Home > Mathematical constant (sorted by continued fraction representation)

Here is a sorting of mathematical constants by their representations as continued fractions:

(Constants known to be irrational have infinite continued fractions: their last term is ....)

Symbol \| Numeral	Number set	Value	Continued fraction representation
		[ Base 10 ]
Λ		> – 2.7 · 10^-9
0		= 0	[0;]
1/2		= 0.5	[0; 1, 1]
C₂		≈ 0.66016 18158 46869 57392 78121 10014 55577	[0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, 2, 4, 21, 2, 405, 2, 1, 33, 1, 1]
γ		≈ 0.57721 56649 01532 86060 65120 90082 40243	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, 11, 3, 7, 1, 7, 1, 1, 5, 1, 49, ...]
Ω or W(1)		≈ 0.56714 32904 09783 87299 99686 62210 35555	[0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, 1, 1, 1, 1, 7, 1, 4, 2]
β^*		≈ 0.70258	[0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5]
K	?	≈ 0.76422 36535 89220 66	[0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, 3, 3, 18, 2, 1, 2, 1, 2, 1, 6]
B₄In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Brun's constant for twin primes and usually denoted by B (sequence in OEIS): : in stark		≈ 0.87058 83800	[0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23]
KCatalan's constant ''K which occasionally appears in estimates in combinatorics, is defined by : or equivalently : Its numerical value is approximately K . 915 965 594 177 219 015 054 603 514 932 384 110 774. It is not known whether K is rational or irrat		≈ 0.91596 55941 77219 01505 46035 14932 38411	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, 9, 3, 1, 1, 1, 1, 1, 1, 2, 2]
1/2		= 0.5	[0; 2]
M₁Real numbers The Meissel-Mertens constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm: : Its		≈ 0.26149 72128 47642 78375 54268 38608 69585	[0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, 1, 5, 19, 1, 5, 1, 1, 1, 1, 1]
1		= 1	[1;]
φ		≈ 1.61803 39887 49894 84820 45868 34365 63811	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...]
E_BThe Erdos-Borwein constant is the sum of the reciprocals of the Mersenne numbers. By definition it is: It can be proved that the following forms are equivalent to the former: where represents a multiplicative function, the number of positive divisors of t		≈ 1.60669 51524 15291 763	[1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, 2, 1, 1, 1, 20, 1, 3, 1, 1, 1, ...]
B₂In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Brun's constant for twin primes and usually denoted by B (sequence in OEIS): : in stark		≈ 1.90216 05823	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, 2, 2]
K		≈ 1.13198 824	[1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2]
B´_L		≈ 1.08366	[1; 11, 1, 20, 2, 1, 12, 2, 2]
√2		≈ 1.41421 35623 73095 04880 16887 24209 69807	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]
μThe Ramanujan-Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function. Its value is approximately x ≈ 1. 451369234883381050283968485892027449493. Mathematical constants Real numbers.		≈ 1.45136 92348 83381 05028 39684 85892 027	[1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, 7, 2, 1, 1, 1, 2, 30, 6, 3, 6]
2		= 2	[2;]
α		≈ 2.50290 78750 95892 82228 39028 73218 21578	[2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, 17, 1, 1, 5, 3, 2, 6, 3, 5, 1]
eThe mathematical constant e (occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function.		≈ 2.71828 18284 59045 23536 02874 71352 66249	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, ...]
K_h		≈ 2.68545 20011 19507 41674	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, 1, 1, 90]
3		= 3	[3;]
π		≈ 3.14159 26535 89793 23846 26433 83279 50288	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, ...]
4		= 4	[4;]
δ		≈ 4.66920 16091 02990 67185 32038 20466 20161	[4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, 1, 33, 1, 1, 53, 1, 1, 1, 1, 1]

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