Mathematical constants by continued fraction representation

This is a list of mathematical constants sorted by their representations as continued fractions.

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	Member of	decimal	Continued fraction	Notes
	$0$	$\mathbb {Z}$	0.00000 00000	[0; ]
Golden ratio conjugate	$\phi ^{-1}$	$\mathbb {A} \setminus \mathbb {Q}$	0.61803 39887	[0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]	irrational
Cahen's constant	$C$	$\mathbb {R}$	0.64341 05463	[0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …]	All terms are squares and truncated at 10 terms due to large size.
First Hardy–Littlewood conjecture	$C_{2}$	$\mathbb {R}$	0.66016 18158	[0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …]	Hardy–Littlewood's twin prime constant. Presumed irrational, but not proved.
Euler-Mascheroni constant	$\gamma$	$\mathbb {R}$	0.57721 56649	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]	Presumed irrational, but not proved.
Omega constant	$\Omega$	$\mathbb {R} \setminus \mathbb {A}$	0.56714 32904	[0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]
Embree–Trefethen constant	$\beta ^{\star }$	$\mathbb {R}$	0.70258	[0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …]	Value only known to 5 decimal places.
Continued fraction constant	Continued fraction constant	$\mathbb {R} \setminus \mathbb {A}$	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to a ratio $I_{1}(2)/I_{2}(2)$ of modified Bessel functions of the first kind evaluated at 2
Landau–Ramanujan constant	$K$	$\mathbb {R} (\setminus \mathbb {Q} ?)$	0.76422 36535	[0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …]	May have been proven irrational.
Gauss's constant	${\frac {1}{M(1,{\sqrt {2}})}}$	$\mathbb {R} \setminus \mathbb {A}$	0.83462 68417	[0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …]	Gauss's constant
Brun's theorem	$B_{4}$	$\mathbb {R}$	0.87058 83800	[0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …]	Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005.
Champernowne constant	$C_{2}$	$\mathbb {R} \setminus \mathbb {A}$	0.86224 01259	[0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …]	Base 2 Champernowne constant. The binary expansion is $C_{2}=0.110111001011101111000\ldots _{2}$
Catalan's constant	$G$	$\mathbb {R}$	0.91596 55942	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]	Presumed irrational, but not proved.
One half	${\frac {1}{2}}$	$\mathbb {Q}$	0.50000 00000	[0; 2]
Bernstein's constant	$\beta$	$\mathbb {R}$	0.28016 94990	[0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …]	Presumed irrational, but not proved.
Meissel–Mertens constant	$M$	$\mathbb {R}$	0.26149 72128	[0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …]	Presumed irrational, but not proved.
MRB constant	$C_{{}_{MRB}}$	$\mathbb {R}$	0.18785 96424	[0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]
Champernowne constant	$C_{10}$	$\mathbb {R} \setminus \mathbb {A}$	0.12345 67891	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, $4.57540\times 10^{165}$, 6, 1, …]	Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C_{10}$ has 2504 digits.
	$1$	$\mathbb {N}$	1.00000 00000	[1; ]
Golden ratio	$\phi$	$\mathbb {A} \setminus \mathbb {Q}$	1.61803 39887	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
Erdős–Borwein constant	$E$	$\mathbb {R} \setminus \mathbb {Q}$	1.60669 51524	[1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …]	Not known whether algebraic or transcendental.
Brun's constant	$B_{2}$	$\mathbb {R}$	1.90216 05831	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]	Brun's twin prime constant. Estimated value; best bounds $1.8304<B_{2}<2.347$.
Square root of 2	${\sqrt {2}}$	$\mathbb {A} \setminus \mathbb {Q}$	1.41421 35624	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]
Ramanujan-Soldner constant	$\mu$	$\mathbb {R}$	1.45136 92349	[1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …]	Presumed irrational, but not proved.
Backhouse's constant	$B$	$\mathbb {R}$	1.45607 49485	[1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, …]
Plastic number	$\rho$	$\mathbb {A} \setminus \mathbb {Q}$	1.32471 95724	[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]
Apéry's constant	$\zeta (3)$	$\mathbb {R} \setminus \mathbb {Q}$	1.20205 69032	[1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]
Random Fibonacci sequence	$V$	$\mathbb {R}$	1.13198 82488	[1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …]	Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica.
	$2$	$\mathbb {N}$	2.00000 00000	[2; ]
Gelfond–Schneider constant	$2^{\sqrt {2}}$	$\mathbb {R} \setminus \mathbb {A}$	2.66514 41426	[2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]
	$\alpha$	$\mathbb {R}$	2.50290 78751	[2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]
Base of the natural logarithm	$e$	$\mathbb {R} \setminus \mathbb {A}$	2.71828 18285	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]
Khinchin's constant	$K_{0}$	$\mathbb {R}$	2.68545 20011	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]
Fransén–Robinson constant	$F$	$\mathbb {R}$	2.80777 02420	[2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]
Universal parabolic constant	$P_{2}$	$\mathbb {R} \setminus \mathbb {A}$	2.29558 71494	[2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]
	$3$	$\mathbb {N}$	3.00000 00000	[3; ]
Reciprocal Fibonacci constant	$\psi$	$\mathbb {R} \setminus \mathbb {Q}$	3.35988 56662	[3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]
	$\pi$	$\mathbb {R} \setminus \mathbb {A}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]
	$4$	$\mathbb {N}$	4.00000 00000	[4; ]
	$\delta$	$\mathbb {R}$	4.66920 16091	[4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]
	$5$	$\mathbb {N}$	5.00000 00000	[5; ]
Gelfond's constant	$e^{\pi }$	$\mathbb {R} \setminus \mathbb {A}$	23.14069 26328	[23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …]	Gelfond's constant. Can also be expressed as $(-1)^{-i}$; from this form, it is transcendental due to the Gelfond–Schneider theorem.

References

Bibliography

• Cuyt, Annie A.M.; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). "Mathematical constants". Handbook of Continued Fractions for Special Functions. Springer Science + Business Media. ISBN 9781402069499. Search this book on
• Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014). Neverending Fractions: An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series. 23. Cambridge University Press. ISBN 9780521186490. ISSN 0950-2815. Search this book on

See also

• Physical constant
• Mathematical constants and functions

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