragintillion	Quinquadragintilliard	1000⁹⁰⁺¹	1000000^45.5
...	...	...		...	...
10³⁰⁰	Novemnonagintillion	Quinquagintillion		1000⁹⁹⁺¹	1000000⁵⁰
10³⁰³	Centillion	Thousand quinquagintillion	Quinquagintilliard	1000¹⁰⁰⁺¹	1000000^50.5
...	...	...		...	...
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	1000000^100.5

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

SI prefixes for powers of 10

Value	1000^m	SI prefix	Name	Binary prefix	1024^m = 2^10m	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	6999433012701892219♠0.433012701892219323381861585376	Area of an equilateral triangle with side length 1.
√5 − 1/2	6999618033988749894♠0.618033988749894848204586834366	Golden ratio conjugate Φ, reciprocal of and one less than the golden ratio.
√3/2	6999866025403784438♠0.866025403784438646763723170753	Height of an equilateral triangle with side length 1.
¹²√2	7000105946309435929♠1.059463094359295264561825294946	Twelfth root of two. Proportion between the frequencies of adjacent semitones in the equal temperament scale.
3√2/4	7000106066017177982♠1.060660171779821286601266543157	The size of the cube that satisfies Prince Rupert's cube.
³√2	7000125992104989487♠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.
—	7000130357726903429♠1.303577269034296391257099112153	Conway's constant, defined as the unique positive real root of a certain polynomial of degree 71.
${\sqrt[ {3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {{\frac {23}{3}}}}}}+{\sqrt[ {3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {{\frac {23}{3}}}}}}$	7000132471795724474♠1.324717957244746025960908854478	Plastic number, the unique real root of the cubic equation x³ = x + 1.
√2	7000141421356237309♠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).
${\frac {1}{3}}+{\frac {2}{3{\sqrt[{3}]{116+12{\sqrt {93}}}}}}+{\frac {1}{6}}{\sqrt[{3}]{116+12{\sqrt {93}}}}$	7000146557123187676♠1.465571231876768026656731225220	The limit to the ratio between subsequent numbers in the binary Look-and-say sequence.
${\frac {\sqrt {5+2{\sqrt {5}}}}{2}}$	7000153884176858762♠1.538841768587626701285145288018	Altitude of a regular pentagon with side length 1.
√17 − 1/2	7000156155281280883♠1.561552812808830274910704927987	The Triangular root of 2.
√5 + 1/2	7000161803398874989♠1.618033988749894848204586834366	Golden ratio (φ), the larger of the two real roots of x² = x + 1.
${\frac {5}{4{\sqrt {5-2{\sqrt {5}}}}}}$	7000172047740058896♠1.720477400588966922759011977389	Area of a regular pentagon with side length 1.
√3	7000173205080756887♠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.
${\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}$	7000183928675521416♠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	7000223606797749978♠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	7000241421356237309♠2.414213562373095048801688724210	Silver ratio (δ_S), the larger of the two real roots of x² = 2x + 1. Altitude of a regular octagon with side length 1.
√6	7000244948974278317♠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.
3√3/2	7000259807611353315♠2.598076113533159402911695122588	Area of a regular hexagon with side length 1.
√7	7000264575131106459♠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	7000282842712474619♠2.828427124746190097603377448419	2√2 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	7000316227766016837♠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	7000331662479035540♠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	7000346410161513775♠3.464101615137754587054892683012	2√3 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

(−1)ⁱ = e^{− $π$} = 6998432139183000000♠0.0432139183...
Liouville constant: c = 6999110001000000000♠0.110001000000000000000001000...
Champernowne constant: C₁₀ = 6999123456789101112♠0.12345678910111213141516...
iⁱ = √e^{− $π$} = 6999207879576000000♠0.207879576...
1/ $π$ = 6999318309886183790♠0.318309886183790671537767526745028724068919291480...^[7]
1/e = 6999367879441171442♠0.367879441171442321595523770161460867445811131031...^[7]
Prouhet–Thue–Morse constant: $τ$ = 6999412454033640000♠0.412454033640...
log₁₀ e = 6999434294481903251♠0.434294481903251827651128918916605082294397005803...^[7]
Omega constant: Ω = 6999567143290409783♠0.5671432904097838729999686622...
Cahen's constant: c = 6999643410546290000♠0.64341054629...
ln 2: 6999693147180559945♠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 = 6999834626800000000♠0.8346268...
$π$ /√12 = 0.9068..., the fraction of the plane covered by the densest possible circle packing^[9]
eⁱ + e⁻ⁱ = 2 cos 1 = 7000108060461000000♠1.08060461...
$π$ ⁴/90 = ζ(4) = 7000108232300000000♠1.082323...^[10]
√2_s: 7000155961046900000♠1.559610469...^[11]
log₂ 3: 7000158496250100000♠1.584962501... (the logarithm of any positive integer to any integer base greater than 1 is either rational or transcendental)
Gaussian integral: √ $π$ = 7000177245385090551♠1.772453850905516...
Komornik–Loreti constant: q = 7000178723165000000♠1.787231650...
Universal parabolic constant: P₂ = 7000229558714939000♠2.29558714939...
Gelfond–Schneider constant: 2^√2 = 7000266514414300000♠2.665144143...
e = 7000271828182845904♠2.718281828459045235360287471353...
$π$ = 7000314159265358979♠3.141592653589793238462643383279...
ⁱ√i = √e^$π$ = 7000481047738100000♠4.810477381...
Tau, or 2 $π$ : $τ$ = 7000628318530717958♠6.283185307179586..., The ratio of the circumference to a radius, and the number of radians in a complete circle^[12]^[13]
Gelfond's constant: 7001231406926327792♠23.14069263277925...
Ramanujan's constant: e^{$π$ √163} = 7017262537412640768♠262537412640768743.99999999999925...

Suspected transcendentals

These are irrational numbers that are thought to be, but have not yet been proved to be, transcendental.

Z(1): 3000263694537132682♠−0.736305462867317734677899828925614672...
Heath-Brown–Moroz constant: C = 6997131764100000000♠0.001317641...
Kepler–Bouwkamp constant: 6999114942044800000♠0.1149420448...
MRB constant: 6999187859000000000♠0.187859...
Meissel–Mertens constant: M = 6999261497212847642♠0.2614972128476427837554268386086958590516...
Bernstein's constant: β = 6999280169499000000♠0.2801694990...
Strongly carefree constant: 6999286747000000000♠0.286747...^[14]
Gauss–Kuzmin–Wirsing constant: λ₁ = 6999303663002900000♠0.3036630029...^[15]
Hafner–Sarnak–McCurley constant: 6999353236371900000♠0.3532363719...
Artin's constant: 6999373955813600000♠0.3739558136...
Prime constant: ρ = 6999414682509851111♠0.414682509851111660248109622...
Carefree constant: 6999428249000000000♠0.428249...^[16]
S(1): 6999438259147390354♠0.438259147390354766076756696625152...
F(1): 6999538079506912768♠0.538079506912768419136387420407556...
Stephens' constant: 6999575959000000000♠0.575959...^[17]
Euler–Mascheroni constant: γ = 6999577215664901532♠0.577215664901532860606512090082...
Golomb–Dickman constant: λ = 6999624329988543550♠0.62432998854355087099293638310083724...
Twin prime constant: C₂ = 6999660161815846869♠0.660161815846869573927812110014...
Copeland–Erdős constant: 6999235711131719232♠0.235711131719232931374143...
Feller–Tornier constant: 6999661317000000000♠0.661317...^[18]
Laplace limit: ε = 6999662743419300000♠0.6627434193...
Taniguchi's constant: 6999678234000000000♠0.678234...^[19]
Continued Fraction Constant: C = 6999697774657964008♠0.697774657964007982006790592551...^[20]
Embree–Trefethen constant: β* = 6999702580000000000♠0.70258...
Sarnak's constant: 6999723648000000000♠0.723648...^[21]
Landau–Ramanujan constant: 6999764223653589220♠0.76422365358922066299069873125...
C(1): 6999779893400376822♠0.77989340037682282947420641365...
1/ζ(3) = 6999831907000000000♠0.831907..., the probability that three random numbers have no common factor greater than 1.^[8]
Brun's constant for prime quadruplets: B₂ = 6999870588380000000♠0.8705883800...
Quadratic class number constant: 6999881513000000000♠0.881513...^[22]
Catalan's constant: G = 6999915965594177219♠0.915965594177219015054603514932384110774...
Viswanath's constant: σ(1) = 7000113198824879430♠1.1319882487943...
Khinchin–Lévy constant: 7000118656911040000♠1.1865691104...
ζ(3) = 7000120205690315959♠1.202056903159594285399738161511449990764986292..., also known as Apéry's constant, known to be irrational, but not known whether or not it is transcendental.^[23]
Vardi's constant: E = 7000126408473530500♠1.264084735305...
Glaisher–Kinkelin constant: A = 7000128242711999999♠1.28242712...
Mills' constant: A = 7000130637788386308♠1.30637788386308069046...
Totient summatory constant: 7000133978400000000♠1.339784...^[24]
Ramanujan–Soldner constant: μ = 7000145136923488338♠1.451369234883381050283968485892027449493...
Backhouse's constant: 7000145607494800000♠1.456074948...
Favard constant: K₁ = 7000157079633000000♠1.57079633...
Erdős–Borwein constant: E = 7000160669515241529♠1.606695152415291763...
Somos' quadratic recurrence constant: σ = 7000166168794963359♠1.661687949633594121296...
Niven's constant: c = 7000170521100000000♠1.705211...
Brun's constant: B₂ = 7000190216058310400♠1.902160583104...
Landau's totient constant: 7000194359600000000♠1.943596...^[25]
exp(−W₀(−ln(³√3))) = 7000247805268028829♠2.47805268028830..., the smaller solution to 3^x = x³ and what, when put to the root of itself, is equal to 3 put to the root of itself.^[26]
Second Feigenbaum constant: α = 2.5029...
Sierpiński's constant: K = 7000258498175957925♠2.5849817595792532170658936...
Barban's constant: 7000259653600000000♠2.596536...^[27]
Khinchin's constant: K₀ = 7000268545200099999♠2.685452001...
Fransén–Robinson constant: F = 7000280777024200000♠2.8077702420...
Murata's constant: 7000282641900000000♠2.826419...^[28]
Lévy's constant: γ = 7000327582291872181♠3.275822918721811159787681882...
Reciprocal Fibonacci constant: ψ = 7000335988566624317♠3.359885666243177553172011302918927179688905133731...
Van der Pauw's constant: $π$ /ln 2 = 7000453236014182719♠4.53236014182719380962...^[29]
First Feigenbaum constant: δ = 4.6692...

Numbers not known with high precision

The constant in the Berry–Esseen Theorem: 0.4097 < C < 0.4748
2nd Landau's constant: 0.4330 < B < 0.472
Bloch's constant: 0.4332 < B < 0.4719
1st Landau's constant: 0.5 < L < 0.5433
3rd Landau's constant: 0.5 < A ≤ 0.7853
Grothendieck constant: 1.57 < k < 1.79

Hypercomplex numbers

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

Algebraic complex numbers

Imaginary unit: i = √−1
nth roots of unity: (ξ_n)^k = cos (2 $π$ k/n) + i sin (2 $π$ k/n), while 0 ≤ k ≤ n−1, GCD(k, n) = 1

Other hypercomplex numbers

The quaternions
The octonions
The sedenions
The dual numbers (with an infinitesimal)

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.

Aleph-null: ℵ₀: the smallest infinite cardinal, and the cardinality of ℕ, the set of natural numbers
Aleph-one: ℵ₁: the cardinality of ω₁, the set of all countable ordinal numbers
Beth-one: ℶ₁ the cardinality of the continuum 2^ℵ₀
ℭ or ${\mathfrak {c}}$ : the cardinality of the continuum 2^ℵ₀
omega: ω, the smallest infinite ordinal

Numbers representing measured quantities

Various terms have arisen to describe commonly used measured quantities.

Pair: 2 (the base of the binary numeral system)
Dozen: 12 (the base of the duodecimal numeral system)
Baker's dozen: 13
Score: 20 (the base of the vigesimal numeral system)
Gross: 144 (= 12²)
Great gross: 1728 (= 12³)

Numbers representing physical quantities

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

Avogadro constant: N_A = 7023602214179300000♠6.0221417930×10²³ mol⁻¹
Coulomb's constant: k_e = 7009898755178736800♠8.987551787368×10⁹ N·m²/C² (m/F)
Electronvolt: eV = 6981160217648740000♠1.60217648740×10⁻¹⁹ J
Electron relative atomic mass: A_r(e) = 6996548579909432299♠0.0005485799094323...
Fine structure constant: α = 6997729735253765000♠0.007297352537650...
Gravitational constant: G = 6989667384000000000♠6.67384×10⁻¹¹ N·(m/kg)²
Molar mass constant: M_u = 0.001 kg/mol
Planck constant: h = 6966662606896330000♠6.6260689633×10⁻³⁴ J · s
Rydberg constant: R_∞ = 7007109737315685277♠10973731.56852773 m⁻¹
Speed of light in vacuum: c = 7008299792458000000♠299792458 m/s
Stefan–Boltzmann constant: σ = 6992567040000000000♠5.670400×10⁻⁸ W · m⁻² · K⁻⁴

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

↑ 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.
↑ Rouse, Margaret. "Mathematical Symbols". Retrieved 1 April 2015.
↑ "Eighty-six – Definition of eighty-six by Merriam-Webster". merriam-webster.com. Archived from the original on 2013-04-08.
↑ Weisstein, Eric W. "Hardy–Ramanujan Number". Archived from the original on 2004-04-08.
1 2 3 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.
↑ Ezard, John (2 Jan 2003). "Tolkien catches up with his hobbit". The Guardian. Retrieved 6 Apr 2018.
1 2 3 "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 27.
1 2 "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
↑ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 30.
↑ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33.
↑ "Nick's Mathematical Puzzles: Solution 29". Archived from the original on 2011-10-18.
↑ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
↑ Sequence A019692.
↑ A065473
↑ Weisstein, Eric W. "Gauss–Kuzmin–Wirsing Constant". MathWorld.
↑ A065464
↑ A065478
↑ A065493
↑ A175639
↑ Weisstein, Eric W. "Continued Fraction Constant". Wolfram Research, Inc. Archived from the original on 2011-10-24.
↑ A065476
↑ A065465
↑ "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
↑ A065483
↑ A082695
↑ A166928
↑ A175640
↑ A065485
↑ A163973
↑ "Bags of Talent, a Touch of Panic, and a Bit of Luck: The Case of Non-Numerical Vague Quantifiers" from Linguista Pragensia, Nov. 2, 2010 Archived 2012-07-31 at Archive.is
↑ Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"

External links

The Database of Number Correlations: 1 to 2000+
What's Special About This Number? A Zoology of Numbers: from 0 to 500
Name of a Number
See how to write big numbers
About big numbers at the Library of Congress Web Archives (archived 2001-11-25)
Robert P. Munafo's Large Numbers page
Different notations for big numbers – by Susan Stepney
Names for Large Numbers, in How Many? A Dictionary of Units of Measurement by Russ Rowlett
What's Special About This Number? (from 0 to 9999)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[Rosen-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.

[mw-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.

[ShipAssistant-5] 1 2 3 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.

[David_Wells_page_27-7] 1 2 3 "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 27.

[David_Wells_page_29-8] 1 2 "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 A019692.

[14] A065473

[15] Weisstein, Eric W. "Gauss–Kuzmin–Wirsing Constant". MathWorld.

[16] A065464

[17] A065478

[18] A065493

[19] A175639

[20] Weisstein, Eric W. "Continued Fraction Constant". Wolfram Research, Inc. Archived from the original on 2011-10-24.

[21] A065476

[22] A065465

[23] "The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33

[24] A065483

[25] A082695

[26] A166928

[27] A175640

[28] A065485

[29] A163973

[30] "Bags of Talent, a Touch of Panic, and a Bit of Luck: The Case of Non-Numerical Vague Quantifiers" from Linguista Pragensia, Nov. 2, 2010 Archived 2012-07-31 at Archive.is

[31] Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"