List Of Numbers

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Contents hide
* (Top)
* Natural numbers Toggle Natural numbers subsection 
** Mathematical significance
**
** Cultural or practical significance
**
* Classes of natural numbers Toggle Classes of natural numbers subsection 
** Prime numbers
**
** Highly composite numbers
**
** Perfect numbers
**
* Integers Toggle Integers subsection 
** SI prefixes
**
* Rational numbers
*
* Irrational numbers Toggle Irrational numbers subsection 
** Algebraic numbers
**
** Transcendental numbers
**
** Irrational but not known to be transcendental
**
* Real numbers Toggle Real numbers subsection 
** Real but not known to be irrational, nor transcendental
**
** Numbers not known with high precision
**
* Hypercomplex numbers Toggle Hypercomplex numbers subsection 
** Algebraic complex numbers
**
** Other hypercomplex numbers
**
* Transfinite numbers
*
* Numbers representing physical quantities
*
* Numbers representing geographical and astronomical distances
*
* Numbers without specific values
*
* Named numbers
*
* See also
*
* References 
*
* Further reading
*
* External links
*
List of numbers
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From Wikipedia, the free encyclopedia
This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.
This is a list of notable numbers and articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities which could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as the interesting 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 a complex number (3+4i), but not when it is in the form of a vector (3,4). This list will also be categorised with the standard convention of types of numbers.
This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five (the noun referring to the number).
Natural numbers
Main article: Natural number
The natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used for counting and often have ethno-cultural significance (see below). Beyond this, natural numbers are widely used as a building block for other number systems including the integers, rational numbers and real 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". Defined by the Peano axioms, the natural numbers form an infinitely large set. Often referred to as "the naturals", the natural numbers are usually symbolised by a boldface N (or blackboard bold
N
{\displaystyle \mathbb {\mathbb {N} } }
, Unicode U+2115 ℕ DOUBLE-STRUCK CAPITAL N).
The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions. In set theory and computer science, 0 is typically considered a natural number. In number 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 as cardinal numbers, which may go by various names. Natural numbers may also be used as ordinal numbers.
Table of small 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 	
310 	311 	312 	313 	 	 	 	 	318 	
 	 	 	 	400 	500 	600 	700 	800 	900
 	1000 	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 	1012 	1012 	1012 	1012 	1012
larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100	larger numbers, including 10100 and 1010100
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 the binary number system, used in almost all modern computers and information systems.
* 3, 22-1, the first Mersenne prime. It is the first odd prime, and it is also the 2 bit integer maximum value.
* 4, the first composite number.
* 6, the first of the series of perfect numbers, whose proper factors sum to the number itself.
* 9, the first odd number that is composite.
* 11, the fifth prime and first palindromic multi-digit number in base 10.
* 12, the first sublime number.
* 17, the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.
* 24, all Dirichlet characters mod n are real if and only if n is a divisor of 24.
* 25, the first centered square number besides 1 that is also a square number.
* 27, the cube of 3, the value of 33.
* 28, the second perfect number.
* 30, the smallest sphenic number.
* 32, the smallest nontrivial fifth power.
* 36, the smallest number which is a perfect power but not a prime power.
* 70, the smallest weird number.
* 72, the smallest Achilles number.
* 108, the second Achilles number.
* 255, 28 − 1, 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.
* 341, the smallest base 2 Fermat pseudoprime.
* 496, the third perfect number.
* 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.[1]
* 8128, the fourth perfect number.
* 142857, the smallest base 10 cyclic number.
* 9814072356, the largest perfect power that contains no repeated digits in base ten.
Cultural or practical significance
Along with their mathematical properties, many integers have cultural 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 in Christianity as the Trinity. Also considered significant in Hinduism (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 a dozen and the number of months in a year, of constellations of the Zodiac and astrological signs and of Apostles of Jesus.
* 13, considered an "unlucky" number in Western superstition. Also known as a "Baker's dozen".[citation needed]
* 17, considered ill-fated in Italy and other countries of Greek and Latin origins.
* 18, considered a "lucky" number due to it being the value for life in Jewish numerology.
* 40, considered a significant number in Tengrism 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 work The Hitchhiker's Guide to the Galaxy.
* 69, a slang term for reciprocal oral 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.[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.
* 420, a code-term that refers to the consumption of cannabis.
* 666, the number of the beast from the Book of Revelation.
* 786, regarded as sacred in the Muslim Abjad numerology.
* 5040, mentioned by Plato in the Laws 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 the decimal number system.
* 12, the number base for measuring time in many civilizations.
* 14, the number of days in a fortnight.
* 16, the number of digits in the hexadecimal number system.
* 24, number of hours in a day.
* 31, the number of days most months of the year have.
* 60, the number base for some ancient counting systems, such as the Babylonians', and the basis for many modern measuring systems.
* 360, the number of sexagesimal degrees in a full circle.
* 365, the number of days in the common year, while there are 366 days in a leap year of the solar Gregorian calendar.
List of integers notable in computing
* 4, the number of bits in a nibble.
* 8, the number of bits in an octet and usually in a byte.
* 256, The number of possible combinations within 8 bits, or an octet.
* 1024, the number of bytes in a kibibyte, and bits in a kibibit.
* 65535, 216 − 1, the maximum value of a 16-bit unsigned integer.
* 65536, 216, the number of possible 16-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-bit True Color computer graphics.
* 2147483647, 231 − 1, the maximum value of a 32-bit signed integer using two's complement representation.
* 9223372036854775807, 263 − 1, the maximum value of a 64-bit signed integer using two'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 at classes of natural numbers.
Prime numbers
Main articles: Prime number and List of prime numbers
A prime number is a positive integer which has exactly two divisors: 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 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
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 a set of numbers commonly encountered in arithmetic and number theory. There are many subsets of the integers, including the natural numbers, prime numbers, perfect numbers, etc. Many integers are notable for their mathematical properties. Integers are usually symbolised by a boldface Z (or blackboard bold
Z
{\displaystyle \mathbb {\mathbb {Z} } }
, Unicode U+2124 ℤ DOUBLE-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, and 0, the additive identity.
As with the natural numbers, the integers may also have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.
SI prefixes
One important use of integers is in orders of magnitude. A power of 10 is a number 10k, where k is an integer. For instance, with k = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, k = -3 gives 1/1000, or 0.001. This is used 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 are used as prefixes in the SI system. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.
Value 	1000m 	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 the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[4] Since q may be equal to 1, every integer is trivially a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold
Q
{\displaystyle \mathbb {Q} }
, Unicode U+211A ℚ DOUBLE-STRUCK CAPITAL Q);[5] it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
Rational numbers such as 0.12 can be represented in infinitely 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 at numeral (linguistics).
Table of notable rational numbers
Decimal expansion	Fraction	Notability
1.0 	1
/
1 	One is the multiplicative identity. One is trivially a rational number, as it is equal to 1/1.
1 	1
/
1 	One is the multiplicative identity. One is trivially a rational number, as it is equal to 1/1.
−0.083 333... 	−
+
1
/
12	The value assigned to the series 1+2+3... by zeta function regularization and Ramanujan 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 for figurate numbers, such as triangular numbers and pentagonal numbers.
3.142 857... 	22
/
7	A widely used approximation for the number
π
{\displaystyle \pi }
. It can be proven that this number exceeds
π
{\displaystyle \pi }
.
0.166 666... 	1
/
6 	One sixth. Often appears in mathematical equations, such as in the sum of squares of the integers and in the solution to the Basel problem.
Irrational numbers
Main article: Irrational number
The irrational numbers are a set of numbers that includes all real numbers that are not rational numbers. The irrational numbers are categorised as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not.
Algebraic numbers
Main article: Algebraic number
Name 	Expression 	Decimal expansion 	Notability
Golden ratio conjugate (
Φ
{\displaystyle \Phi }
)	5
−
1
2
{\displaystyle {\frac {{\sqrt {5}}-1}{2}}} 	0.618033988749894848204586834366	Reciprocal of (and one less than) the golden ratio.
Twelfth root of two 	2
12
{\displaystyle {\sqrt[{12}]{2}}} 	1.059463094359295264561825294946	Proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale.
Cube root of two 	2
3
{\displaystyle {\sqrt[{3}]{2}}} 	1.259921049894873164767210607278	Length of the edge of a cube with volume two. See doubling 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 binary Look-and-say sequence (OEIS: A014715).
Plastic ratio 	1
2
+
1
6
23
3
3
+
1
2
−
1
6
23
3
3
{\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}	1.324717957244746025960908854478	The unique real root of the cubic equation x3 = x + 1.(OEIS: A060006) The limit ratio between subsequent numbers in the Van der Laan sequence.(OEIS: A182097)
Square root of two 	2
{\displaystyle {\sqrt {2}}} 	1.414213562373095048801688724210	√2 = 2 sin 45° = 2 cos 45° Square root of two a.k.a. Pythagoras' constant. Ratio of diagonal to side length in a square. Proportion between the sides of paper sizes in the ISO 216 series (originally DIN 476 series).
Supergolden ratio 	1
+
29
+
3
93
2
3
+
29
−
3
93
2
3
3
{\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {93}}}{2}}}}{3}}}	1.465571231876768026656731225220	The only real solution of
x
3
=
x
2
+
1
{\displaystyle x^{3}=x^{2}+1}
.(OEIS: A092526) The limit ratio between subsequent numbers in Narayana's cows sequence.(OEIS: A000930)
Triangular root of 2	17
−
1
2
{\displaystyle {\frac {{\sqrt {17}}-1}{2}}} 	1.561552812808830274910704927987	
Golden ratio (φ) 	5
+
1
2
{\displaystyle {\frac {{\sqrt {5}}+1}{2}}} 	1.618033988749894848204586834366	The larger of the two real roots of x2 = x + 1.
Square root of three	3
{\displaystyle {\sqrt {3}}} 	1.732050807568877293527446341506	√3 = 2 sin 60° = 2 cos 30° . A.k.a. the measure of the fish or Theodorus' constant. Length of the space diagonal of a cube with edge length 1. Altitude of an equilateral triangle with side length 2. Altitude of a regular hexagon with side length 1 and diagonal length 2.
Tribonacci constant 	1
+
19
+
3
33
3
+
19
−
3
33
3
3
{\displaystyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}} 	1.839286755214161132551852564653	Appears in the volume and coordinates of the snub cube and some related polyhedra. It satisfies the equation x + x−3 = 2.
Square root of five 	5
{\displaystyle {\sqrt {5}}} 	2.236067977499789696409173668731	Length of the diagonal of a 1 × 2 rectangle.
Silver ratio (δS) 	2
+
1
{\displaystyle {\sqrt {2}}+1} 	2.414213562373095048801688724210	The larger of the two real roots of x2 = 2x + 1.
Altitude of a regular octagon with side length 1.
Bronze ratio (S3) 	13
+
3
2
{\displaystyle {\frac {{\sqrt {13}}+3}{2}}} 	3.302775637731994646559610633735	The larger of the two real roots of x2 = 3x + 1.
Transcendental numbers
Main article: Transcendental number
Name 	Symbol
or
Formula	Decimal expansion 	Notes and notability
Gelfond's constant 	e
π
{\displaystyle e^{\pi }}	23.14069263277925... 	
Ramanujan's constant 	e
π
163
{\displaystyle e^{\pi {\sqrt {163}}}}	262537412640768743.99999999999925... 	
Gaussian integral 	π
{\displaystyle {\sqrt {\pi }}}	1.772453850905516... 	
Komornik–Loreti constant 	q
{\displaystyle q}	1.787231650... 	
Universal parabolic constant	P
2
{\displaystyle P_{2}}	2.29558714939... 	
Gelfond–Schneider constant 	2
2
{\displaystyle 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
{\displaystyle -1}
.
Pi 	π
{\displaystyle \pi }	3.141592653589793238462643383279502884197169399375...	Pi is an irrational number that is the result of dividing the circumference of a circle by its diameter.
Super square-root of 2 	2
s
{\textstyle {\sqrt {2_{s}}}}
[6]	1.559610469...[7] 	
Liouville constant 	L
{\textstyle L}	0.110001000000000000000001000... 	
Champernowne constant 	C
10
{\textstyle C_{10}}	0.12345678910111213141516... 	
Prouhet–Thue–Morse constant 	τ
{\textstyle \tau }	0.412454033640... 	
Omega constant 	Ω
{\displaystyle \Omega }	0.5671432904097838729999686622... 	
Cahen's constant 	C
{\textstyle C}	0.64341054629... 	
Natural logarithm of 2 	ln 2 	0.693147180559945309417232121458 	
Gauss's constant 	G
{\textstyle G}	0.8346268... 	
Tau 	2π: τ	6.283185307179586476925286766559... 	The ratio of the circumference to a radius, and the number of radians in a complete circle;[8][9] 2
×
{\displaystyle \times }
π
Irrational but not known to be transcendental
Some numbers are known to be irrational 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 as Apéry's constant	1.202056903159594285399738161511449990764986292 	[10] 	[11]
Erdős–Borwein constant, E 	1.606695152415291763... 	[12][13] 	[citation needed]
Copeland–Erdős constant 	0.235711131719232931374143... 	Can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare'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 at prime constant. 	[citation needed]
Reciprocal Fibonacci constant, ψ 	3.359885666243177553172011302918927179688905133731...	[14][15] 	[16]
Real numbers
The real numbers are a superset containing the algebraic and the transcendental numbers. The real numbers, sometimes referred to as "the reals", are usually symbolised by a boldface R (or blackboard bold
R
{\displaystyle \mathbb {\mathbb {R} } }
, Unicode U+211D ℝ DOUBLE-STRUCK CAPITAL R). 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.
Real but not known to be irrational, nor transcendental
Name and symbol 	Decimal expansion 	Notes
Euler–Mascheroni constant, γ 	0.577215664901532860606512090082...[17] 	Believed to be transcendental but not proven to be so. However, it was shown that at least one of
γ
{\displaystyle \gamma }
and the Euler-Gompertz constant
δ
{\displaystyle \delta }
is transcendental.[18][19] It was also shown that all but at most one number in an infinite list containing
γ
4
{\displaystyle {\frac {\gamma }{4}}}
have to be transcendental.[20][21]
Euler–Gompertz constant, δ 	0.596 347 362 323 194 074 341 078 499 369...[22]	It was shown that at least one of the Euler-Mascheroni constant
γ
{\displaystyle \gamma }
and the Euler-Gompertz constant
δ
{\displaystyle \delta }
is transcendental.[18][19]
Catalan's constant, G 	0.915965594177219015054603514932384110774... 	It is not known whether this number is irrational.[23]
Khinchin's constant, K0 	2.685452001...[24] 	It is not known whether this number is irrational.[25]
1st Feigenbaum constant, δ 	4.6692... 	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[26]
2nd Feigenbaum constant, α 	2.5029... 	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[26]
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...[27] 	
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 of twin primes.
Landau's totient constant 	1.943596...[28] 	
Brun's constant for prime quadruplets, B4	0.8705883800... 	
Viswanath's constant 	1.1319882487943... 	
Khinchin–Lévy constant 	1.1865691104...[29] 	This number represents the probability that three random numbers have no common factor greater than 1.[30]
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...[31] 	
Hafner–Sarnak–McCurley constant,σ 	0.3532363719... 	
Artin's constant,CArtin 	0.3739558136... 	
S(1) 	0.438259147390354766076756696625152... 	
F(1) 	0.538079506912768419136387420407556... 	
Stephens' constant 	0.575959...[32] 	
Golomb–Dickman constant, λ 	0.62432998854355087099293638310083724... 	
Twin prime constant, C2 	0.660161815846869573927812110014... 	
Feller–Tornier constant 	0.661317...[33] 	
Laplace limit, ε 	0.6627434193...[34] 	
Embree–Trefethen constant 	0.70258... 	
Numbers not known with high precision
See also: Normal number and Uncomputable number
Some real numbers, including transcendental numbers, are not known with high precision.
* The constant in the Berry–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 (also 2nd 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 in Romanov'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 an element of a unital algebra over the field of real numbers. The complex numbers are often symbolised by a boldface C (or blackboard bold
C
{\displaystyle \mathbb {\mathbb {C} } }
, Unicode U+2102 ℂ DOUBLE-STRUCK CAPITAL C), while the set of quaternions is denoted by a boldface H (or blackboard bold
H
{\displaystyle \mathbb {H} }
, Unicode U+210D ℍ DOUBLE-STRUCK CAPITAL H).
Algebraic complex numbers
* Imaginary unit: i = − 1 {\textstyle i={\sqrt {-1}}} 
* nth roots of unity: ξ n k = cos ⁡ ( 2 π k n ) + i sin ⁡ ( 2 π k n ) {\textstyle \xi _{n}^{k}=\cos {\bigl (}2\pi {\frac {k}{n}}{\bigr )}+i\sin {\bigl (}2\pi {\frac {k}{n}}{\bigr )}} , while 0 ≤ k ≤ n − 10 {\textstyle 0\leq k\leq n-10} , GCD(k, n) = 1
Other hypercomplex numbers
* The quaternions
* The octonions
* The sedenions
* The dual numbers (with an infinitesimal)
Transfinite numbers
Main article: Transfinite number
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 {\displaystyle \mathbb {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 2א0
* ℭ or c {\displaystyle {\mathfrak {c}}} : the cardinality of the continuum 2א0
* Omega: ω, the smallest infinite ordinal
Numbers representing physical quantities
Main articles: Physical constant and List of physical constants
Physical quantities that appear in the universe are often described using physical constants.
* Avogadro constant: NA = 6.02214076×1023 mol−1[35]
* Electron mass: me = 9.1093837015(28)×10−31 kg[36]
* Fine-structure constant: α = 7.2973525693(11)×10−3[37]
* Gravitational constant: G = 6.67430(15)×10−11 m3⋅kg−1⋅s−2[38]
* Molar mass constant: Mu = 0.99999999965(30)×10−3 kg⋅mol−1[39]
* Planck constant: h = 6.62607015×10−34 J⋅Hz−1[40]
* Rydberg constant: R∞ = 10973731.568160(21) m−1[41]
* Speed of light in vacuum: c = 299792458 m⋅s−1[42]
* Vacuum electric permittivity: ε0 = 8.8541878128(13)×10−12 F⋅m−1[43]
Numbers representing geographical and astronomical distances
* 6378.137, the average equatorial radius of Earth in kilometers (following GRS 80 and WGS 84 standards).
* 40075.0167, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards).
* 384399, the semi-major axis of the orbit 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 or Astronomical Unit (AU), in meters.
* 9460730472580800, one light-year, the distance travelled by light in one Julian year, in meters.
* 30856775814913673, the distance of one parsec, another astronomical unit, in whole meters.
Numbers without specific values
Main article: Indefinite and fictitious numbers
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".[44] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".[45]
Named numbers
* Centillion, 10303
* Eddington number, ~1080
* Googol, 10100
* Googolplex, 10(10100)
* Graham's number
* Hardy–Ramanujan number, 1729
* Kaprekar's constant, 6174
* Mega/Circle(2)
* Moser's number
* Rayo's number
* Shannon number
* Skewes's number
* SSCG(3)
* TREE(3)
See also
	Mathematics portal
* Absolute Infinite
* English numerals
* Floating-point arithmetic
* Fraction
* Integer sequence
* Interesting number paradox
* Large numbers
* List of mathematical constants
* List of prime numbers
* List of types of numbers
* Mathematical constant
* Metric prefix
* Names of large numbers
* Names of small numbers
* Negative number
* Numeral (linguistics)
* Numeral prefix
* Order of magnitude
* Orders of magnitude (numbers)
* Ordinal number
* The Penguin Dictionary of Curious and Interesting Numbers
* Power of two
* Power of 10
* Surreal number
* Table of prime factors
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.
# "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". Retrieved 1 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, retrieved 2021-10-28
# "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 OEIS: A019692.
# See Apéry 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. Retrieved 2020-10-14.
# Rivoal, 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.
# Lagarias, 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. Retrieved 2020-10-14.
# 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.
# Briggs, 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".
# "2018 CODATA Value: Avogadro constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: molar mass constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2021-04-28.
# "2018 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
# "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.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 ζ ( 2 ) {\displaystyle \zeta (2)} et ζ ( 3 ) {\displaystyle \zeta (3)} ", Astérisque, 61: 11–13.
Further reading
* Kingdom of Infinite Number: A Field Guide by Bryan Bunch, W.H. Freeman & Company, 2001. ISBN 0-7167-4447-3
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 the Wayback Machine (archived 27 November 2010)
* 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)
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