List of numbers explained

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 that 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 categorized 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

, Unicode).

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	309
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
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.

Cultural or practical significance

Along with their mathematical properties, many integers have cultural significance^[1] 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.

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

See main article: 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		-	31	37	41	43	47	53	59	61	67		-	73	79	83	89	97	101	103	107	109		-	127	131	137	139	149	151	157	163	167		-	179	181	191	193	197	199	211	223	227		-	233	239	241	251	257	263	269	271	277		-	283	293	307	311	313	317	331	337	347		-	353	359	367	373	379	383	389	397	401		-	419	421	431	433	439	443	449	457	461		-	467	479	487	491	499	503	509	521	523

Highly composite numbers

See 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

See 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:

Integers

See main article: Integer.

, Unicode); 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 10^k, where k is an integer. For instance, with k = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, k = -3 gives 1/1000, or 0.001. 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.

1000^m	Name	Symbol
1000¹	Kilo	k
1000²	Mega	M
1000³	Giga	G
1000⁴	Tera	T
1000⁵	Peta	P
1000⁶	Exa	E
1000⁷	Zetta	Z
1000⁸	Yotta	Y
1000⁹	Ronna	R
1000¹⁰	Quetta	Q

Rational numbers

, Unicode);^[3] 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, nine seventy-fifths, 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).

+ class="nowrap"	Table of notable rational numbers	Decimal expansion	Fraction	Notability
1.0		One is the multiplicative identity. One is a rational number, as it is equal to 1/1.
1
−0.083 333...		The value assigned to the series 1+2+3... by zeta function regularization and Ramanujan summation.
0.5		One half occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle: × base × perpendicular height and in the formulae for figurate numbers, such as triangular numbers and pentagonal numbers.
3.142 857...		A widely used approximation for the number \pi . It can be proven that this number exceeds \pi .
0.166 666...		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

See 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

See main article: Algebraic number.

Name

Expression

Decimal expansion

Notability

Golden ratio conjugate (

\Phi

)

	\sqrt{5
	-1}{2}

Reciprocal of (and one less than) the golden ratio.

Twelfth root of two

\sqrt[12]{2}

Proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale.

Cube root of two

\sqrt[3]{2}

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)

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 .

Plastic ratio

\sqrt[3]{	1	+
	2

	1	\sqrt{
	6

	23
	3

}} +\sqrt[3]

The only real solution of

x³=x+1

. The limit ratio between subsequent numbers in the Van der Laan sequence.

\sqrt{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

\dfrac{1+\sqrt[3]{\dfrac{29+3\sqrt{3 ⋅ 31}}{2}}+\sqrt[3]{\dfrac{29-3\sqrt{3 ⋅ 31}}{2}}}{3}

The only real solution of

x³=x²+1

. The limit ratio between subsequent numbers in Narayana's cows sequence.

Triangular root of 2

	\sqrt{17
	-1}{2}

Golden ratio (φ)

	\sqrt{5
	+1}{2}

The larger of the two real roots of x = x + 1.

\sqrt{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+\sqrt[3]{19+3\sqrt{3 ⋅ 11

} +\sqrt[3]}

The only real solution of

x³=x²+x+1

. The limit ratio between subsequent numbers in the Tribonacci sequence. Appears in the volume and coordinates of the snub cube and some related polyhedra.

Supersilver ratio

\dfrac{2+\sqrt[3]{\dfrac{43+3\sqrt{3 ⋅ 59}}{2}}+\sqrt[3]{\dfrac{43-3\sqrt{3 ⋅ 59}}{2}}}{3}

The only real solution of

x³=2x²+1

. The limit ratio between subsequent numbers in the third-order Pell sequence.

Square root of five

\sqrt{5}

Length of the diagonal of a 1 × 2 rectangle.

Silver ratio (δ)

\sqrt{2}+1

The larger of the two real roots of x = 2x + 1.
Altitude of a regular octagon with side length 1.

Bronze ratio (S)

	\sqrt{13
	+3}{2}

The larger of the two real roots of x = 3x + 1.

Transcendental numbers

See main article: Transcendental number.

Name	Symbolor Formula	Decimal expansion	Notes and notability
Gelfond's constant	e^\pi	...
Ramanujan's constant	e^\pi\sqrt{163 }	...
Gaussian integral	\sqrt{\pi}	...
Komornik–Loreti constant	q	...
Universal parabolic constant	P₂	...
Gelfond–Schneider constant	2^\sqrt{2 }	...
Euler's number	e	...	Raising e to the power of i will result in -1 .
Pi	\pi	...	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	$\sqrt$	...^[4]
Liouville constant	$L$	...
Champernowne constant	$C_$	...	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	$\tau$	...
Omega constant	\Omega	...
Cahen's constant	$C$	...
Natural logarithm of 2	ln 2
Gauss's constant	$G$	...
Tau	2 :	...	The ratio of the circumference to a radius, and the number of radians in a complete circle;^[5] ^[6] 2 x

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		^[7]	^[8]
Erdős–Borwein constant, E	...
Copeland–Erdős constant	...	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.
Prime constant, ρ	...	Proof of the number's irrationality is given at prime constant.
Reciprocal Fibonacci constant, ψ	...	^[9] ^[10]	^[11]

Real numbers

, Unicode). 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

...^[12]

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.^[13] ^[14] It was also shown that all but at most one number in an infinite list containing

	\gamma
	4

have to be transcendental.^[15] ^[16]

Euler–Gompertz constant, δ

0.596 347 362 323 194 074 341 078 499 369...^[17]

It was shown that at least one of the Euler-Mascheroni constant

\gamma

and the Euler-Gompertz constant

\delta

is transcendental.

Catalan's constant, G

...

It is not known whether this number is irrational.

Khinchin's constant, K₀

...^[18]

It is not known whether this number is irrational.

1st Feigenbaum constant, δ

4.6692...

Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.^[19]

2nd Feigenbaum constant, α

2.5029...

Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.

Glaisher–Kinkelin constant, A

...

Backhouse's constant

...

Fransén–Robinson constant, F

...

Lévy's constant,β

1.18656 91104 15625 45282...

Mills' constant, A

...

It is not known whether this number is irrational.

Ramanujan–Soldner constant, μ

...

Sierpiński's constant, K

...

Totient summatory constant

...

Vardi's constant, E

...

Somos' quadratic recurrence constant, σ

...

Niven's constant, C

...

Brun's constant, B₂

...

The irrationality of this number would be a consequence of the truth of the infinitude of twin primes.

Landau's totient constant

...

Brun's constant for prime quadruplets, B₄

...

Viswanath's constant

...

Khinchin–Lévy constant

...^[20]

This number represents the probability that three random numbers have no common factor greater than 1.^[21]

Landau–Ramanujan constant

...

C(1)

...

Z(1)

...

Heath-Brown–Moroz constant, C

...

Kepler–Bouwkamp constant,K'

...

MRB constant,S

...

It is not known whether this number is irrational.

Meissel–Mertens constant, M

...

Bernstein's constant, β

...

Gauss–Kuzmin–Wirsing constant, λ₁

...

Hafner–Sarnak–McCurley constant,σ

...

...

...

...

...

Golomb–Dickman constant, λ

...

Twin prime constant, C₂

...

Feller–Tornier constant

...

Laplace limit, ε

...^[22]

Embree–Trefethen constant

...

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

\Complex

, Unicode), while the set of quaternions is denoted by a boldface (or blackboard bold

, Unicode).

Algebraic complex numbers

Imaginary unit: $i=\sqrt$
nth roots of unity: $\xi_^=\cos\bigl(2\pi\frac\bigr)+i\sin\bigl(2\pi\frac\bigr)$ , while $0 \leq k \leq n-10$ , GCD(k, n) = 1

Other hypercomplex numbers

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

Transfinite numbers

See 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

, the set of natural numbers

Aleph-one: ℵ₁: the cardinality of ω₁, the set of all countable ordinal numbers
Beth-one:

\beth₀

: the cardinality of the continuum 2

ℭ or

akc

: the cardinality of the continuum 2

Omega: ω, the smallest infinite ordinal

Numbers representing physical quantities

See main article: Physical constant and List of physical constants. Physical quantities that appear in the universe are often described using physical constants.

Numbers representing geographical and astronomical distances

, the average equatorial radius of Earth in kilometers (following GRS 80 and WGS 84 standards).
, the length of the Equator in kilometers (following GRS 80 and WGS 84 standards).
, 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.
, the average distance between the Earth and the Sun or Astronomical Unit (AU), in meters.
, one light-year, the distance travelled by light in one Julian year, in meters.
, the distance of one parsec, another astronomical unit, in whole meters.

Numbers without specific values

See 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".^[23] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".^[24]

Named numbers

Hardy–Ramanujan number, 1729
Kaprekar's constant, 6174
Eddington number, ~10⁸⁰
Googol, 10¹⁰⁰
Shannon number
Centillion, 10³⁰³
Skewes's number
Googolplex, 10⁽¹⁰¹⁰⁰)
Mega/Circle(2)
Moser's number
Graham's number
TREE(3)
SSCG(3)
Rayo's number

References

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
- 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)

Notes and References

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. 10.1080/09540261.2020.1769289. 0954-0261. 32527165. 219605482.
Book: Rosen, Kenneth. Discrete Mathematics and its Applications. McGraw-Hill. 2007. 978-0-07-288008-3. 6th. New York, NY. 105, 158–160.
Web site: Mathematical Symbols. Rouse. Margaret. 1 April 2015.
Web site: Nick's Mathematical Puzzles: Solution 29. https://web.archive.org/web/20111018184029/http://www.qbyte.org/puzzles/p029s.html. 2011-10-18. live.
"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 69
Sequence .
See .
"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 33
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';
Web site: A001620 - OEIS. 2020-10-14. oeis.org.
Rivoal. Tanguy. 2012. On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant. Michigan Mathematical Journal. EN. 61. 2. 239–254. 10.1307/mmj/1339011525. 0026-2285. free.
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. 10.1090/S0273-0979-2013-01423-X. 1303.1856. 0273-0979. free.
Murty. M. Ram. Saradha. N.. 2010-12-01. Euler–Lehmer constants and a conjecture of Erdös. Journal of Number Theory. en. 130. 12. 2671–2682. 10.1016/j.jnt.2010.07.004. 0022-314X. 10.1.1.261.753.
Murty. M. Ram. Zaytseva. Anastasia. 2013-01-01. Transcendence of Generalized Euler Constants. The American Mathematical Monthly. 120. 1. 48–54. 10.4169/amer.math.monthly.120.01.048. 20495981. 0002-9890.
Web site: A073003 - OEIS. 2020-10-14. oeis.org.
Web site: Khinchin's Constant.
Keith. Briggs. Feigenbaum scaling in discrete dynamical systems. University of Melbourne. 1997. PhD.
Web site: Lévy Constant.
"The Penguin Dictionary of Curious and Interesting Numbers" by David Wells, page 29.
Web site: Laplace Limit.
http://versita.metapress.com/content/t98071387u726916/?p=1ad6a085630c432c94528c5548f5c2c4&pi=1 "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
https://www.bostonglobe.com/ideas/2016/07/13/the-surprising-history-indefinite-hyperbolic-numerals/qYTKpkP9lyWVfItLXuTHdM/story.html Boston Globe, July 13, 2016: "The surprising history of indefinite hyperbolic numerals"

List of numbers explained

Natural numbers

Mathematical significance

Cultural or practical significance

Classes of natural numbers

Prime numbers

Highly composite numbers

Perfect numbers

Integers

SI prefixes

Rational numbers

Irrational numbers

Algebraic numbers

Transcendental numbers

Irrational but not known to be transcendental

Real numbers

Real but not known to be irrational, nor transcendental

Numbers not known with high precision

Hypercomplex numbers

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

Notes and References