List of numbers

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 ${\displaystyle \mathbb {\mathbb {N} } }$, 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.

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.

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 ${\displaystyle \mathbb {\mathbb {Z} } }$, 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.

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.^[5] 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 ${\displaystyle \mathbb {Q} }$, Unicode U+211A ℚ DOUBLE-STRUCK CAPITAL Q);^[6] 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).

More information A widely used approximation for the number ...

Table of notable rational numbers

Decimal expansion	Fraction	Notability
1.0	⁠1/1⁠	One is the multiplicative identity. One is a rational number, as it is equal to 1/1.
1
−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 ${\displaystyle \pi }$. It can be proven that this number exceeds ${\displaystyle \pi }$.
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.

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

Algebraic numbers

Main article: Algebraic number

More information ( ...

Name	Expression	Decimal expansion	Notability
Golden ratio conjugate (${\displaystyle \Phi }$)	${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$	0.618033988749894848204586834366	Reciprocal of (and one less than) the golden ratio.
Twelfth root of two	${\displaystyle {\sqrt[{12}]{2}}}$	1.059463094359295264561825294946	Proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale.
Cube root of two	${\displaystyle {\sqrt[{3}]{2}}}$	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	${\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}$	1.324717957244746025960908854478	The only real solution of ${\displaystyle x^{3}=x+1}$.(OEIS: A060006) The limit ratio between subsequent numbers in the Van der Laan sequence. (OEIS: A182097)
Square root of two	${\displaystyle {\sqrt {2}}}$	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	${\displaystyle {\dfrac {1+{\sqrt[{3}]{\dfrac {29+3{\sqrt {3\cdot 31}}}{2}}}+{\sqrt[{3}]{\dfrac {29-3{\sqrt {3\cdot 31}}}{2}}}}{3}}}$	1.465571231876768026656731225220	The only real solution of ${\displaystyle x^{3}=x^{2}+1}$.(OEIS: A092526) The limit ratio between subsequent numbers in Narayana's cows sequence. (OEIS: A000930)
Triangular root of 2	${\displaystyle {\frac {{\sqrt {17}}-1}{2}}}$	1.561552812808830274910704927987
Golden ratio (φ)	${\displaystyle {\frac {{\sqrt {5}}+1}{2}}}$	1.618033988749894848204586834366	The larger of the two real roots of x2 = x + 1.
Square root of three	${\displaystyle {\sqrt {3}}}$	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	${\displaystyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {3\cdot 11}}}}+{\sqrt[{3}]{19-3{\sqrt {3\cdot 11}}}}}{3}}}$	1.839286755214161132551852564653	The only real solution of ${\displaystyle x^{3}=x^{2}+x+1}$.(OEIS: A058265) The limit ratio between subsequent numbers in the Tribonacci sequence.(OEIS: A000073) Appears in the volume and coordinates of the snub cube and some related polyhedra.
Supersilver ratio	${\displaystyle {\dfrac {2+{\sqrt[{3}]{\dfrac {43+3{\sqrt {3\cdot 59}}}{2}}}+{\sqrt[{3}]{\dfrac {43-3{\sqrt {3\cdot 59}}}{2}}}}{3}}}$	2.20556943040059031170202861778	The only real solution of ${\displaystyle x^{3}=2x^{2}+1}$.(OEIS: A356035) The limit ratio between subsequent numbers in the third-order Pell sequence. (OEIS: A008998)
Square root of five	${\displaystyle {\sqrt {5}}}$	2.236067977499789696409173668731	Length of the diagonal of a 1 × 2 rectangle.
Silver ratio (δS)	${\displaystyle {\sqrt {2}}+1}$	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)	${\displaystyle {\frac {{\sqrt {13}}+3}{2}}}$	3.302775637731994646559610633735	The larger of the two real roots of x2 = 3x + 1.

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Transcendental numbers

Main article: Transcendental number

More information , ...

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

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Real but not known to be irrational, nor transcendental

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

More information Believed to be transcendental but not proven to be so. However, it was shown that at least one of ...

Name and symbol	Decimal expansion	Notes
Euler–Mascheroni constant, γ	0.577215664901532860606512090082...[18]	Believed to be transcendental but not proven to be so. However, it was shown that at least one of ${\displaystyle \gamma }$ and the Euler-Gompertz constant ${\displaystyle \delta }$ is transcendental.[19][20] It was also shown that all but at most one number in an infinite list containing ${\displaystyle {\frac {\gamma }{4}}}$ have to be transcendental.[21][22]
Euler–Gompertz constant, δ	0.596 347 362 323 194 074 341 078 499 369...[23]	It was shown that at least one of the Euler-Mascheroni constant ${\displaystyle \gamma }$ and the Euler-Gompertz constant ${\displaystyle \delta }$ is transcendental.[19][20]
Catalan's constant, G	0.915965594177219015054603514932384110774...	It is not known whether this number is irrational.[24]
Khinchin's constant, K0	2.685452001...[25]	It is not known whether this number is irrational.[26]
1st Feigenbaum constant, δ	4.6692...	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[27]
2nd Feigenbaum constant, α	2.5029...	Both Feigenbaum constants are believed to be transcendental, although they have not been proven to be so.[27]
Glaisher–Kinkelin constant, A	1.28242712...
Backhouse's constant	1.456074948...
Fransén–Robinson constant, F	2.8077702420...
Lévy's constant,β	1.18656 91104 15625 45282...
Mills' constant, A	1.30637788386308069046...	It is not known whether this number is irrational.(Finch 2003)
Ramanujan–Soldner constant, μ	1.451369234883381050283968485892027449493...
Sierpiński's constant, K	2.5849817595792532170658936...
Totient summatory constant	1.339784...[28]
Vardi's constant, E	1.264084735305...
Somos' quadratic recurrence constant, σ	1.661687949633594121296...
Niven's constant, C	1.705211...
Brun's constant, B2	1.902160583104...	The irrationality of this number would be a consequence of the truth of the infinitude of twin primes.
Landau's totient constant	1.943596...[29]
Brun's constant for prime quadruplets, B4	0.8705883800...
Viswanath's constant	1.1319882487943...
Khinchin–Lévy constant	1.1865691104...[30]	This number represents the probability that three random numbers have no common factor greater than 1.[31]
Landau–Ramanujan constant	0.76422365358922066299069873125...
C(1)	0.77989340037682282947420641365...
Z(1)	−0.736305462867317734677899828925614672...
Heath-Brown–Moroz constant, C	0.001317641...
Kepler–Bouwkamp constant,K'	0.1149420448...
MRB constant,S	0.187859...	It is not known whether this number is irrational.
Meissel–Mertens constant, M	0.2614972128476427837554268386086958590516...
Bernstein's constant, β	0.2801694990...
Gauss–Kuzmin–Wirsing constant, λ1	0.3036630029...[32]
Hafner–Sarnak–McCurley constant,σ	0.3532363719...
Artin's constant,CArtin	0.3739558136...
S(1)	0.438259147390354766076756696625152...
F(1)	0.538079506912768419136387420407556...
Stephens' constant	0.575959...[33]
Golomb–Dickman constant, λ	0.62432998854355087099293638310083724...
Twin prime constant, C2	0.660161815846869573927812110014...
Feller–Tornier constant	0.661317...[34]
Laplace limit, ε	0.6627434193...[35]
Embree–Trefethen constant	0.70258...

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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 ${\displaystyle \mathbb {\mathbb {C} } }$, Unicode U+2102 ℂ DOUBLE-STRUCK CAPITAL C), while the set of quaternions is denoted by a boldface H (or blackboard bold ${\displaystyle \mathbb {H} }$, Unicode U+210D ℍ DOUBLE-STRUCK CAPITAL H).

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

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

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

• 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.

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".^[45] Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".^[46]

• 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