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List of numbers

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This is adynamic list and may never be able to satisfy particular standards for completeness. You can help byediting the page to add missing items, with references toreliable sources.

This is a list of notablenumbers and articles about them. The list does not contain all numbers in existence as most of thenumber sets are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities that could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as theinteresting number paradox.

The definition of what is classed as a number is rather diffuse and based on historical distinctions. For example, the pair of numbers (3,4) is commonly regarded as a number when it is in the form of acomplex number (3+4i), but not when it is in the form of avector (3,4). This list will also be categorized with the standard convention oftypes of numbers.

This list focuses on numbers asmathematical objects and isnot a list ofnumerals, which are linguistic devices: nouns, adjectives, or adverbs thatdesignate numbers. The distinction is drawn between thenumber five (anabstract object equal to 2+3), and thenumeral five (thenoun referring to the number).

Natural numbers

Main article:Natural number

Natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used forcounting and often have ethno-cultural significance (see below). Beyond this, natural numbers are widely used as a building block for other number systems including theintegers,rational numbers andreal numbers. Natural numbers are those used forcounting (as in "there aresix (6) coins on the table") andordering (as in "this is thethird (3rd) largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers". Defined by thePeano axioms, the natural numbers form an infinitely large set. Often referred to as "the naturals", the natural numbers are usually symbolised by a boldfaceN (orblackboard bold $\mathbb {\mathbb {N} }$ , UnicodeU+2115 ℕDOUBLE-STRUCK CAPITAL N).

The inclusion of0 in the set of natural numbers is ambiguous and subject to individual definitions. Inset theory andcomputer science, 0 is typically considered a natural number. Innumber theory, it usually is not. The ambiguity can be solved with the terms "non-negative integers", which includes 0, and "positive integers", which does not.

Natural numbers may be used ascardinal numbers, which may go byvarious names. Natural numbers may also be used asordinal numbers.

Table of notable 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	261	262	263	264	265	266	267	268	269
270	271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288	289
290	291	292	293	294	295	296	297	298	299
300	301	302	303	304	305	306	307	308	309
310	311	312	313	314	315	316		318
			323		325
	341
			353						359
360			363		365				369
							377
				384
400
420
440
					495	496
500	501
	511	512
					555
600
610			613			616
						666
			693
700
720
			743	744
							777
						786
800	801
						836
840
880	881							888
900
	911
	971
							987
									999
1000	1001
			1023	1024
									1089
			1093
					1105
				1234
									1289
								1458
		2000	3000	4000	5000	6000	7000	8000	9000
	10,000	20,000	30,000	40,000	50,000	60,000	70,000	80,000	90,000
10⁵	10⁶	10⁷	10⁸	10⁹	10¹⁰	10¹¹	10¹²	10¹³
larger numbers, including10¹⁰⁰ and10^10¹⁰⁰

Mathematical significance

Natural numbers may have properties specific to the individual number or may be part of a set (such as prime numbers) of numbers with a particular property.

List of mathematically significant natural numbers

1, the multiplicative identity. Also the only natural number (not including 0) that is not prime or composite.
2, the base of thebinary number system, used in almost all modern computers and information systems. Also the only natural even number to also be prime.
3, 2²-1, the firstMersenne prime and firstFermat number. It is the first odd prime, and it is also the 2 bit integer maximum value.
4, the firstcomposite number.
5, the sum of the first two primes and only prime which is the sum of 2 consecutive primes. The ratio of the length from the side to a diagonal of a regular pentagon is thegolden ratio.
6, the first of the series ofperfect numbers, whose proper factors sum to the number itself.
7, the first and onlyprime number preceding acube.
8, the largest cube in theFibonacci sequence.
9, the firstodd number that iscomposite.
10, the base of thedecimal system, used in everyday life.
11, the fifth prime and first palindromic multi-digit number in base 10.
12, the firstsublime number.
17, the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.
24, allDirichlet characters modn arereal if and only ifn is a divisor of 24.
25, the firstcentered square number besides 1 that is also a square number.
27, the cube of 3, the value of 3³.
28, the secondperfect number.
30, the smallestsphenic number.
32, the smallest nontrivialfifth power.
36, the smallest number which is aperfect power but not aprime power.
70, the smallestweird number.
72, the smallestAchilles number.
108, the secondAchilles number.
255, 2⁸ − 1, the smallestperfect totient number that is neither a power of three nor thrice a prime; it is also the largest number that can be represented using an8-bit unsignedinteger.
341, the smallest base 2Fermat pseudoprime.
496, the thirdperfect number.
1729, theHardy–Ramanujan number, also known as the secondtaxicab number; that is, the smallest positive integer that can be written as the sum of two positive cubes in two different ways.^[1]
8128, the fourth perfect number.
142857, the smallestbase 10 cyclic number.
9814072356, the largestperfect power that contains no repeated digits in base ten.

Cultural or practical significance

Along with their mathematical properties, many integers havecultural significance^[2] or are also notable for their use in computing and measurement. As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.

List of integers notable for their cultural meanings

3, significant inChristianity as theTrinity. Also considered significant inHinduism (Trimurti,Tridevi). Holds significance in a number of ancient mythologies.
4, considered an"unlucky" number in modern China, Japan and Korea due to its audible similarity to the word "death" in their respective languages.
7, the number of days in a week, and considered a "lucky" number in Western cultures.
8, considered a"lucky" number in Chinese culture due to its aural similarity to the Chinese term for prosperity.
12, a common grouping known as adozen and the number of months in a year, of constellations of theZodiac andastrological signs and ofApostles ofJesus.
13, considered an"unlucky" number in Western superstition. Also known as a "Baker's dozen".^[3]
17, consideredill-fated in Italy and other countries of Greek and Latin origins.
18, considered a "lucky" number due to it being the value for theHebrew word for life inJewish numerology.
40, considered a significant number inTengrism and Turkish folklore. Multiple customs, such as those relating to how many days one must visit someone after a death in the family, include the number forty.
42, the "answer to the ultimate question of life, the universe, and everything" in the popular 1979 science fiction workThe Hitchhiker's Guide to the Galaxy.
69, a slang term for reciprocaloral sex.
86, a slang term that is used in the American popular culture as a transitive verb to mean throw out or get rid of.^[4]
108, considered sacred by theDharmic religions. Approximately equal to the ratio of the distance from Earth to Sun and diameter of the Sun.
420, a code-term that refers to the consumption ofcannabis.
666, thenumber of the beast from theBook of Revelation.
786, regarded as sacred in the MuslimAbjad numerology.
5040, mentioned byPlato in theLaws as one of the most important numbers for the city.

List of integers notable for their use in units, measurements and scales

10, the number of digits in thedecimal number system.
12, thenumber base for measuring time in many civilizations.
14, the number of days in afortnight.
16, the number of digits in thehexadecimal number system.
24, number ofhours in aday.
31, the number of days most months of the year have.
60, thenumber base for some ancient counting systems, such as theBabylonians', and the basis for many modern measuring systems.
360, the number ofsexagesimal degrees in a fullcircle.
365, the number of days in the common year, while there are 366 days in aleap year of the solarGregorian calendar.
1000, the scale factor of mostmetric prefixes.

List of integers notable in computing

2, radix ofbinary numbers used in most digital electronics.
4, the number ofbits in anibble.
8, the number of bits in anoctet and usually in abyte.
16, used as aradix inhexadecimal notation. Also frequently usedmemory bus width (in bits) in older systems.
32 and64, typicalmemory bus widths (in bits) in contemporary systems.
256, The number of possible combinations within8 bits, or an octet.
1024, the number of bytes in akibibyte, and bits in akibibit.
65535, 2¹⁶ − 1, the maximum value of a16-bit unsigned integer.
65536, 2¹⁶, the number of possible16-bit combinations.
65537, 2¹⁶ + 1, the most popular RSA public key prime exponent in most SSL/TLS certificates on the Web/Internet.
16777216, 2²⁴, or 16⁶; the hexadecimal "million" (0x1000000), and the total number of possible color combinations in 24/32-bitTrue Color computer graphics.
2147483647, 2³¹ − 1, the maximum value of a32-bit signed integer usingtwo's complement representation.
9223372036854775807, 2⁶³ − 1, the maximum value of a64-bit signed integer usingtwo's complement representation.

Classes of natural numbers

Subsets of the natural numbers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members. Infinitely many such sets are possible. A list of notable classes of natural numbers may be found atclasses of natural numbers.

Prime numbers

Main articles:Prime number andList of prime numbers

A prime number is a positive integer which has exactly twodivisors: 1 and itself.

The first 100 prime numbers are:

Table of 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

Highly composite numbers

Main article:Highly composite number

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used ingeometry, 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

Main article:Perfect number

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

The first 10 perfect numbers:

6
28
496
8128
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

Integers

Main article:Integer

The integers are aset of numbers commonly encountered inarithmetic andnumber theory. There are manysubsets of the integers, including thenatural numbers,prime numbers,perfect numbers, etc. Many integers are notable for their mathematical properties. Integers are usually symbolised by a boldfaceZ (orblackboard bold $\mathbb {\mathbb {Z} }$ , UnicodeU+2124 ℤDOUBLE-STRUCK CAPITAL Z); this became the symbol for the integers based on the German word for "numbers" (Zahlen).

Notable integers include−1, the additive inverse of unity, and0, theadditive identity.

As with the natural numbers, the integers may also have cultural or practical significance. For instance,−40 is the equal point in theFahrenheit andCelsius scales.

SI prefixes

One important use of integers is inorders of magnitude. Apower of 10 is a number 10^k, wherek is an integer. For instance, withk = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance,k = -3 gives 1/1000, or 0.001. This is used inscientific notation, real numbers are written in the formm × 10ⁿ. The number 394,000 is written in this form as 3.94 × 10⁵.

Integers are used asprefixes in theSI system. Ametric prefix is aunit prefix that precedes a basic unit of measure to indicate amultiple orfraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefixkilo-, for example, may be added togram to indicatemultiplication by one thousand: one kilogram is equal to one thousand grams. The prefixmilli-, likewise, may be added tometre to indicatedivision by one thousand; one millimetre is equal to one thousandth of a metre.

Value	1000^m	Name	Symbol
1000	1000¹	Kilo	k
1000000	1000²	Mega	M
1000000000	1000³	Giga	G
1000000000000	1000⁴	Tera	T
1000000000000000	1000⁵	Peta	P
1000000000000000000	1000⁶	Exa	E
1000000000000000000000	1000⁷	Zetta	Z
1000000000000000000000000	1000⁸	Yotta	Y
1000000000000000000000000000	1000⁹	Ronna	R
1000000000000000000000000000000	1000¹⁰	Quetta	Q

Rational numbers

Main article:Rational number

A rational number is any number that can be expressed as thequotient orfractionp/q of twointegers, anumeratorp and a non-zerodenominatorq.^[5] Sinceq may be equal to 1, every integer is trivially a rational number. Theset 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 boldfaceQ (orblackboard bold $\mathbb {Q}$ , UnicodeU+211A ℚDOUBLE-STRUCK CAPITAL Q);^[6] it was thus denoted in 1895 byGiuseppe Peano afterquoziente, Italian for "quotient".

Rational numbers such as 0.12 can be represented ininfinitely many ways, e.g.zero-point-one-two (0.12),three twenty-fifths (⁠3/25⁠),nine seventy-fifths (⁠9/75⁠), etc. This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction.

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

Table of notable rational numbers
Decimal expansion	Fraction	Notability
1.0	⁠1/1⁠	One is the multiplicative identity. One is a rational number, as it is equal to 1/1.
1	⁠1/1⁠
−0.083 333...	⁠−+1/12⁠	The value assigned to the series1+2+3... byzeta function regularization andRamanujan summation.
0.5	⁠1/2⁠	One half occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle:⁠1/2⁠ × base × perpendicular height and in the formulae forfigurate numbers, such astriangular numbers andpentagonal numbers.
3.142 857...	⁠22/7⁠	A widely used approximation for the number $\pi$ . It can beproven that this number exceeds $\pi$ .
0.166 666...	⁠1/6⁠	One sixth. Often appears in mathematical equations, such as in thesum of squares of the integers and in the solution to the Basel problem.

Real numbers

Real numbers are least upper bounds of sets of rational numbers that are bounded above, or greatest lower bounds of sets of rational numbers that are bounded below, or limits of convergent sequences of rational numbers. Real numbers that are not rational numbers are calledirrational numbers. The real numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not; all rational numbers are algebraic.

Algebraic numbers

Main article:Algebraic number

Name	Expression	Decimal expansion	Notability
Golden ratio conjugate ( $\Phi$ )	${\frac {{\sqrt {5}}-1}{2}}$	0.618033988749894848204586834366	Reciprocal of (and one less than) thegolden ratio.
Twelfth root of two	${\sqrt[{12}]{2}}$	1.059463094359295264561825294946	Proportion between the frequencies of adjacentsemitones in the12 tone equal temperament scale.
Cube root of two	${\sqrt[{3}]{2}}$	1.259921049894873164767210607278	Length of the edge of acube with volume two. Seedoubling the cube for the significance of this number.
Conway's constant	(cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots)	1.303577269034296391257099112153	Defined as the unique positive real root of a certain polynomial of degree 71. The limit ratio between subsequent numbers in the binaryLook-and-say sequence (OEIS: A014715).
Plastic ratio	${\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}$	1.324717957244746025960908854478	The only real solution of $x^{3}=x+1$ .(OEIS: A060006) The limit ratio between subsequent numbers in theVan der Laan sequence. (OEIS: A182097)
Square root of two	${\sqrt {2}}$	1.414213562373095048801688724210	√2 = 2 sin 45° = 2 cos 45°Square root of two a.k.a.Pythagoras' constant. Ratio ofdiagonal to side length in asquare. Proportion between the sides ofpaper sizes in theISO 216 series (originallyDIN 476 series).
Supergolden ratio	${\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {3\cdot 31}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {3\cdot 31}}}{2}}}}{3}}$	1.465571231876768026656731225220	The only real solution of $x^{3}=x^{2}+1$ .(OEIS: A092526) The limit ratio between subsequent numbers inNarayana's cows sequence. (OEIS: A000930)
Triangular root of 2	${\frac {{\sqrt {17}}-1}{2}}$	1.561552812808830274910704927987
Golden ratio (φ)	${\frac {{\sqrt {5}}+1}{2}}$	1.618033988749894848204586834366	The larger of the two real roots ofx² =x + 1.
Square root of three	${\sqrt {3}}$	1.732050807568877293527446341506	√3 = 2 sin 60° = 2 cos 30° . A.k.a.the measure of the fish or Theodorus' constant. Length of thespace diagonal of acube with edge length 1.Altitude of anequilateral triangle with side length 2. Altitude of aregular hexagon with side length 1 and diagonal length 2.
Tribonacci constant	${\frac {1+{\sqrt[{3}]{19+3{\sqrt {3\cdot 11}}}}+{\sqrt[{3}]{19-3{\sqrt {3\cdot 11}}}}}{3}}$	1.839286755214161132551852564653	The only real solution of $x^{3}=x^{2}+x+1$ .(OEIS: A058265) The limit ratio between subsequent numbers in theTribonacci sequence.(OEIS: A000073) Appears in the volume and coordinates of thesnub cube and some related polyhedra.
Supersilver ratio	${\dfrac {2+{\sqrt[{3}]{\dfrac {43+3{\sqrt {3\cdot 59}}}{2}}}+{\sqrt[{3}]{\dfrac {43-3{\sqrt {3\cdot 59}}}{2}}}}{3}}$	2.20556943040059031170202861778	The only real solution of $x^{3}=2x^{2}+1$ .(OEIS: A356035) The limit ratio between subsequent numbers in thethird-order Pell sequence. (OEIS: A008998)
Square root of five	${\sqrt {5}}$	2.236067977499789696409173668731	Length of thediagonal of a 1 × 2rectangle.
Silver ratio (δ_S)	${\sqrt {2}}+1$	2.414213562373095048801688724210	The larger of the two real roots ofx² = 2x + 1. Altitude of aregular octagon with side length 1.
Bronze ratio (S₃)	${\frac {{\sqrt {13}}+3}{2}}$	3.302775637731994646559610633735	The larger of the two real roots ofx² = 3x + 1.

Transcendental numbers

Main article:Transcendental number

Name	Symbol or Formula	Decimal expansion	Notes and notability
Gelfond's constant	$e^{\pi }$	23.14069263277925...
Ramanujan's constant	$e^{\pi {\sqrt {163}}}$	262537412640768743.99999999999925...
Gaussian integral	${\sqrt {\pi }}$	1.772453850905516...
Komornik–Loreti constant	$q {\displaystyle q}$	1.787231650...
Universal parabolic constant	$P_{2}$	2.29558714939...
Gelfond–Schneider constant	$2^{\sqrt {2}}$	2.665144143...
Euler's number	$e {\displaystyle e}$	2.718281828459045235360287471352662497757247...	Raising e to the power of $i {\displaystyle i}$ π will result in $-1$ .
Pi	$\pi$	3.141592653589793238462643383279502884197169399375...	Pi is a constant irrational number that is the result of dividing the circumference of a circle by its diameter.
Super square-root of 2	${\textstyle {\sqrt {2}}_{s}}$ ^[7]	1.559610469...^[8]
Liouville constant	${\textstyle L}$	0.110001000000000000000001000...	The decimal number with 1 in thenth position after the decimal point ifn is afactorial and 0 elsewhere.
Champernowne constant	${\textstyle C_{10}}$	0.12345678910111213141516...	This constant contains every number string inside it, as its decimals are just every number in order. (1,2,3,etc.)
Prouhet–Thue–Morse constant	${\textstyle \tau }$	0.412454033640...
Omega constant	$\Omega$	0.5671432904097838729999686622...
Cahen's constant	${\textstyle C}$	0.64341054629...
Natural logarithm of 2	ln 2	0.693147180559945309417232121458
Lemniscate constant	${\textstyle \varpi }$	2.622057554292119810464839589891...	The ratio of the perimeter ofBernoulli's lemniscate to its diameter.
Tau	$\tau =2\pi$	6.283185307179586476925286766559...	The ratio of thecircumference to aradius, and the number ofradians in a complete circle;^[9]^[10] 2 $\times$ π

Irrational but not known to be transcendental

Some numbers are known to beirrational numbers, but have not been proven to be transcendental. This differs from the algebraic numbers, which are known not to be transcendental.

Name	Decimal expansion	Proof of irrationality	Reference of unknown transcendentality
ζ(3), also known asApéry's constant	1.202056903159594285399738161511449990764986292	^[11]	^[12]
Erdős–Borwein constant, E	1.606695152415291763...	^[13]^[14]	^{[citation needed]}
Copeland–Erdős constant	0.235711131719232931374143...	Can be proven withDirichlet's theorem on arithmetic progressions orBertrand's postulate (Hardy and Wright, p. 113) orRamare's theorem that every even integer is a sum of at most six primes. It also follows directly from its normality.	^{[citation needed]}
Prime constant, ρ	0.414682509851111660248109622...	Proof of the number's irrationality is given atprime constant.	^{[citation needed]}
Reciprocal Fibonacci constant, ψ	3.359885666243177553172011302918927179688905133731...	^[15]^[16]	^[17]

Real but not known to be irrational, nor transcendental

For some numbers, it is not known whether they are algebraic or transcendental. The following list includes real numbers that have not been proved to be irrational, nor transcendental.

Name and symbol	Decimal expansion	Notes
Euler–Mascheroni constant, γ	0.577215664901532860606512090082...^[18]	Believed to be transcendental but not proven to be so. However, it was shown that at least one of $\gamma$ and the Euler-Gompertz constant $\delta$ is transcendental.^[19]^[20] It was also shown that all but at most one number in an infinite list containing ${\frac {\gamma }{4}}$ have to be transcendental.^[21]^[22]
Euler–Gompertz constant, δ	0.596 347 362 323 194 074 341 078 499 369...^[23]	It was shown that at least one of the Euler-Mascheroni constant $\gamma$ and the Euler-Gompertz constant $\delta$ is transcendental.^[19]^[20]
Catalan's constant, G	0.915965594177219015054603514932384110774...	It is not known whether this number is irrational.^[24]
Khinchin's constant, K₀	2.685452001...^[25]	It is not known whether this number is irrational.^[26]
1stFeigenbaum constant, δ	4.6692...	Both Feigenbaum constants are believed to betranscendental, although they have not been proven to be so.^[27]
2ndFeigenbaum constant, α	2.5029...	Both Feigenbaum constants are believed to betranscendental, although they have not been proven to be so.^[27]
Glaisher–Kinkelin constant, A	1.28242712...
Backhouse's constant	1.456074948...
Fransén–Robinson constant, F	2.8077702420...
Lévy's constant,β	1.18656 91104 15625 45282...
Mills' constant, A	1.30637788386308069046...	It is not known whether this number is irrational.(Finch 2003)
Ramanujan–Soldner constant, μ	1.451369234883381050283968485892027449493...
Sierpiński's constant, K	2.5849817595792532170658936...
Totient summatory constant	1.339784...^[28]
Vardi's constant, E	1.264084735305...
Somos' quadratic recurrence constant, σ	1.661687949633594121296...
Niven's constant, C	1.705211...
Brun's constant, B₂	1.902160583104...	The irrationality of this number would be a consequence of the truth of the infinitude oftwin primes.
Landau's totient constant	1.943596...^[29]
Brun's constant for prime quadruplets, B₄	0.8705883800...
Viswanath's constant	1.1319882487943...
Khinchin–Lévy constant	1.1865691104...^[30]	This number represents the probability that three random numbers have nocommon factor greater than 1.^[31]
Landau–Ramanujan constant	0.76422365358922066299069873125...
C(1)	0.77989340037682282947420641365...
Z(1)	−0.736305462867317734677899828925614672...
Heath-Brown–Moroz constant, C	0.001317641...
Kepler–Bouwkamp constant,K'	0.1149420448...
MRB constant,S	0.187859...	It is not known whether this number is irrational.
Meissel–Mertens constant, M	0.2614972128476427837554268386086958590516...
Bernstein's constant, β	0.2801694990...
Gauss–Kuzmin–Wirsing constant, λ₁	0.3036630029...^[32]
Hafner–Sarnak–McCurley constant,σ	0.3532363719...
Artin's constant,C_Artin	0.3739558136...
S(1)	0.438259147390354766076756696625152...
F(1)	0.538079506912768419136387420407556...
Stephens' constant	0.575959...^[33]
Golomb–Dickman constant, λ	0.62432998854355087099293638310083724...
Twin prime constant, C₂	0.660161815846869573927812110014...
Feller–Tornier constant	0.661317...^[34]
Laplace limit, ε	0.6627434193...^[35]
Embree–Trefethen constant	0.70258...

Numbers not known with high precision

Some real numbers, including transcendental numbers, are not known with high precision.

The constant in theBerry–Esseen Theorem: 0.4097 <C < 0.4748
De Bruijn–Newman constant: 0 ≤ Λ ≤ 0.2
Chaitin's constants Ω, which are transcendental and provably impossible to compute.
Bloch's constant (also2nd Landau'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: 1.67 <k < 1.79
Romanov's constant inRomanov's theorem: 0.107648 <d < 0.49094093, Romanov conjectured that it is 0.434

Hypercomplex numbers

Main article:Hypercomplex number

Hypercomplex number is a term for anelement of a unitalalgebra over thefield ofreal numbers. Thecomplex numbers are often symbolised by a boldfaceC (orblackboard bold $\mathbb {\mathbb {C} }$ , UnicodeU+2102 ℂDOUBLE-STRUCK CAPITAL C), while the set ofquaternions is denoted by a boldfaceH (orblackboard bold $\mathbb {H}$ , UnicodeU+210D ℍDOUBLE-STRUCK CAPITAL H).

Algebraic complex numbers

Imaginary unit: ${\textstyle i={\sqrt {-1}}}$
nthroots of unity: ${\textstyle \xi _{n}^{k}=\cos {\bigl (}2\pi {\frac {k}{n}}{\bigr )}+i\sin {\bigl (}2\pi {\frac {k}{n}}{\bigr )}}$ , while ${\textstyle 0\leq k\leq n-10}$ ,GCD(k,n) = 1

Other hypercomplex numbers

Transfinite numbers

Main article:Transfinite number

Transfinite numbers are numbers that are "infinite" in the sense that they are larger than allfinite numbers, yet not necessarilyabsolutely infinite.

Aleph-null: ℵ₀, the smallest infinite cardinal, and the cardinality of $\mathbb {N}$ , the set ofnatural numbers
Aleph-one: ℵ₁, the cardinality of ω₁, the set of all countable ordinal numbers
Beth-one: $\beth _{1}$ or ${\mathfrak {c}}$ , thecardinality of the continuum 2^ℵ₀
Omega: ω, the smallestinfinite ordinal

Numbers representing physical quantities

Main articles:Physical constant andList of physical constants

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

Avogadro constant:N_A = 6.02214076×10²³ mol⁻¹‍^[36]
Electron mass:m_e = 9.1093837139(28)×10⁻³¹ kg‍^[37]
Fine-structure constant:α = 0.0072973525643(11)‍^[38]
Gravitational constant:G = 6.67430(15)×10⁻¹¹ m³⋅kg⁻¹⋅s⁻²‍^[39]
Molar mass constant:M_u = 1.00000000105(31)×10⁻³ kg⋅mol⁻¹‍^[40]
Planck constant:h = 6.62607015×10⁻³⁴ J⋅Hz⁻¹‍^[41]
Rydberg constant:R_∞ = 10973731.568157(12) m⁻¹‍^[42]
Speed of light in vacuum:c = 299792458 m⋅s⁻¹‍^[43]
Vacuum permittivity:ε₀ = 8.8541878188(14)×10⁻¹² F⋅m⁻¹‍^[44]

Numbers representing geographical and astronomical distances

6378.137, the average equatorial radius of Earth inkilometers (followingGRS 80 andWGS 84 standards).
40075.0167, the length of theEquator in kilometers (following GRS 80 and WGS 84 standards).
384399, the semi-major axis of theorbit of the Moon, in kilometers, roughly the distance between the center of Earth and that of the Moon.
149597870700, the average distance between the Earth and the Sun orAstronomical Unit (AU), in meters.
9460730472580800, onelight-year, the distance travelled by light in oneJulian year, in meters.
30856775814913673, the distance of oneparsec, another astronomical unit, in whole meters.

Numbers without specific values

Main article:Indefinite and fictitious numbers

Many languages have words expressingindefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, asplaceholder names, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier".^[45] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".^[46]

Named numbers

References

^Weisstein, Eric W."Hardy–Ramanujan Number".Archived from the original on 2004-04-08.
^Ayonrinde, Oyedeji A.; Stefatos, Anthi; Miller, Shadé; Richer, Amanda; Nadkarni, Pallavi; She, Jennifer; Alghofaily, Ahmad; Mngoma, Nomusa (2020-06-12). "The salience and symbolism of numbers across cultural beliefs and practice".International Review of Psychiatry.33 (1–2):179–188.doi:10.1080/09540261.2020.1769289.ISSN 0954-0261.PMID 32527165.S2CID 219605482.
^"Demystified | Why a baker's dozen is thirteen".www.britannica.com. Retrieved2024-06-05.
^"Eighty-six – Definition of eighty-six".Merriam-Webster.Archived from the original on 2013-04-08.
^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". Retrieved1 April 2015.
^Lipscombe, Trevor Davis (2021-05-06),"Super Powers: Calculate Squares, Square Roots, Cube Roots, and More",Quick(er) Calculations, Oxford University Press, pp. 103–124,doi:10.1093/oso/9780198852650.003.0010,ISBN 978-0-19-885265-0, retrieved2021-10-28
^"Nick's Mathematical Puzzles: Solution 29".Archived from the original on 2011-10-18.
^"The Penguin Dictionary of Curious and Interesting Numbers" byDavid Wells, page 69
^SequenceOEIS: A019692.
^SeeApéry 1979.
^"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
^Erdős, P. (1948),"On arithmetical properties of Lambert series"(PDF),J. Indian Math. Soc., New Series,12:63–66,MR 0029405
^Borwein, Peter B. (1992), "On the irrationality of certain series",Mathematical Proceedings of the Cambridge Philosophical Society,112 (1):141–146,Bibcode:1992MPCPS.112..141B,CiteSeerX 10.1.1.867.5919,doi:10.1017/S030500410007081X,MR 1162938,S2CID 123705311
^André-Jeannin, Richard; 'Irrationalité de la somme des inverses de certaines suites récurrentes.';Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 308, issue 19 (1989), pp. 539-541.
^S. Kato, 'Irrationality of reciprocal sums of Fibonacci numbers', Master's thesis, Keio Univ. 1996
^Duverney, Daniel, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa; 'Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers';
^"A001620 - OEIS".oeis.org. Retrieved2020-10-14.
^^a ^bRivoal, Tanguy (2012)."On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant".Michigan Mathematical Journal.61 (2):239–254.doi:10.1307/mmj/1339011525.ISSN 0026-2285.
^^a ^bLagarias, Jeffrey C. (2013-07-19)."Euler's constant: Euler's work and modern developments".Bulletin of the American Mathematical Society.50 (4):527–628.arXiv:1303.1856.doi:10.1090/S0273-0979-2013-01423-X.ISSN 0273-0979.
^Murty, M. Ram; Saradha, N. (2010-12-01)."Euler–Lehmer constants and a conjecture of Erdös".Journal of Number Theory.130 (12):2671–2682.CiteSeerX 10.1.1.261.753.doi:10.1016/j.jnt.2010.07.004.ISSN 0022-314X.
^Murty, M. Ram; Zaytseva, Anastasia (2013-01-01)."Transcendence of Generalized Euler Constants".The American Mathematical Monthly.120 (1):48–54.doi:10.4169/amer.math.monthly.120.01.048.ISSN 0002-9890.S2CID 20495981.
^"A073003 - OEIS".oeis.org. Retrieved2020-10-14.
^Nesterenko, Yu. V. (January 2016), "On Catalan's constant",Proceedings of the Steklov Institute of Mathematics,292 (1):153–170,doi:10.1134/s0081543816010107,S2CID 124903059
^"Khinchin's Constant".
^Weisstein, Eric W."Khinchin's constant".MathWorld.
^^a ^bBriggs, Keith (1997).Feigenbaum scaling in discrete dynamical systems(PDF) (PhD thesis).University of Melbourne.
^OEIS: A065483
^OEIS: A082695
^"Lévy Constant".
^"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
^Weisstein, Eric W."Gauss–Kuzmin–Wirsing Constant".MathWorld.
^OEIS: A065478
^OEIS: A065493
^"Laplace Limit".
^"2022 CODATA Value: Avogadro constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: electron mass".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: fine-structure constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: Newtonian constant of gravitation".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: molar mass constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: Planck constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: Rydberg constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: speed of light in vacuum".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"2022 CODATA Value: vacuum electric permittivity".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
^"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 atarchive.today
^Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"

Finch, Steven R. (2003), "Anmol Kumar Singh",Mathematical Constants (Encyclopedia of Mathematics and its Applications, Series Number 94), Cambridge University Press, pp. 130–133,ISBN 0521818052
Apéry, Roger (1979), "Irrationalité de $\zeta (2)$ et $\zeta (3)$ ",Astérisque,61:11–13.

External links

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 theWayback Machine (archived 27 November 2010)
Robert P. Munafo's Large Numbers page
Different notations for big numbers – by Susan Stepney
Names for Large Numbers, inHow Many? A Dictionary of Units of Measurement by Russ Rowlett
What's Special About This Number? (from 0 to 9999)

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