Mathematical Constants And Functions

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. 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.

List

{, class="wikitable sortable"
, -
! Name
! Symbol
! Decimal expansion
! Formula
! Year
! Set
, -
, One
, 1
, 1
,
, data-sort-value="-2000", Prehistory
, data-sort-value="1", $\mathbb{N}$
, -
, Two
, 2
, 2
,
, data-sort-value="-2000", Prehistory
, data-sort-value="1", $\mathbb{N}$
, -
, One half
, 1/2
, data-sort-value="0.50000", 0.5
,
, data-sort-value="-2000", Prehistory
, data-sort-value="3", $\mathbb{Q}$
, -
, Pi
, $\pi$
, 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", $\mathbb{T}$
, -
, Square root of 2,
Pythagoras constant.
, $\sqrt{2}$
, 1.41421 35623 73095 04880
, Positive root of $x^2=2$
, data-sort-value="-1800", 1800 to 1600 BCE
, data-sort-value="4", $\mathbb{A}$
, -
, Square root of 3,
Theodorus' constant
, $\sqrt{3}$
, 1.73205 08075 68877 29352
, Positive root of $x^2=3$
, data-sort-value="-465", 465 to 398 BCE
, data-sort-value="4", $\mathbb{A}$
, -
, Square root of 5
, $\sqrt{5}$
, 2.23606 79774 99789 69640
, Positive root of $x^2=5$
, data-sort-value="-464",
, data-sort-value="4", $\mathbb{A}$
, -
, Phi, Golden ratio
, $\varphi$ or $\phi$
, 1.61803 39887 49894 84820
, $\frac{1+\sqrt{5{2}$
, data-sort-value="-301", ~300 BCE
, data-sort-value="4", $\mathbb{A}$
, -
, Silver ratio
, $\delta_S$
, 2.41421 35623 73095 04880
, $\sqrt{2}+1$
, data-sort-value="-301", ~300 BCE
, data-sort-value="4", $\mathbb{A}$
, -
, Zero
, 0
, 0
,
, data-sort-value="-300", 300 to 100 BCE
, data-sort-value="2", $\mathbb{Z}$
, -
, Negative one
, −1
, −1
,
, data-sort-value="-300", 300 to 200 BCE
, data-sort-value="2", $\mathbb{Z}$
, -
, Cube root of 2
, \sqrt 2}
, 1.25992 10498 94873 16476
, Real root of $x^3=2$
, 46 to 120 CE
, data-sort-value="4", $\mathbb{A}$
, -
, Cube root of 3
, \sqrt 3}
, 1.44224 95703 07408 38232
, Real root of $x^3=3$
, data-sort-value="47",
, data-sort-value="4", $\mathbb{A}$
, -
, Twelfth root of 2
, \sqrt 22}
, 1.05946 30943 59295 26456
, Real root of $x^{12}=2$
, data-sort-value="47",
, data-sort-value="4", $\mathbb{A}$
, -
, Supergolden ratio
, $\psi$
, 1.46557 12318 76768 02665
, \frac{1 + \sqrt \frac{29 + 3\sqrt{93{2 + \sqrt \frac{29 - 3\sqrt{93{2}{3}
Real root of $x^{3} = x^{2} + 1$
, data-sort-value="47",
, data-sort-value="4", $\mathbb{A}$
, -
, Imaginary unit
, $i$
, data-sort-value="0",
, Either of the two roots of $x^2=-1$
, 1501 to 1576
, data-sort-value="8", $\mathbb{C}$
, -
, Connective constant for the hexagonal lattice
, $\mu$
, 1.84775 90650 22573 51225
, $\sqrt{2 + \sqrt{2$, as a root of the polynomial $x ^ 4-4 x ^ 2 + 2=0$
, 1593
, data-sort-value="4", $\mathbb{A}$
, -
, Kepler–Bouwkamp constant
, $K'$
, 0.11494 20448 53296 20070
, $\prod_{n=3}^\infty \cos\left(\frac{\pi}{n} \right) = \cos\left(\frac{\pi}{3} \right) \cos\left(\frac{\pi}{4} \right) \cos\left(\frac{\pi}{5}\right) ...$
, 1596
, data-sort-value="7",
, -
, Wallis's constant
,
, 2.09455 14815 42326 59148
, \sqrt \frac{45-\sqrt{1929{18+\sqrt \frac{45+\sqrt{1929{18
Real root of $x^{3} - 2x - 5 = 0$
, 1616 to 1703
, data-sort-value="4", $\mathbb{A}$
, -
, Euler's number
, $e$
, 2.71828 18284 59045 23536
, $\lim_{n \to \infty} \left( 1 + \frac {1}{n}\right)^n = \sum_{n=0}^{\infty}\frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} \cdots$
, 1618
, data-sort-value="5", $\mathbb{T}$
, -
, Natural logarithm of 2
, $\ln 2$
, 0.69314 71805 59945 30941
, Real root of $e^{x} = 2$
$\sum_{n=1}^\infty \frac{(-1)^{n+1{n} = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$
, 1619 & 1668
, data-sort-value="5", $\mathbb{T}$
, -
, Lemniscate constant
, $\varpi$
, 2.62205 75542 92119 81046
, $\pi \, {G} = 4 \sqrt{\tfrac2\pi}\,\Gamma{\left(\tfrac54 \right)^2} = \tfrac14 \sqrt{\tfrac{2}{\pi\,\Gamma {\left(\tfrac14 \right)^2}$
where $G$ is Gauss's constant
, 1718 to 1798
, data-sort-value="5", $\mathbb{T}$
, -
, Euler's constant
, $\gamma$
, 0.57721 56649 01532 86060
, $\lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx$
, 1735
, data-sort-value="7",
, -
, Erdős–Borwein constant
, $E$
, 1.60669 51524 15291 76378
, $\sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} \! + \! \frac{1}{3} \! + \! \frac{1}{7} \! + \! \frac{1}{15} \! + \! \cdots$
, 1749
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Omega constant
, $\Omega$
, 0.56714 32904 09783 87299
, $W(1)=\frac{1}{\pi}\int_0^\pi\log\left(1+\frac{\sin t}{t}e^{t\cot t}\right)dt$
where W is the Lambert W function
, 1758 & 1783
, data-sort-value="5", $\mathbb{T}$
, -
, Apéry's constant
, $\zeta(3)$
, 1.20205 69031 59594 28539
, $\sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots$
, 1780
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Laplace limit
,
, 0.66274 34193 49181 58097
, Real root of $\frac{ x e^\sqrt{x^2+1{\sqrt{x^2+1}+1} = 1$
, data-sort-value="1782", ~1782
, data-sort-value="5", $\mathbb{T}$
, -
, Ramanujan–Soldner constant
, $\mu$
, 1.45136 92348 83381 05028
, $\mathrm{li}(x) = \int_0^x \frac{dt}{\ln t} = 0$; root of the logarithmic integral function.
, 1792
, data-sort-value="7",
, -
, Gauss's constant
, $G$
, 0.83462 68416 74073 18628
, $\frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{\Gamma(\frac{1}{4})^2}{2\sqrt{2 \pi^3 = \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4$
where ''agm'' is the arithmetic–geometric mean
, 1799
, data-sort-value="5", $\mathbb{T}$
, -
, Second Hermite constant
, $\gamma_{2}$
, 1.15470 05383 79251 52901
, $\frac{2}{\sqrt{3$
, 1822 to 1901
, data-sort-value="4", $\mathbb{A}$
, -
, Liouville's constant
, $L$
, 0.11000 10000 00000 00000 0001
, $\sum_{n=1}^\infty \frac{1}{10^{n! = \frac {1}{10^{1! + \frac{1}{10^{2! + \frac{1}{10^{3! + \frac{1}{10^{4! + \cdots$
, data-sort-value="1844", Before 1844
, data-sort-value="5", $\mathbb{T}$
, -
, First continued fraction constant
, $C_1$
, 0.69777 46579 64007 98201
, $\tfrac 1 {1+\tfrac 1{2+\tfrac 1{3+\tfrac 1{4+\tfrac 1{5+\cdots }$
$\frac{I_1(2)}{I_0(2)}$, where $I_{\alpha}(x)$ is the modified Bessel function
, 1855
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Ramanujan's constant
,
, 262 53741 26407 68743
.99999 99999 99250 073
, $e^{\pi\sqrt{163$
, 1859
, data-sort-value="5", $\mathbb{T}$
, -
, Glaisher–Kinkelin constant
, $A$
, 1.28242 71291 00622 63687
, $e^{\frac{1}{12}-\zeta^\prime(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^\infty \frac{1}{n+1} \sum\limits_{k=0}^n \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)}$
, 1860
, data-sort-value="7",
, -
, Catalan's constant
, $G$
, 0.91596 55941 77219 01505
, $\int_0^1 \!\! \int_0^1 \!\! \frac{dx \, dy}{1{+}x^2 y^2} = \! \sum_{n = 0}^\infty \! \frac{(-1)^n}{(2n{+}1)^2} \! = \! \frac{1}{1^2}{-}\frac{1}{3^2}{+}{\cdots}$
, 1864
, data-sort-value="7",
, -
, Dottie number
,
, 0.73908 51332 15160 64165
, Real root of $\cos x = x$
, 1865
, data-sort-value="5", $\mathbb{T}$
, -
, Meissel–Mertens constant
, $M$
, 0.26149 72128 47642 78375
, $\lim_{n\to\infty}\left(\sum_{p\le n}\frac{1}{p}\ln\ln n\right) = \gamma + \sum_{p}\left(\ln\left(1 - \frac{1}{p}\right) + \frac{1}{p}\right)$
where ''γ'' is the Euler–Mascheroni constant and ''p'' is prime
, 1866 & 1873
, data-sort-value="7",
, -
, Universal parabolic constant
, $P$
, 2.29558 71493 92638 07403
, $\ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arsinh}(1)+\sqrt{2}$
, data-sort-value="1891", Before 1891
, data-sort-value="5", $\mathbb{T}$
, -
, Cahen's constant
, $C$
, 0.64341 05462 88338 02618
, $\sum_{k=1}^{\infty} \frac{(-1)^{k{s_k-1} = \frac{1}{1} - \frac{1}{2} + \frac{1}{6} - \frac{1}{42} + \frac{1}{1806} {\,\pm \cdots}$
where ''s_k'' is the ''k''th term of '' Sylvester's sequence'' 2, 3, 7, 43, 1807, ...
, 1891
, data-sort-value="5", $\mathbb{T}$
, -
, Gelfond's constant
, $e^{\pi}$
, 23.14069 26327 79269 0057
, $(-1)^{-i} = i^{-2i} = \sum_{n=0}^\infty \frac{\pi^{n{n!} = 1 + \frac{\pi^{1{1} + \frac{\pi^{2{2} + \frac{\pi^{3{6} + \cdots$
, 1900
, data-sort-value="5", $\mathbb{T}$
, -
, Gelfond–Schneider constant
, $2^{\sqrt{2$
, 2.66514 41426 90225 18865
,
, data-sort-value="1902", Before 1902
, data-sort-value="5", $\mathbb{T}$
, -
, Second Favard constant
, $K_{2}$
, 1.23370 05501 36169 82735
, $\frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots$
, 1902 to 1965
, data-sort-value="5", $\mathbb{T}$
, -
, Golden angle
, $g$
, 2.39996 32297 28653 32223
, $\frac{2\pi}{\varphi^2} = \pi (3-\sqrt{5})$ or

$180 (3-\sqrt{5})=137.50776\ldots$ in degrees
, 1907
, data-sort-value="5", $\mathbb{T}$
, -
, Sierpiński's constant
, $K$
, 2.58498 17595 79253 21706
, $\begin{align} &\pi\left(2\gamma+\ln\frac{4\pi^3}{\Gamma(\tfrac{1}{4})^4}\right) = \pi (2 \gamma + 4 \ln\Gamma(\tfrac{3}{4}) - \ln\pi) \\ &= \pi \left(2 \ln 2+3 \ln \pi + 2 \gamma - 4 \ln \Gamma (\tfrac{1}{4})\right) \end{align}$
, 1907
, data-sort-value="7",
, -
, Landau–Ramanujan constant
, $K$
, 0.76422 36535 89220 66299
, $\frac1{\sqrt2}\prod_ {\left(1-\frac1{p^2}\right)^{-\frac{1}{2}\!\!=\frac\pi4\prod_ {\left(1-\frac1{p^2}\right)^\frac{1}{2$
, 1908
, data-sort-value="7",
, -
, First Nielsen– Ramanujan constant
, $a_{1}$
, 0.82246 70334 24113 21823
, $\frac{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \cdots$
, 1909
, data-sort-value="5", $\mathbb{T}$
, -
, Gieseking constant
, $G$
, 1.01494 16064 09653 62502
, $\frac{3\sqrt{3{4} \left(1- \sum_{n=0}^\infty \frac{1}{(3n+2)^2}+ \sum_{n=1}^\infty\frac{1}{(3n+1)^2} \right)=$

$\textstyle \frac{3\sqrt{3{4} \left( 1 - \frac{1}{2^2} + \frac{1}{4^2}-\frac{1}{5^2}+\frac{1}{7^2}-\frac{1}{8^2}+\frac{1}{10^2} \pm \cdots \right)$.
, 1912
, data-sort-value="7",
, -
, Bernstein's constant
, $\beta$
, 0.28016 94990 23869 13303
, $\lim_{n\to\infty} 2n E_{2n}(f)$, where ''E''_''n''(f) is the error of the best uniform approximation to a real function ''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
, \frac{1+\sqrt 19+3\sqrt{33+\sqrt 19-3\sqrt{33}{3} = \frac{1+4\cosh\left(\frac{1}{3}\cosh^{-1}\left(2+\frac{3}{8}\right)\right)}{3}
Real root of $x^{3} - x^{2} - x - 1 = 0$
, 1914 to 1963
, data-sort-value="4", $\mathbb{A}$
, -
, Brun's constant
, $B_{2}$
, 1.90216 05831 04
, $\textstyle {\sum\limits_p(\frac1{p}+\frac1{p+2})} = (\frac1{3} \! + \! \frac1{5}) + (\tfrac1{5} \! + \! \tfrac1{7}) + (\tfrac1{11} \! + \! \tfrac1{13}) + \cdots$
where the sum ranges over all primes ''p'' such that ''p'' + 2 is also a prime
, 1919
, data-sort-value="7",
, -
, Twin primes constant
, $C_{2}$
, 0.66016 18158 46869 57392
, $\prod_{\textstyle{p\;{\rm prime}\atop p \ge 3 \left(1 - \frac{1}{(p-1)^2}\right)$
, 1922
, data-sort-value="7",
, -
, Plastic number
, $\rho$
, 1.32471 79572 44746 02596
, \sqrt 1 + \! \sqrt 1 + \! \sqrt 1 + \cdots} = \textstyle \sqrt \frac{1}{2}+\frac{\sqrt{69{18+\sqrt \frac{1}{2}-\frac{\sqrt{69{18
Real root of $x^{3} = x + 1$
, 1924
, data-sort-value="4", $\mathbb{A}$
, -
, Bloch's constant
, $B$
, data-sort-value=0.43320, $0.4332\leq B\leq 0.4719$
, The best known bounds are $\frac{\sqrt{3{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2\cdot\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$
, 1925
, data-sort-value="7",
, -
, Z score for the 97.5 percentile point
, $z_{.975}$
, 1.95996 39845 40054 23552
, $\sqrt{2}\operatorname{erf}^{-1}(0.95)$ where is the inverse error function
Real number $z$ such that $\frac{1}{\sqrt{2\pi\int_{-\infty}^{z} e^{-x^2/2} \, \mathrm{d}x = 0.975$
, 1925
, data-sort-value="7",
, -
, Landau's constant
, $L$
, data-sort-value=0.50000, $0.5 < L \le 0.54326$
, The best known bounds are $0.5 < L \le \frac{\Gamma(\frac{1}{3})\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6})}$
, 1929
, data-sort-value="7",
, -
, Landau's third constant
, $A$
, data-sort-value=0.50000, $0.5 < A \le 0.7853$
,
, 1929
, data-sort-value="7",
, -
, Prouhet–Thue–Morse constant
, $\tau$
, 0.41245 40336 40107 59778
, \sum_{n=0}^{\infty} \frac{t_n}{2^{n+1 = \frac{1}{4}\left -\prod_{n=0}^{\infty}\left(1-\frac{1}{2^{2^n\right)\right/math>
where ${t_n}$ is the ''n''^th term of the Thue–Morse sequence
, 1929
, data-sort-value="5", $\mathbb{T}$
, -
, Golomb–Dickman constant
, $\lambda$
, 0.62432 99885 43550 87099
, $\int_{0}^{1} e^{\mathrm{Li}(t)} dt = \int_0^{\infty} \frac{\rho(t)}{t+2} dt$
where Li(''t'') is the logarithmic integral, and ''ρ''(''t'') is the Dickman function
, 1930 & 1964
, data-sort-value="7",
, -
, Constant related to the asymptotic behavior of Lebesgue constants
, $c$
, 0.98943 12738 31146 95174
, $\lim_{n\to\infty}\!\! \left(\!{L_n{-}\frac{4}{\pi^2}\ln(2n{+}1)}\!\!\right)\!{=} \frac{4}{\pi^2}\!\left({\sum_{k=1}^\infty \!\frac{2\ln k}{4k^2{-}1 {-}\frac{\Gamma'(\tfrac12)}{\Gamma(\tfrac12)}\!\!\right)$
, 1930
, data-sort-value="7",
, -
, Feller–Tornier constant
, $\mathcal{C}_{\mathrm{FT$
, 0.66131 70494 69622 33528
, ${\frac{1}{2}\prod_{p\text{ prime \left(1-\frac{2}{p^2}\right) + \frac{1}{2 =\frac{3}{\pi^2}\prod_{p\text{ prime \left(1-\frac{1}{p^2-1}\right) + \frac{1}{2}$
, 1932
, data-sort-value="7",
, -
, Base 10 Champernowne constant
, $C_{10}$
, 0.12345 67891 01112 13141
, Defined by concatenating representations of successive integers:
0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
, 1933
, data-sort-value="5", $\mathbb{T}$
, -
, Salem constant
, $\sigma_{10}$
, 1.17628 08182 59917 50654
, Largest real root of $x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1=0$
, 1933
, data-sort-value="4", $\mathbb{A}$
, -
, Khinchin's constant
, $K_{0}$
,  
, \prod_{n=1}^\infty \left 1+{1\over n(n+2)\right{\log_2(n)}
, 1934
, data-sort-value="7",
, -
, Lévy's constant (1)
, $\beta$
, 1.18656 91104 15625 45282
, $\frac {\pi^2}{12\,\ln 2}$
, 1935
, data-sort-value="7",
, -
, Lévy's constant (2)
, $e^{\beta}$
, 3.27582 29187 21811 15978
, $e^{\pi^2/(12\ln2)}$
, 1936
, data-sort-value="7",
, -
, Copeland–Erdős constant
, $\mathcal{C}_{CE}$
, 0.23571 11317 19232 93137
, Defined by concatenating representations of successive prime numbers:
0.2 3 5 7 11 13 17 19 23 29 31 37 ...
, 1946
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Mills' constant
, $A$
, 1.30637 78838 63080 69046
, Smallest positive real number ''A'' such that $\lfloor A^{3^{n \rfloor$ is prime for all positive integers ''n''
, 1947
, data-sort-value="7",
, -
, Gompertz constant
, $\delta$
, 0.59634 73623 23194 07434
, $\int_0^\infty \!\! \frac{e^{-x{1+x} \, dn = \!\! \int_0^1 \!\! \frac{dx}{1-\ln x} = {\tfrac 1 {1+\tfrac 1{1+\tfrac 1{1+\tfrac 2{1+\tfrac 2{1+\tfrac 3{1+3{/\cdots$
, data-sort-value="1948", Before 1948
, data-sort-value="7",
, -
, de Bruijn–Newman constant
, $\Lambda$
, data-sort-value="0", $0\le\Lambda\le0.2$
, The number Λ where for where $H(\lambda,z)=\int^{\infty}_0e^{\lambda u^2}\Phi(u)\cos(zu)du$ has real zeros if and only if λ ≥ Λ.

where $\Phi(u)=\sum_{n=1}^{\infty}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u$.
, 1950
, data-sort-value="7",
, -
, Van der Pauw constant
, $\frac{\pi}{\ln 2}$
, 4.53236 01418 27193 80962
,
, data-sort-value="1958", Before 1958
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Magic angle
, $\theta_{\mathrm{m$
, 0.95531 66181 245092 78163
, $\arctan \sqrt{2} = \arccos \tfrac{1}{\sqrt 3} \approx \textstyle {54.7356} ^{ \circ }$
, data-sort-value="1959", Before 1959
, data-sort-value="5", $\mathbb{T}$
, -
, Artin's constant
, $C_{\mathrm{Artin$
, 0.37395 58136 19202 28805
, $\prod_{p\text{ prime \left(1-\frac{1}{p(p-1)}\right)$
, data-sort-value="1961", Before 1961
, data-sort-value="7",
, -
, Porter's constant
, $C$
, 1.46707 80794 33975 47289
, $\frac{6\ln 2}{\pi ^2} \left(3 \ln 2 + 4 \,\gamma -\frac{24}{\pi ^2} \,\zeta '(2)-2 \right)-\frac{1}{2}$
where ''γ'' is the Euler–Mascheroni constant and is the derivative of the Riemann zeta function evaluated at ''s'' = 2
, 1961
, data-sort-value="7",
, -
, Lochs constant
, $L$
, 0.97027 01143 92033 92574
, $\frac {6 \ln 2 \ln 10}{ \pi^2}$
, 1964
, data-sort-value="7",
, -
, DeVicci's tesseract constant
,
, 1.00743 47568 84279 37609
, The largest cube that can pass through in an 4D hypercube.
Positive root of $4x^8{-}28x^6{-}7x^4{+}16x^2{+}16=0$
, 1966
, data-sort-value="4", $\mathbb{A}$
, -
, Lieb's square ice constant
,
, 1.53960 07178 39002 03869
, $\left(\frac{4}{3}\right)^\frac{3}{2}=\frac{8}{3\sqrt3}$
, 1967
, data-sort-value="4", $\mathbb{A}$
, -
, Niven's constant
, $C$
, 1.70521 11401 05367 76428
, $1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)$
, 1969
, data-sort-value="7",
, -
, Stephens' constant
,
, 0.57595 99688 92945 43964
, $\prod_{p\text{ prime \left(1 - \frac{p}{p^3-1}\right)$
, 1969
, data-sort-value="7",
, -
, Regular paperfolding sequence
, $P$
, 0.85073 61882 01867 26036
, $\sum_{n=0}^{\infty} \frac {8^{2^n{2^{2^{n+2-1} = \sum_{n=0}^{\infty} \cfrac {\tfrac {1}{2^{2^n} {1-\tfrac{1}{2^{2^{n+2$
, 1970
, data-sort-value="5", $\mathbb{T}$
, -
, Reciprocal Fibonacci constant
, $\psi$
, 3.35988 56662 43177 55317
, $\sum_{n=1}^{\infty} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \cdots$
where ''F_n'' is the ''n''^th Fibonacci number
, 1974
, data-sort-value="6", $\mathbb{R} \setminus \mathbb{Q}$
, -
, Chvátal–Sankoff constant for the binary alphabet
, $\gamma_2$
, data-sort-value=0.78807 10000, $0.788071 \le \gamma_2 \le 0.826280$
, $\lim_{n\to\infty}\frac{\operatorname{E}$lambda_{n,2}{n}
where is the expected longest common subsequence of two random length-''n'' binary strings
, 1975
, data-sort-value="7",
, -
, Feigenbaum constant δ
, $\delta$
, 4.66920 16091 02990 67185
, $\lim_{n \to \infty}\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1$
where the sequence ''x_n'' is given by $x_{n+1} = ax_n(1-x_n)$
, 1975
, data-sort-value="7",
, -
, Chaitin's constants
, $\Omega$
, data-sort-value=0.00787 49969 97812 3844, In general they are uncomputable numbers.
But one such number is 0.00787 49969 97812 3844.

, $\sum_{p \in P} 2^{-, p$

*: Halted program
*}: Size in bits of program
*: Domain of all programs that stop.

, 1975
, data-sort-value="5", $\mathbb{T}$
, -
, Robbins constant
, $\Delta(3)$
, 0.66170 71822 67176 23515
, $\frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{5}$
, 1977
, data-sort-value="5", $\mathbb{T}$
, -
, Weierstrass constant
,
, 0.47494 93799 87920 65033
, $\frac{2^{5/4} \sqrt{\pi} \, e^{\pi/8{\Gamma(\frac{1}{4})^{2$
, data-sort-value="1978", Before 1978
, data-sort-value="5", $\mathbb{T}$
, -
, Fransén–Robinson constant
, $F$
, 2.80777 02420 28519 36522
, $\int_{0}^\infty \frac{dx}{\Gamma(x)} = e + \int_0^\infty \frac{e^{-x{\pi^2 + \ln^2 x}\, dx$
, 1978
, data-sort-value="7",
, -
, Feigenbaum constant α
, $\alpha$
, 2.50290 78750 95892 82228
, Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram
, 1979
, data-sort-value="7" ,
, -
, Second du Bois-Reymond constant
, $C_2$
, 0.19452 80494 65325 11361
, $\frac{e^2-7}{2} = \int_0^\infty \left, {\frac{d}{dt}\left(\frac{\sin t}{t}\right)^2}\\,dt-1$
, 1983
, data-sort-value="5" , $\mathbb{T}$
, -
, Erdős–Tenenbaum–Ford constant
, $\delta$
, 0.86071 33205 59342 06887
, $1-\frac{1+\log\log2}{\log2}$
, 1984
, data-sort-value="7" ,
, -
, Conway's constant
, $\lambda$
, 1.30357 72690 34296 39125
, Real root of the polynomial:
$\begin{smallmatrix} x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\ -x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\ +x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\ -12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\ -10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}\\ +3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\ +5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0 \quad\quad\quad \end{smallmatrix}$
, 1987
, data-sort-value="4" , $\mathbb{A}$
, -
, Hafner–Sarnak–McCurley constant
, $\sigma$
, 0.35323 63718 54995 98454
, $\prod_{p\text{ prime{\left(1- \left(1-\prod_{n\ge1}\left(1-\frac{1}{p^n}\right)\right)^2 \right)} \!$
, 1991
, data-sort-value="7" ,
, -
, Backhouse's constant
, $B$
, 1.45607 49485 82689 67139
, $\lim_{k \to \infty}\left , \frac{q_{k+1{q_k} \right \vert \quad \scriptstyle \text {where:} \displaystyle \;\; Q(x)=\frac{1}{P(x)}= \! \sum_{k=1}^\infty q_k x^k$
$P(x) = 1+\sum_{k=1}^\infty {p_k x^k} = 1+2x+3x^2+5x^3+\cdots$where ''p_k'' is the ''k''^th prime number
, 1995
, data-sort-value="7" ,
, -
, Viswanath constant
,
, 1.13198 82487 943
, $\lim_{n \to \infty}, f_n, ^\frac{1}{n}$      where ''f''_''n'' = ''f''_''n''−1 ± ''f''_''n''−2, where the signs + or − are chosen at random with equal probability 1/2
, 1997
, data-sort-value="7" ,
, -
, Komornik–Loreti constant
, $q$
, 1.78723 16501 82965 93301
, Real number $q$ such that $1 = \sum_{n=1}^\infty \frac{t_k}{q^k}$, or $\prod_{n=0}^\infty\left (1-\frac{1}{q^{2^n\right )+\frac{q-2}{q-1}=0$
where ''t_k'' is the ''k''^th term of the Thue–Morse sequence
, 1998
, data-sort-value="5" , $\mathbb{T}$
, -
, Embree–Trefethen constant
, $\beta^{\star}$
, 0.70258
,
, 1999
, data-sort-value="7" ,
, -
, Heath-Brown–Moroz constant
, $C$
, 0.00131 76411 54853 17810
, $\prod_{p\text{ prime \left(1-\frac{1}{p}\right)^7\left(1+\frac{7p+1}{p^2}\right)$
, 1999
, data-sort-value="7" ,
, -
, MRB constant
, $S$
, 0.18785 96424 62067 12024 MRB constant
, \sum_{n=1}^{\infty} (-1)^n (n^{1/n}-1) = - \sqrt 1} + \sqrt 2} - \sqrt 3} + \cdots
, 1999
, data-sort-value="7" ,
, -
, Prime constant
, $\rho$
, 0.41468 25098 51111 66024
, $\sum_{p\text{ prime \frac{1}{2^p}= \frac{1}{4} + \frac{1}{8} + \frac{1}{32} + \cdots$
, 1999
, data-sort-value="6" , $\mathbb{R} \setminus \mathbb{Q}$
, -
, Somos' quadratic recurrence constant
, $\sigma$
, 1.66168 79496 33594 12129
, $\prod_{n=1}^\infty n^} = 1^{1/2} \; 2^{1/4} \; 3^{1/8} \cdots$
, 1999
, data-sort-value="7" ,
, -
, Foias constant
, $\alpha$
, 1.18745 23511 26501 05459
, $x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n\text{ for }n=1,2,3,\ldots$
Foias constant is the unique real number such that if ''x''₁ = ''α'' then the sequence diverges to infinity
, 2000
, data-sort-value="7" ,
, -
, Logarithmic capacity of the unit disk
,
, 0.59017 02995 08048 11302
, $\frac{\Gamma(\tfrac14)^2}{4 \pi^{3/2$
, data-sort-value="2003" , Before 2003
, data-sort-value="5" , $\mathbb{T}$
, -
, Taniguchi constant
,
, 0.67823 44919 17391 97803
, $\prod_{p\text{ prime \left(1 - \frac{3}{p^3}+\frac{2}{p^4}+\frac{1}{p^5}-\frac{1}{p^6}\right)$
, data-sort-value="2005" , Before 2005
, data-sort-value="7" ,

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.

{, class="wikitable sortable"
, -
!Name
! Symbol
! Set
! Decimal expansion
! Continued fraction
! Notes
, -
, Zero
, 0
, , $\mathbb{Z}$
, , 0.00000 00000
, , ; , ,
, -
, Golomb–Dickman constant
, $\lambda$
, ,
, , 0.62432 99885
, , ; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …
, , E. Weisstein noted that the continued fraction has an unusually large number of 1s.
, -
, Cahen's constant
, $C_2$
, , $\mathbb{T}$
, , 0.64341 05463
, , ; 1, 1, 1, 2², 3², 13², 129², 25298², 420984147², 269425140741515486², …
, , 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
, $\gamma$
, ,
, , 0.57721 56649
, , ; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …
, , Using the continued fraction expansion, it was shown that if is rational, its denominator must exceed 10²⁴⁴⁶⁶³.
, -
, First continued fraction constant
, $C_1$
, , $\mathbb{R} \setminus \mathbb{Q}$
, , 0.69777 46579
, , ; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …, , Equal to the ratio $I_{1}(2)/I_{0}(2)$ of modified Bessel functions of the first kind evaluated at 2.
, -
, Catalan's constant
, $G$
, ,
, , 0.91596 55942
, , ; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …
, , Computed up to terms by E. Weisstein.
, -
, One half
, 1/2
, , $\mathbb{Q}$
, , 0.50000 00000
, , ; 2, ,
, -
, Prouhet–Thue–Morse constant
, $\tau$
, , $\mathbb{T}$
, , 0.41245 40336
, , ; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …
, , Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
, -
, Copeland–Erdős constant
, $\mathcal{C}_{CE}$
, , $\mathbb{R} \setminus \mathbb{Q}$
, , 0.23571 11317
, , ; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …
, , Computed up to 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.
, -
, Base 10 Champernowne constant
, $C_{10}$
, , $\mathbb{T}$
, , 0.12345 67891
, , ; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, , 6, 1, …
, , Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C_{10}$ has 2504 digits.
, -
, One
, 1
, , $\mathbb{N}$
, , 1.00000 00000
, , ; , ,
, -
, Phi, Golden ratio
, $\varphi$
, , $\mathbb{A}$
, , 1.61803 39887
, , ; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
, , The convergents are ratios of successive Fibonacci numbers.
, -
, Brun's constant
, $B_2$
, ,
, , 1.90216 05831
, , ; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …, , The ''n''^th roots of the denominators of the ''n''^th convergents are close to Khinchin's constant, suggesting that $B_2$ is irrational. If true, this will prove the twin prime conjecture.
, -
, Square root of 2
, $\sqrt 2$
, , $\mathbb{A}$
, , 1.41421 35624
, , ; 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
, , $\mathbb{N}$
, , 2.00000 00000
, , ; , ,
, -
, Euler's number
, $e$
, , $\mathbb{T}$
, , 2.71828 18285
, , ; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …
, , The continued fraction expansion has the pattern ; 1, 2, 1, 1, 4, 1, ..., 1, 2''n'', 1, ...
, -
, Khinchin's constant
, $K_0$
, ,
, , 2.68545 20011
, , ; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …
, , 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
, , $\mathbb{N}$
, , 3.00000 00000
, , ; , ,
, -
, Pi
, $\pi$
, , $\mathbb{T}$
, , 3.14159 26536
, , ; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …
, , The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of .

Sequences from constants

{, class="wikitable sortable"
, -
! Name
! Symbol
! Formula
! Year
! Set
, -
, Harmonic number
, $H_n$
, $\sum^n_{k=1}\frac{1}{k}$
, data-sort-value="-400" , Antiquity
, data-sort-value="3" , $\mathbb{Q}$
, -
, Gregory coefficients
, $G_n$
, $\frac 1 {n!} \int_0^1 x(x-1)(x-2)\cdots(x-n+1)\, dx = \int_0^1 \binom x n \, dx$
, 1670
, data-sort-value="3" , $\mathbb{Q}$
, -
, Bernoulli number
, $B^\pm_n$
, $\frac{t}{2} \left( \operatorname{coth} \frac{t}{2} \pm 1 \right) = \sum_{m=0}^\infty \frac{B^{\pm{_m t^m}{m!}$
, 1689
, data-sort-value="3" , $\mathbb{Q}$
, -
, Hermite constants
, $\gamma_{n}$
, For a lattice L in Euclidean space R^''n'' with unit covolume, i.e. vol(R^''n''/''L'') = 1, let λ(''L'') denote the least length of a nonzero element of L. Then √γn is the maximum of λ(L) over all such lattices L.
, data-sort-value="1822" , 1822 to 1901
, data-sort-value="7" , $\mathbb{R}$
, -
, Hafner–Sarnak–McCurley constant
, $D(n)$
, D(n)= \prod^\infty_{k=1}\left\{1-\left -\prod^n_{j=1}(1-p_k^{-j}) \right2 \right\}
, data-sort-value="1883" , 1883
, data-sort-value="7" , $\mathbb{R}$
, -
, Stieltjes constants
, $\gamma_n$
, ${\frac {(-1)^{n}n!}{2\pi \int _{0}^e^\zeta \left(e^+1\right)dx.$
, data-sort-value="1894", before 1894
, data-sort-value="7" , $\mathbb{R}$
, -
, Favard constants
, $K_{r}$
, $\frac{4}{\pi}\sum_{n = 0}^\infty \left(\frac{(-1)^n}{2n+1} \right)^{\!r+1}=\frac{4}{\pi}\left( \frac{(-1)^{0(r+1){1^r}+\frac{(-1)^{1(r+1){3^r}+\frac{(-1)^{2(r+1){5^r}+\frac{(-1)^{3(r+1){7^r}+\cdots\right)$
, data-sort-value="1902" , 1902 to 1965
, data-sort-value="7" , $\mathbb{R}$
, -
, Generalized Brun's Constant
, $B_{n}$
, ${\sum\limits_p(\frac1{p}+\frac1{p+n})}$where the sum ranges over all primes ''p'' such that ''p'' + n is also a prime
, data-sort-value="1919" , 1919
, data-sort-value="7" , $\mathbb{R}$
, -
, Champernowne constants
, $C_{b}$
, Defined by concatenating representations of successive integers in base b.
$C_b=\sum^\infty_{n=1}\frac{n}{b^{\left(\sum^n_{k=1}\lceil\log_b(k+1)\rceil\right)$
, 1933
, data-sort-value="5" , $\mathbb{T}$
, -
, Lagrange number
, $L(n)$
, $\sqrt{9-\frac{4}$ where $m_n$ is the nth smallest number such that $m^2+x^2+y^2=3mxy\,$ has positive (x,y).
, data-sort-value="1957", before 1957
, data-sort-value="4" , $\mathbb{A}$
, -
, Feller's coin-tossing constants
, $\alpha_k,\beta_k$
, $\alpha_k$ is the smallest positive real root of $x^{k+1}=2^{k+1}(x-1),\beta_k=\frac{2-\alpha_k}{k+1-k\alpha_k}$
, 1968
, data-sort-value="7" , $\mathbb{A}$
, -
, Stoneham number
, $\alpha_{b,c}$
, $\sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{c^k}c^k}$ where b,c are coprime integers.
, 1973
, data-sort-value="7" , $\mathbb{R} \setminus \mathbb{Q}$
, -
, Beraha constants
, $B(n)$
, $2+2\cos\left(\frac{2\pi}{n}\right)$
, 1974
, data-sort-value="7" , $\mathbb{A}$
, -
, Chvátal–Sankoff constants
, $\gamma_k$
, \lim_{n\to\infty}\frac{E lambda_{n,k}{n}
, 1975
, data-sort-value="7" , $\mathbb{R}$
, -
, Hyperharmonic number
, $H^{(r))}_n$
, $\sum^n_{k=1}H^{(r-1)}_k$ and $H^{(0)}_n=\frac{1}{n}$
, 1995
, data-sort-value="3" , $\mathbb{Q}$
, -
, Gregory number
, $G_x$
, $\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}$ for rational x greater than one.
, data-sort-value="1996", before 1996
, data-sort-value="7" , $\mathbb{R}$
, -
, Metallic mean
,
, $\frac{n+\sqrt{n^2+4{2}$
, data-sort-value="1998" , before 1998
, data-sort-value="4" , $\mathbb{A}$

See also

* Invariant (mathematics)
* List of mathematical symbols
*List of mathematical symbols by subject
* List of numbers
* List of physical constants
*Particular values of the Riemann zeta function
*Physical constant

Notes

References

Site MathWorld Wolfram.com

Site OEIS.org

Site OEIS Wiki

Bibliography

* English translation by Catriona and David Lischka.
*
*
*

Further reading

*

External links

Inverse Symbolic Calculator, Plouffe's Inverter



On-Line Encyclopedia of Integer Sequences (OEIS)

Steven Finch's page of mathematical constants

{{DEFAULTSORT:Mathematical Constants and Functions
*
mathematical constants
Constants
Articles containing video clips
constants
Continued fractions