A
mathematical constant is a key
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an
alphabet letter), or by mathematicians' names to facilitate using it across multiple
mathematical problems. For example, the constant
π may be defined as the ratio of the length of a circle's
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
. The following list includes a
decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
and set containing each number, ordered by year of discovery.
The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
List
{, class="wikitable sortable"
, -
! Name
! Symbol
! Decimal expansion
! Formula
! Year
! Set
, -
,
One
, 1
, 1
,
, data-sort-value="-2000", Prehistory
, data-sort-value="1",
, -
,
Two
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
, 2
, 2
,
, data-sort-value="-2000", Prehistory
, data-sort-value="1",
, -
,
One half
, 1/2
, data-sort-value="0.50000", 0.5
,
, data-sort-value="-2000", Prehistory
, data-sort-value="3",
, -
,
Pi
,
, 3.14159 26535 89793 23846
, Ratio of a circle's circumference to its diameter.
, data-sort-value="-1900", 1900 to 1600 BCE
, data-sort-value="5",
, -
,
Square root of 2,
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
constant.
,
, 1.41421 35623 73095 04880
, Positive root of
, data-sort-value="-1800",
1800 to 1600 BCE
, data-sort-value="4",
, -
,
Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...
,
Theodorus' constant
,
, 1.73205 08075 68877 29352
, Positive root of
, data-sort-value="-465", 465 to 398 BCE
, data-sort-value="4",
, -
,
Square root of 5
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This numbe ...
,
, 2.23606 79774 99789 69640
, Positive root of
, data-sort-value="-464",
, data-sort-value="4",
, -
, Phi,
Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
,
or
, 1.61803 39887 49894 84820
,
, data-sort-value="-301", ~300 BCE
, data-sort-value="4",
, -
,
Silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
,
, 2.41421 35623 73095 04880
,
, data-sort-value="-301", ~300 BCE
, data-sort-value="4",
, -
,
Zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
, 0
, 0
,
, data-sort-value="-300", 300 to 100 BCE
, data-sort-value="2",
, -
,
Negative one
, −1
, −1
,
, data-sort-value="-300", 300 to 200 BCE
, data-sort-value="2",
, -
,
Cube root of 2
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related probl ...
,
, 1.25992 10498 94873 16476
, Real root of
, 46 to 120 CE
, data-sort-value="4",
, -
,
Cube root of 3
,
, 1.44224 95703 07408 38232
, Real root of
, data-sort-value="47",
, data-sort-value="4",
, -
,
Twelfth root of 2
The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semi ...
,
, 1.05946 30943 59295 26456
, Real root of
, data-sort-value="47",
, data-sort-value="4",
, -
,
Supergolden ratio
In mathematics, two quantities are in the supergolden ratio if the quotient of the larger number divided by the smaller one is equal to
:\psi = \frac
which is the only real solution to the equation x^3 = x^2+1. It can also be represented using ...
,
, 1.46557 12318 76768 02665
,
Real root of
, data-sort-value="47",
, data-sort-value="4",
, -
,
Imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
,
, data-sort-value="0",
, Either of the two roots of
, 1501 to 1576
, data-sort-value="8",
, -
,
Connective constant In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. While the connective con ...
for the hexagonal lattice
,
, 1.84775 90650 22573 51225
,
, as a root of the polynomial
, 1593
, data-sort-value="4",
, -
,
Kepler–Bouwkamp constant
In plane geometry, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following sequence. Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. I ...
,
, 0.11494 20448 53296 20070
,
, 1596
, data-sort-value="7",
, -
,
Wallis's constant
,
, 2.09455 14815 42326 59148
,
Real root of
, 1616 to 1703
, data-sort-value="4",
, -
,
Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
,
, 2.71828 18284 59045 23536
,
, 1618
, data-sort-value="5",
, -
,
Natural logarithm of 2
,
, 0.69314 71805 59945 30941
, Real root of
, 1619 & 1668
, data-sort-value="5",
, -
,
Lemniscate constant
In mathematics, the lemniscate constant p. 199 is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of for the circle. Equivalently, the perimeter ...
,
, 2.62205 75542 92119 81046
,
where
is
Gauss's constant
, 1718 to 1798
, data-sort-value="5",
, -
,
Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
,
, 0.57721 56649 01532 86060
,
, 1735
, data-sort-value="7",
, -
,
Erdős–Borwein constant
,
, 1.60669 51524 15291 76378
,
, 1749
, data-sort-value="6",
, -
,
Omega constant
,
, 0.56714 32904 09783 87299
,
where W is the
Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential function ...
, 1758 & 1783
, data-sort-value="5",
, -
,
Apéry's constant
In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number
:
\begin
\zeta(3) &= \sum_^\infty \frac \\
&= \lim_ \left(\frac + \frac + \cdots + \frac\right),
\end
...
,
, 1.20205 69031 59594 28539
,
, 1780
, data-sort-value="6",
, -
,
Laplace limit In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately
: 0.66274 34193 49181 58097 47420 97109 25290.
K ...
,
, 0.66274 34193 49181 58097
, Real root of
, data-sort-value="1782", ~1782
, data-sort-value="5",
, -
,
Ramanujan–Soldner constant
,
, 1.45136 92348 83381 05028
,
; root of the
logarithmic integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
function.
, 1792
, data-sort-value="7",
, -
,
Gauss's constant
,
, 0.83462 68416 74073 18628
,
where ''agm'' is the
arithmetic–geometric mean
, 1799
, data-sort-value="5",
, -
, Second
Hermite constant
,
, 1.15470 05383 79251 52901
,
, 1822 to 1901
, data-sort-value="4",
, -
,
Liouville's constant
,
, 0.11000 10000 00000 00000 0001
,
, data-sort-value="1844", Before 1844
, data-sort-value="5",
, -
, First
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
constant
,
, 0.69777 46579 64007 98201
,
, where
is the
modified Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
, 1855
, data-sort-value="6",
, -
,
Ramanujan's constant In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factoriza ...
,
, 262 53741 26407 68743
.99999 99999 99250 073
,
, 1859
, data-sort-value="5",
, -
,
Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those ...
,
, 1.28242 71291 00622 63687
,
, 1860
, data-sort-value="7",
, -
,
Catalan's constant
In mathematics, Catalan's constant , is defined by
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function. Its numerical value is approximately
:
It is not known whether is irra ...
,
, 0.91596 55941 77219 01505
,
, 1864
, data-sort-value="7",
, -
,
Dottie number
The Dottie number is the unique real fixed point of the cosine function.
In mathematics, the Dottie number is a constant that is the unique real root of the equation
: \cos x = x ,
where the argument of \cos is in radians. The decimal expan ...
,
, 0.73908 51332 15160 64165
, Real root of
, 1865
, data-sort-value="5",
, -
,
Meissel–Mertens constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard– de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in ...
,
, 0.26149 72128 47642 78375
,
where ''γ'' is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
and ''p'' is prime
, 1866 & 1873
, data-sort-value="7",
, -
,
Universal parabolic constant
,
, 2.29558 71493 92638 07403
,
, data-sort-value="1891", Before 1891
, data-sort-value="5",
, -
,
Cahen's constant
,
, 0.64341 05462 88338 02618
,
where ''s
k'' is the ''k''th term of ''
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are
:2, 3, 7, 43, 1807, 3263443, 10650056950807, 11342371305542184 ...
'' 2, 3, 7, 43, 1807, ...
, 1891
, data-sort-value="5",
, -
,
Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is , that is, raised to the power . Like both and , this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application ...
,
, 23.14069 26327 79269 0057
,
, 1900
, data-sort-value="5",
, -
,
Gelfond–Schneider constant
,
, 2.66514 41426 90225 18865
,
, data-sort-value="1902", Before 1902
, data-sort-value="5",
, -
, Second
Favard constant
,
, 1.23370 05501 36169 82735
,
, 1902 to 1965
, data-sort-value="5",
, -
,
Golden angle
,
, 2.39996 32297 28653 32223
,
or
in degrees
, 1907
, data-sort-value="5",
, -
,
Sierpiński's constant
,
, 2.58498 17595 79253 21706
,
, 1907
, data-sort-value="7",
, -
,
Landau–Ramanujan constant In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number ''b'' that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x, the number of positive integers below x that are t ...
,
, 0.76422 36535 89220 66299
,
, 1908
, data-sort-value="7",
, -
, First
Nielsen–
Ramanujan constant
,
, 0.82246 70334 24113 21823
,
, 1909
, data-sort-value="5",
, -
,
Gieseking constant
,
, 1.01494 16064 09653 62502
,
.
, 1912
, data-sort-value="7",
, -
,
Bernstein's constant
,
, 0.28016 94990 23869 13303
,
, where ''E''
''n''(f) is the error of the best
uniform approximation to a
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
''f''(''x'') on the interval
minus;1, 1by real polynomials of no more than degree ''n'', and ''f''(''x''
) = , ''x''
,
, 1913
, data-sort-value="7",
, -
,
Tribonacci constant
,
, 1.83928 67552 14161 13255
,
Real root of
, 1914 to 1963
, data-sort-value="4",
, -
,
Brun's constant
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by V ...
,
, 1.90216 05831 04
,
where the sum ranges over all primes ''p'' such that ''p'' + 2 is also a prime
, 1919
, data-sort-value="7",
, -
,
Twin primes constant
,
, 0.66016 18158 46869 57392
,
, 1922
, data-sort-value="7",
, -
,
Plastic number
In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic ...
,
, 1.32471 79572 44746 02596
,
Real root of
, 1924
, data-sort-value="4",
, -
,
Bloch's constant
,
, data-sort-value=0.43320,
, The best known bounds are
, 1925
, data-sort-value="7",
, -
,
Z score for the 97.5 percentile point
,
, 1.95996 39845 40054 23552
,
where is the
inverse error function
Real number
such that
, 1925
, data-sort-value="7",
, -
,
Landau's constant
,
, data-sort-value=0.50000,
, The best known bounds are
, 1929
, data-sort-value="7",
, -
,
Landau's third constant
,
, data-sort-value=0.50000,
,
, 1929
, data-sort-value="7",
, -
,
Prouhet–Thue–Morse constant In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
,
, 0.41245 40336 40107 59778
,