Set Of All Natural Numbers

This is a listing of notable numbers and articles about notable numbers. The listing does not contain all numbers in being as near of the number sets are space. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers accept qualities which could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known every bit 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 (three,4) is commonly regarded as a number when it is in the form of a complex number (iii+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 fatigued between the number 5 (an abstract object equal to 2+3), and the numeral five (the noun referring to the number).

Natural numbers [edit]

The natural numbers are a subset of the integers and are of historical and pedagogical value every bit they can be used for counting and oft have ethno-cultural significance (meet beneath). Beyond this, natural numbers are widely used every bit a building block for other number systems including the integers, rational numbers and real numbers. Natural numbers are those used for counting (equally in "there are six (vi) coins on the table") and ordering (as in "this is the tertiary (3rd) largest city in the land"). 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 class an infinitely large set. Often referred to as "the naturals", the natural numbers are commonly symbolised by a boldface $N$ (or blackboard bold $\mathbb {\mathbb {N} }$ , Unicode U+2115 ℕ DOUBLE-STRUCK Uppercase North).

The inclusion of 0 in the ready of natural numbers is ambiguous and discipline to individual definitions. In ready theory and informatics, 0 is typically considered a natural number. In number theory, information technology unremarkably 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 every bit cardinal numbers, which may get past various names. Natural numbers may also be used as ordinal numbers.

Tabular array of minor natural numbers
0	1	two	3	4	five	6	7	8	9
10	11	12	13	14	15	sixteen	17	eighteen	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
fifty	51	52	53	54	55	56	57	58	59
sixty	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
xc	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		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						269
270	271		273			276	277
280	281							288
290			300	400	500	600	700	800	900
	1000	2000	3000	4000	5000	6000	7000	8000	9000
	10,000	twenty,000	30,000	twoscore,000	fifty,000	sixty,000	seventy,000	80,000	90,000
10⁵	10⁶	x⁷	10⁸	10⁹	ten¹²
larger numbers, including 10¹⁰⁰ and 10^ten¹⁰⁰

Mathematical significance [edit]

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

Listing of mathematically significant natural numbers

1, the multiplicative identity. Besides the only natural number (not including 0) that isn't prime or blended.
2, the base of operations of the binary number system, used in almost all modern computers and data systems.
iii, 2²-1, the commencement Mersenne prime. Information technology is the showtime odd prime, and it is too the two bit integer maximum value.
4, the beginning composite number
half-dozen, the first of the series of perfect numbers, whose proper factors sum to the number itself.
9, the first odd number that is blended
11, the fifth prime and first palindromic multi-digit number in base 10.
12, the first sublime number.
17, the sum of the start four prime numbers, and the only prime which is the sum of 4 consecutive primes.
24, all Dirichlet characters mod n are existent if and only if n is a divisor of 24.
25, the get-go centered square number besides 1 that is also a square number.
27, the cube of iii, the value of iii^three.
28, the 2d perfect number.
30, the smallest sphenic number.
32, the smallest nontrivial fifth power.
36, the smallest number which is perfect ability merely non prime power.
72, the smallest Achilles number.
255, 2^eight − one, 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-flake 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 2 dissimilar ways.^[1]
8128, the quaternary perfect number.
142857, the smallest base of operations x cyclic number.
9814072356, the largest perfect power that contains no repeated digits in base x.

Cultural or practical significance [edit]

Along with their mathematical properties, many integers have cultural significance^[two] or are as well notable for their utilize in computing and measurement. Every bit mathematical properties (such as divisibility) can confer applied utility, there may be coaction and connections between the cultural or practical significance of an integer and its mathematical properties.

Listing 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.
iv, considered an "unlucky number" in modernistic China, Japan and Korea due to its audible similarity to the word "expiry."
7, the number of days in a week, and considered a "lucky" number in Western cultures.
8, considered a "lucky" number in Chinese civilization due to its aural similarity to the 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.
xiii, considered an "unlucky" number in Western superstition. Also known as a "Baker'due south Dozen".
18, considered a "lucky" number due to it beingness the value for life in Jewish numerology.
40, considered a significant number in Tengriism and Turkish sociology. Multiple customs, such as those relating to how many days one must visit someone after a decease in the family, include the number xl.
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, used as slang to refer to a sexual human action.
86, a slang term that is used in the American pop culture as a transitive verb to mean throw out or become rid of.^[3]
108, considered sacred past the Dharmic Religions. Approximately equal to the ratio of the distance from World to Sun and bore of the Sun.
420, a code-term that refers to the consumption of cannabis.
666, the Number of the Beast from the Volume of Revelation.
786, regarded as sacred in the Muslim Abjad numerology.
5040, mentioned past Plato in the Laws as 1 of the well-nigh important numbers for the city.

Listing of integers notable for their employ in units, measurements and scales

10, the number of digits in the decimal number organization.
12, the number base of operations for measuring fourth dimension in many civilizations.
fourteen, the number of days in a fortnight.
sixteen, the number of digits in the hexadecimal number system.
24, number of hours in a twenty-four hour period
31, the number of days about months of the year have.
60, the number base for some ancient counting systems, such as the Babylonians', and the basis for many mod measuring systems.
360, the number of sexagesimal degrees in a full circumvolve.
365, the number of days in the common year, while at that place are 366 days in a leap year of the solar Gregorian calendar.

Listing of integers notable in computing

8, the number of bits in a byte
256, The number of possible combinations within 8 bits, or a byte.
1024, the number of bytes in a kibibyte. It's also the number of $.25 in a kibibit.
65535, two^sixteen − one, the maximum value of a xvi-bit unsigned integer.
65536, ii¹⁶, the number of possible 16-fleck combinations.
65537, 2¹⁶ + 1, the most popular RSA public primal prime exponent in most SSL/TLS certificates on the Web/Internet.
16777216, two²⁴, or 16^six; the hexadecimal "1000000" (0x1000000), and the total number of possible colour combinations in 24/32-scrap True Color figurer graphics.
2147483647, 2³¹ − 1, the maximum value of a 32-bit signed integer using two's complement representation.
9223372036854775807, 2⁶³ − ane, the maximum value of a 64-bit signed integer using 2's complement representation.

Classes of natural numbers [edit]

Subsets of the natural numbers, such every bit 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 [edit]

A prime number is a positive integer which has exactly two divisors: one 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 [edit]

A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are oftentimes used in geometry, grouping and time measurement.

The showtime 20 highly composite numbers are:

one, 2, 4, six, 12, 24, 36, 48, sixty, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560

Perfect numbers [edit]

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

The outset ten 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 [edit]

The integers are a gear up of numbers commonly encountered in arithmetics and number theory. There are many subsets of the integers, including the natural numbers, prime number numbers, perfect numbers, etc. Many integers are notable for their mathematical backdrop. Integers are commonly symbolised past a boldface $Z$ (or blackboard bold $\mathbb {\mathbb {Z} }$ , Unicode U+2124 ℤ DOUBLE-STRUCK Capital letter Z); this became the symbol for the integers based on the German word for "numbers" (Zahlen).

Notable integers include −i, the additive inverse of unity, and 0, the additive identity.

As with the natural numbers, the integers may besides have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.

SI prefixes [edit]

1 important use of integers is in orders of magnitude. A ability of 10 is a number 10^k, where k is an integer. For instance, with k = 0, 1, ii, 3, ..., the appropriate powers of ten are one, x, 100, 1000, ... Powers of ten can too be fractional: for instance, chiliad = -3 gives 1/k, 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 every bit 3.94 × 10⁵.

Integers are used as prefixes in the SI organisation. A metric prefix is a unit prefix that precedes a basic unit of measurement of measure to indicate a multiple or fraction of the unit of measurement. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may exist added to gram to indicate multiplication by one yard: one kilogram is equal to grand grams. The prefix milli-, besides, may exist added to metre to signal division past thousand; one millimetre is equal to one thousandth of a metre.

Value	chiliad^m	Name	Symbol
i000	1000¹	Kilo	one thousand
1000 000	g²	Mega	M
i000 000 000	thou^three	Giga	Grand
1000 000 000 000	1000^iv	Tera	T
1000 000 000 000 000	1000⁵	Peta	P
1000 000 000 000 000 000	1000^vi	Exa	E
1000 000 000 000 000 000 000	k^vii	Zetta	Z
1000 000 000 000 000 000 000 000	g⁸	Yotta	Y

Rational numbers [edit]

A rational number is any number that can be expressed every bit the caliber or fraction $p / q$ of two integers, a numerator $p$ and a non-zero denominator $q$ .^[4] Since $q$ may exist equal to 1, every integer is trivially a rational number. The set up of all rational numbers, frequently referred to every bit "the rationals", the field of rationals or the field of rational numbers is normally denoted by a boldface $Q$ (or blackboard bold $\mathbb {Q}$ , Unicode U+211A ℚ DOUBLE-STRUCK Capital letter Q);^[5] it was thus denoted in 1895 by Giuseppe Peano later quoziente, Italian for "quotient".

Rational numbers such as 0.12 tin exist represented in infinitely many ways, due east.grand. aught-point-one-two (0.12), iii twenty-fifths ( 3 / 25 ), nine seventy-fifths ( 9 / 75 ), etc. This can exist 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 plant at numeral (linguistics).

Table of notable rational numbers
Decimal expansion	Fraction	Notability
ane.0	1 / i	1 is the multiplicative identity. I is trivially a rational number, as information technology is equal to 1/1.
1	1 / i
-0.083 333...	−+ one / 12	The value assigned to the series i+2+three... by zeta function regularization and Ramanujan summation.
0.5	1 / 2	Ane half occurs usually in mathematical equations and in existent globe proportions. Ane one-half appears in the formula for the area of a triangle: ane / ii × base × perpendicular height and in the formulae for figurate numbers, such every bit triangular numbers and pentagonal numbers.
iii.142 857...	22 / 7	A widely used approximation for the number $\pi$ . It can be proven that this number exceeds $\pi$ .
0.166 666...	one / 6	One sixth. Often appears in mathematical equations, such equally in the sum of squares of the integers and in the solution to the Basel trouble.

Irrational numbers [edit]

The irrational numbers are a set of numbers that includes all real numbers that are non 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 [edit]

Name	Expression	Decimal expansion	Notability
Golden ratio conjugate ( $\Phi$ )	${\frac {{\sqrt {5}}-ane}{2}}$	0.618033 988 749 894 848 204 586 834 366	Reciprocal of (and ane less than) the golden ratio.
Twelfth root of two	${\sqrt[{12}]{ii}}$	1.059463 094 359 295 264 561 825 294 946	Proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale.
Cube root of two	${\sqrt[{iii}]{2}}$	1.259921 049 894 873 164 767 210 607 278	Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.
Conway'due south abiding	(cannot exist written as expressions involving integers and the operations of addition, subtraction, multiplication, segmentation, and the extraction of roots)	1.303577 269 034 296 391 257 099 112 153	Defined as the unique positive existent root of a sure polynomial of degree 71.
Plastic number	${\sqrt[{iii}]{{\frac {i}{ii}}+{\frac {ane}{half dozen}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{half-dozen}}{\sqrt {\frac {23}{3}}}}}$	1.324717 957 244 746 025 960 908 854 478	The unique real root of the cubic equation x ³ = x + i.
Square root of two	${\sqrt {2}}$	1.414213 562 373 095 048 801 688 724 210	√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 newspaper sizes in the ISO 216 serial (originally DIN 476 series).
Supergolden ratio	${\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {93}}}{ii}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {93}}}{ii}}}}{3}}$	1.465571 231 876 768 026 656 731 225 220	The just real solution of $10^{3}=ten^{ii}+i$ . Besides the limit to the ratio between subsequent numbers in the binary Look-and-say sequence and the Narayana's cows sequence (OEIS: A000930).
Triangular root of two	${\frac {{\sqrt {17}}-i}{2}}$	1.561552 812 808 830 274 910 704 927 987
Golden ratio (φ)	${\frac {{\sqrt {five}}+ane}{2}}$	one.618033 988 749 894 848 204 586 834 366	The larger of the two real roots of ten ² = ten + 1.
Square root of three	${\sqrt {3}}$	1.732050 807 568 877 293 527 446 341 506	√3 = 2 sin 60° = 2 cos 30° . A.1000.a. the mensurate of the fish or Theodorus' abiding. 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	${\frac {1+{\sqrt[{iii}]{xix+iii{\sqrt {33}}}}+{\sqrt[{iii}]{19-3{\sqrt {33}}}}}{three}}$	1.839286 755 214 161 132 551 852 564 653	Appears in the book and coordinates of the snub cube and some related polyhedra. Information technology satisfies the equation 10 + x ⁻³ = 2.
Square root of five	${\sqrt {5}}$	two.236067 977 499 789 696 409 173 668 731	Length of the diagonal of a ane × 2 rectangle.
Silver ratio (δ_S)	${\sqrt {ii}}+1$	ii.414213 562 373 095 048 801 688 724 210	The larger of the two real roots of x ² = 2x + 1. Altitude of a regular octagon with side length 1.
Bronze ratio (S₃)	${\frac {{\sqrt {thirteen}}+3}{2}}$	iii.302775 637 731 994 646 559 610 633 735	The larger of the two existent roots of x ² = iiix + 1.

Transcendental numbers [edit]

Name	Symbol or Formula	Decimal expansion	Notes and notability
Gelfond'due south abiding	$e^{\pi }$	23.140692 632 779 25 ...
Ramanujan's abiding	$e^{\pi {\sqrt {163}}}$	262537 412 640 768 743.999999 999 999 25 ...
Gaussian integral	${\sqrt {\pi }}$	1.772453 850 905 516 ...
Komornik–Loreti constant	$q$	1.787231 650 ...
Universal parabolic abiding	$P_{two}$	two.295587 149 39 ...
Gelfond–Schneider constant	$2^{\sqrt {2}}$	2.665144 143 ...
Euler'south number	$e$	2.718281 828 459 045 235 360 287 471 352 662 497 757 247 ...	Raising east to the power of $i$ $π$ will effect in $-ane$ .
Pi	$\pi$	3.141592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 ...	Pi is an irrational number that is the consequence of dividing the circumference of a circle by its diameter.
Super square-root of ii	${\textstyle {\sqrt {2_{s}}}}$ ^[6]	1.559610 469 ...^[7]
Liouville abiding	${\textstyle 50}$	0.110001 000 000 000 000 000 001 000 ...
Champernowne abiding	${\textstyle C_{x}}$	0.123456 789 101 112 131 415 16 ...
Prouhet–Thue–Morse constant	${\textstyle \tau }$	0.412454 033 640 ...
Omega constant	$\Omega$	0.567143 290 409 783 872 999 968 6622 ...
Cahen's constant	${\textstyle C}$	0.643410 546 29 ...
Natural logarithm of 2	ln 2	0.693147 180 559 945 309 417 232 121 458
Gauss'southward constant	${\textstyle G}$	0.8346268 ...
Tau	2 $π$ : $τ$	6.283185 307 179 586 476 925 286 766 559 ...	The ratio of the circumference to a radius, and the number of radians in a complete circle;^[eight] ^[9] two $\times$ $π$

Irrational but not known to be transcendental [edit]

Some numbers are known to be irrational numbers, merely have non been proven to be transcendental. This differs from the algebraic numbers, which are known not to exist transcendental.

Name	Decimal expansion	Proof of irrationality	Reference of unknown transcendentality
ζ(3), besides known as Apéry's constant	1.202056 903 159 594 285 399 738 161 511 449 990 764 986 292	^[10]	^[11]
Erdős–Borwein constant, E	1.606695 152 415 291 763 ...	^[12] ^[13]	^{[ citation needed ]}
Copeland–Erdős constant	0.235711 131 719 232 931 374 143 ...	Can exist proven with Dirichlet'southward theorem on arithmetic progressions or Bertrand'southward postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most 6 primes. It also follows directly from its normality.	^{[ citation needed ]}
Prime constant, ρ	0.414682 509 851 111 660 248 109 622 ...	Proof of the number's irrationality is given at prime number abiding.	^{[ commendation needed ]}
Reciprocal Fibonacci constant, ψ	3.359885 666 243 177 553 172 011 302 918 927 179 688 905 133 731 ...	^[14] ^[15]	^[16]

Real numbers [edit]

The real numbers are a superset containing the algebraic and the transcendental numbers. The real numbers, sometimes referred to as "the reals", are ordinarily symbolised by a boldface $R$ (or blackboard bold $\mathbb {\mathbb {R} }$ , Unicode U+211D ℝ DOUBLE-STRUCK Uppercase R). For some numbers, it is not known whether they are algebraic or transcendental. The post-obit listing includes real numbers that have not been proved to be irrational, nor transcendental.

Real but not known to be irrational, nor transcendental [edit]


Name and symbol	Decimal expansion	Notes
Euler–Mascheroni abiding, γ	0.577215 664 901 532 860 606 512 090 082 ...^[17]	Believed to be transcendental but non proven to be and so. Notwithstanding, it was shown that at to the lowest degree one of $\gamma$ and the Euler-Gompertz constant $\delta$ is transcendental.^[18] ^[19] It was too shown that all but at nigh one number in an infinite list containing ${\frac {\gamma }{4}}$ have to be transcendental.^[20] ^[21]
Euler–Gompertz constant, δ	0.596 347 362 323 194 074 341 078 499 369...^[22]	Information technology was shown that at least one of the Euler-Mascheroni constant $\gamma$ and the Euler-Gompertz constant $\delta$ is transcendental.^[18] ^[19]
Catalan's abiding, K	0.915965 594 177 219 015 054 603 514 932 384 110 774 ...	It is not known whether this number is irrational.^[23]
Khinchin's constant, Yard₀	2.685452 001 ...^[24]	It is not known whether this number is irrational.^[25]
1st Feigenbaum abiding, δ	4.6692...	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.^[26]
2nd Feigenbaum abiding, α	2.5029...	Both Feigenbaum constants are believed to exist transcendental, although they have not been proven to exist so.^[26]
Glaisher–Kinkelin constant, A	1.282427 12 ...
Backhouse'due south constant	1.456074 948 ...
Fransén–Robinson constant, F	2.807770 2420 ...
Lévy's constant,β	1.18656 91104 15625 45282...
Mills' constant, A	one.306377 883 863 080 690 46 ...	It is not known whether this number is irrational.(Finch 2003)
Ramanujan–Soldner constant, μ	1.451369 234 883 381 050 283 968 485 892 027 449 493 ...
Sierpiński'south constant, 1000	2.584981 759 579 253 217 065 8936 ...
Totient summatory abiding	1.339784 ...^[27]
Vardi'southward constant, E	ane.264084 735 305 ...
Somos' quadratic recurrence constant, σ	one.661687 949 633 594 121 296 ...
Niven'southward constant, C	1.705211 ...
Brun's constant, B₂	1.902160 583 104 ...	The irrationality of this number would be a consequence of the truth of the infinitude of twin primes.
Landau's totient constant	one.943596 ...^[28]
Brun's constant for prime quadruplets, B_four	0.870588 3800 ...
Viswanath's constant	1.131988 248 7943 ...
Khinchin–Lévy constant	1.186569 1104 ...^[29]	This number represents the probability that three random numbers have no common gene greater than ane.^[30]
Landau–Ramanujan constant	0.764223 653 589 220 662 990 698 731 25 ...
C(1)	0.779893 400 376 822 829 474 206 413 65 ...
Z(ane)	−0.736305 462 867 317 734 677 899 828 925 614 672 ...
Heath-Chocolate-brown–Moroz constant, C	0.001317 641 ...
Kepler–Bouwkamp abiding,K'	0.114942 0448 ...
MRB constant,Due south	0.187859 ...	It is not known whether this number is irrational.
Meissel–Mertens constant, Chiliad	0.261497 212 847 642 783 755 426 838 608 695 859 0516 ...
Bernstein's abiding, β	0.280169 4990 ...
Gauss–Kuzmin–Wirsing constant, λ₁	0.303663 0029 ...^[31]
Hafner–Sarnak–McCurley abiding,σ	0.353236 3719 ...
Artin's constant,C_Artin	0.373955 8136 ...
South(one)	0.438259 147 390 354 766 076 756 696 625 152 ...
F(i)	0.538079 506 912 768 419 136 387 420 407 556 ...
Stephens' constant	0.575959 ...^[32]
Golomb–Dickman constant, λ	0.624329 988 543 550 870 992 936 383 100 837 24 ...
Twin prime number abiding, C₂	0.660161 815 846 869 573 927 812 110 014 ...
Feller–Tornier abiding	0.661317 ...^[33]
Laplace limit, ε	0.662743 4193 ...^[34]
Embree–Trefethen abiding	0.70258 ...

Numbers not known with high precision [edit]

Some existent 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 incommunicable to compute.
Bloch'due south abiding (also second Landau's abiding): 0.4332 < B < 0.4719
1st Landau's constant: 0.5 < L < 0.5433
3rd Landau'due south abiding: 0.5 < A ≤ 0.7853
Grothendieck constant: 1.67 < chiliad < 1.79
Romanov'due south abiding in Romanov's theorem: 0.107648 < d < 0.49094093, Romanov conjectured that it is 0.434

Hypercomplex numbers [edit]

Hypercomplex number is a term for an element of a unital algebra over the field of existent numbers. The complex numbers are often symbolised by a boldface $C$ (or blackboard assuming $\mathbb {\mathbb {C} }$ , Unicode U+2102 ℂ DOUBLE-STRUCK CAPITAL C), while the prepare of quaternions is denoted by a boldface $H$ (or blackboard bold $\mathbb {H}$ , Unicode U+210D ℍ DOUBLE-STRUCK Upper-case letter H).

Algebraic circuitous numbers [edit]

Other hypercomplex numbers [edit]

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

Transfinite numbers [edit]

Transfinite numbers are numbers that are "space" in the sense that they are larger than all finite numbers, notwithstanding not necessarily absolutely infinite.

Numbers representing physical quantities [edit]

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

Avogadro abiding: Due north _A = 6.022140 76 ×10²³ mol⁻¹ ^[35]
Electron mass: g _e = 9.109383 7015(28)×10⁻³¹ kg ^[36]
Fine-construction constant: α = 7.297352 5693(11)×10⁻³ ^[37]
Gravitational constant: G = half-dozen.674thirty(15)×x^−eleven mⁱⁱⁱ⋅kg^−one⋅s⁻² ^[38]
Molar mass constant: Thousand _u = 0.999999 999 65(xxx)×ten⁻ⁱⁱⁱ kg⋅mol⁻¹ ^[39]
Planck abiding: h = 6.626070 xv ×10⁻³⁴ J⋅Hz^−one ^[40]
Rydberg abiding: R _∞ = x973 731.568160(21) m⁻¹ ^[41]
Speed of calorie-free in vacuum: c = 299792 458 grand⋅s⁻¹ ^[42]
Vacuum electric permittivity: ε ₀ = 8.854187 8128(xiii)×10⁻¹² F⋅m⁻¹ ^[43]

Numbers representing geographical and astronomical distances [edit]

6378.137, the average equatorial radius of Earth in kilometers (following GRS eighty and WGS 84 standards).
40075.0167, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards).
384399 , the semi-major centrality of the orbit of the Moon, in kilometers, roughly the distance betwixt the middle of Earth and that of the Moon.
149597 870 700 , the average distance between the Earth and the Sunday or Astronomical Unit (AU), in meters.
9460 730 472 580 800 , 1 low-cal-yr, the distance travelled by lite in i Julian year, in meters.
30856 775 814 913 673 , the altitude of one parsec, another astronomical unit of measurement, in whole meters.

Numbers without specific values [edit]

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. 1 technical term for such words is "not-numerical vague quantifier".^[44] Such words designed to signal big quantities tin can be called "indefinite hyperbolic numerals".^[45]

Named numbers [edit]

Eddington number, ~10^fourscore
Googol, x¹⁰⁰
Googolplex, 10^(10¹⁰⁰)
Graham'due south number
Hardy–Ramanujan number, 1729
Kaprekar's constant, 6174
Moser'southward number
Rayo's number
Shannon number
Skewes'due south number
TREE(iii)

Run into likewise [edit]

Absolute infinity
English-language numerals
Floating point
Fraction (mathematics)
Integer sequence
Interesting number paradox
Large numbers
Listing of mathematical constants
List of numbers in diverse languages
List of prime numbers
Listing of types of numbers
Mathematical constant
Names of large numbers
Names of small numbers
Negative number
Number prefix
Numeral (linguistics)
Society of magnitude
Orders of magnitude (numbers)
Ordinal number
The Penguin Lexicon of Curious and Interesting Numbers
Power of two
Power of 10
SI prefix
Surreal number
Tabular array of prime number factors

References [edit]

^ 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 beyond cultural beliefs and practice". International Review of Psychiatry. 33 (ane–2): 179–188. doi:ten.1080/09540261.2020.1769289. ISSN 0954-0261. PMID 32527165. S2CID 219605482.
^ "Eighty-six – Definition of eighty-six by Merriam-Webster". merriam-webster.com. Archived from the original on 2013-04-08.
^ Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Loma. pp. 105, 158–160. ISBN978-0-07-288008-3.
^ Rouse, Margaret. "Mathematical Symbols". Retrieved 1 April 2015.
^ Lipscombe, Trevor Davis (2021-05-06), "Super Powers: Calculate Squares, Foursquare Roots, Cube Roots, and More", Quick(er) Calculations, Oxford University Press, pp. 103–124, doi:10.1093/oso/9780198852650.003.0010, ISBN978-0-xix-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, folio 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 serial" (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 Club, 112 (1): 141–146, Bibcode:1992MPCPS.112..141B, CiteSeerX10.1.one.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, effect 19 (1989), pp. 539-541.
^ South. Kato, 'Irrationality of reciprocal sums of Fibonacci numbers', Principal'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-x-fourteen .
^ ^a ^b Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's abiding, and Gompertz'southward abiding". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
^ ^a ^b Lagarias, Jeffrey C. (2013-07-nineteen). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. fifty (4): 527–628. arXiv:1303.1856. doi:x.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.
^ Murty, M. Ram; Saradha, North. (2010-12-01). "Euler–Lehmer constants and a conjecture of Erdös". Periodical of Number Theory. 130 (12): 2671–2682. CiteSeerXten.ane.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-fourteen .
^ Nesterenko, Yu. Five. (January 2016), "On Catalan'south constant", Proceedings of the Steklov Constitute of Mathematics, 292 (1): 153–170, doi:ten.1134/s0081543816010107, S2CID 124903059
^ "Khinchin'south Abiding".
^ Weisstein, Eric W. "Khinchin'due south abiding". MathWorld.
^ ^a ^b Briggs, Keith (1997). Feigenbaum scaling in detached dynamical systems (PDF) (PhD thesis). University of Melbourne.
^ OEIS: A065483
^ OEIS: A082695
^ "Lévy Constant".
^ "The Penguin Lexicon 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 abiding". 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. xx May 2019. Retrieved 2019-05-20 .
^ "2018 CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. xx 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-xx .
^ "2018 CODATA Value: molar mass constant". The NIST Reference on Constants, Units, and Incertitude. NIST. twenty 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-xx .
^ "2018 CODATA Value: speed of lite 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 Doubtfulness. NIST. twenty May 2019. Retrieved 2019-05-20 .
^ "Numberless of Talent, a Touch on of Panic, and a Bit of Luck: The Case of Non-Numerical Vague Quantifiers" from Linguista Pragensia, November. ii, 2010 Archived 2012-07-31 at archive.today
^ Boston Globe, July thirteen, 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 Printing, pp. 130–133, ISBN0521818052
Apéry, Roger (1979), "Irrationalité de $\zeta (2)$ et $\zeta (3)$ ", Astérisque, 61: xi–13 .

Farther reading [edit]

Kingdom of Infinite Number: A Field Guide by Bryan Bunch, W.H. Freeman & Company, 2001. ISBN 0-7167-4447-3

External links [edit]

The Database of Number Correlations: ane 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
Most big numbers at the Wayback Machine (archived 27 Nov 2010)
Robert P. Munafo'southward Large Numbers page
Dissimilar notations for big numbers – past Susan Stepney
Names for Large Numbers, in How Many? A Dictionary of Units of Measurement by Russ Rowlett
What'due south Special About This Number? (from 0 to 9999)