# List of mathematical constants

A mathematical constant is a key number 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.[1] For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion 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.

## Mathematical constants sorted by their representations as continued fractions

The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known.

Name Symbol Set Decimal expansion Continued fraction Notes
Zero 0 ${\displaystyle \mathbb {Z} }$ 0.00000 00000 [0; ]
Golomb–Dickman constant ${\displaystyle \lambda }$ 0.62432 99885 [0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …][OEIS 95] E. Weisstein noted that the continued fraction has an unusually large number of 1s.[Mw 83]
Cahen's constant ${\displaystyle C_{2}}$ ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$ 0.64341 05463 [0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …][OEIS 96] All terms are squares and truncated at 10 terms due to large size. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant ${\displaystyle \gamma }$ 0.57721 56649[107] [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …] [107][OEIS 97] Using the continued fraction expansion, it was shown that if γ is rational, its denominator must exceed 10244663.
First continued fraction constant ${\displaystyle C_{1}}$ ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$ 0.69777 46579 [0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …] Equal to the ratio ${\displaystyle I_{1}(2)/I_{0}(2)}$ of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant ${\displaystyle G}$ 0.91596 55942[108] [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …] [108][OEIS 98] Computed up to 4851389025 terms by E. Weisstein.[Mw 84]
One half 1/2 ${\displaystyle \mathbb {Q} }$ 0.50000 00000 [0; 2]
Prouhet–Thue–Morse constant ${\displaystyle \tau }$ ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$ 0.41245 40336 [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …][OEIS 99] Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.[109]
Copeland–Erdős constant ${\displaystyle {\mathcal {C}}_{CE}}$ ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$ 0.23571 11317 [0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …][OEIS 100] Computed up to 1011597392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.[Mw 85]
Base 10 Champernowne constant ${\displaystyle C_{10}}$ ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$ 0.12345 67891 [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4.57540×10165, 6, 1, …] [OEIS 101] Champernowne constants in any base exhibit sporadic large numbers; the 40th term in ${\displaystyle C_{10}}$ has 2504 digits.
One 1 ${\displaystyle \mathbb {N} }$ 1.00000 00000 [1; ]
Phi, Golden ratio ${\displaystyle \varphi }$ ${\displaystyle \mathbb {A} }$ 1.61803 39887[110] [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] [111] The convergents are ratios of successive Fibonacci numbers.
Brun's constant ${\displaystyle B_{2}}$ 1.90216 05831 [1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …] The nth roots of the denominators of the nth convergents are close to Khinchin's constant, suggesting that ${\displaystyle B_{2}}$ is irrational. If true, this will prove the twin prime conjecture.[112]
Square root of 2 ${\displaystyle {\sqrt {2}}}$ ${\displaystyle \mathbb {A} }$ 1.41421 35624 [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …] The convergents are ratios of successive Pell numbers.
Two 2 ${\displaystyle \mathbb {N} }$ 2.00000 00000 [2; ]
Euler's number ${\displaystyle e}$ ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$ 2.71828 18285[113] [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …] [114][OEIS 102] The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2n, 1, ...].
Khinchin's constant ${\displaystyle K_{0}}$ 2.68545 20011[115] [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …] [116][OEIS 103] For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three 3 ${\displaystyle \mathbb {N} }$ 3.00000 00000 [3; ]
Pi ${\displaystyle \pi }$ ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$ 3.14159 26536 [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …] [OEIS 104] The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π.

## Sequences of constants

Name Symbol Formula Year Set
Harmonic number ${\displaystyle H_{n}}$ ${\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}}$ Antiquity ${\displaystyle \mathbb {Q} }$
Gregory coefficients ${\displaystyle G_{n}}$ ${\displaystyle {\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx}$ 1670 ${\displaystyle \mathbb {Q} }$
Bernoulli number ${\displaystyle B_{n}^{\pm }}$ ${\displaystyle {\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}}$ 1689 ${\displaystyle \mathbb {Q} }$
Hermite constants[Mw 86] ${\displaystyle \gamma _{n}}$ For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γnn is the maximum of λ1(L) over all such lattices L. 1822 to 1901 ${\displaystyle \mathbb {R} }$
Hafner–Sarnak–McCurley constant[117] ${\displaystyle D(n)}$ ${\displaystyle D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}}$ 1883[Mw 87] ${\displaystyle \mathbb {R} }$
Stieltjes constants ${\displaystyle \gamma _{n}}$ ${\displaystyle {\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.}$ before 1894 ${\displaystyle \mathbb {R} }$
Favard constants[47][Mw 88] ${\displaystyle K_{r}}$ ${\displaystyle {\frac {4}{\pi }}\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{\!r+1}={\frac {4}{\pi }}\left({\frac {(-1)^{0(r+1)}}{1^{r}}}+{\frac {(-1)^{1(r+1)}}{3^{r}}}+{\frac {(-1)^{2(r+1)}}{5^{r}}}+{\frac {(-1)^{3(r+1)}}{7^{r}}}+\cdots \right)}$ 1902 to 1965 ${\displaystyle \mathbb {R} }$
Generalized Brun's Constant[55] ${\displaystyle B_{n}}$ ${\displaystyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+n}})}}$where the sum ranges over all primes p such that p + n is also a prime 1919[OEIS 45] ${\displaystyle \mathbb {R} }$
Champernowne constants[66] ${\displaystyle C_{b}}$ Defined by concatenating representations of successive integers in base b.

${\displaystyle C_{b}=\sum _{n=1}^{\infty }{\frac {n}{b^{\left(\sum _{k=1}^{n}\lceil \log _{b}(k+1)\rceil \right)}}}}$

1933 ${\displaystyle \mathbb {R} \setminus \mathbb {A} }$
Lagrange number ${\displaystyle L(n)}$ ${\displaystyle {\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}}$ where ${\displaystyle m_{n}}$ is the nth smallest number such that ${\displaystyle m^{2}+x^{2}+y^{2}=3mxy\,}$ has positive (x,y). before 1957 ${\displaystyle \mathbb {A} }$
Feller's coin-tossing constants ${\displaystyle \alpha _{k},\beta _{k}}$ ${\displaystyle \alpha _{k}}$ is the smallest positive real root of ${\displaystyle x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}}$ 1968 ${\displaystyle \mathbb {A} }$
Stoneham number ${\displaystyle \alpha _{b,c}}$ ${\displaystyle \sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}}$ where b,c are coprime integers. 1973 ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$
Beraha constants ${\displaystyle B(n)}$ ${\displaystyle 2+2\cos \left({\frac {2\pi }{n}}\right)}$ 1974 ${\displaystyle \mathbb {A} }$
Chvátal–Sankoff constants ${\displaystyle \gamma _{k}}$ ${\displaystyle \lim _{n\to \infty }{\frac {E[\lambda _{n,k}]}{n}}}$ 1975 ${\displaystyle \mathbb {R} }$
Hyperharmonic number ${\displaystyle H_{n}^{(r)}}$ ${\displaystyle \sum _{k=1}^{n}H_{k}^{(r-1)}}$ and ${\displaystyle H_{n}^{(0)}={\frac {1}{n}}}$ 1995 ${\displaystyle \mathbb {Q} }$
Gregory number ${\displaystyle G_{x}}$ ${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}}}}$ for rational x greater than one. before 1996 ${\displaystyle \mathbb {R} }$
Metallic mean ${\displaystyle {\frac {n+{\sqrt {n^{2}+4}}}{2}}}$ before 1998 ${\displaystyle \mathbb {A} }$

## Notes

1. ^ Both i and i are roots of this equation, though neither root is truly "positive" nor more fundamental than the other as they are algebraically equivalent. The distinction between signs of i and i is in some ways arbitrary, but a useful notational device. See imaginary unit for more information.

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