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Encyclopedia > Mathematical constants (sorted by continued fraction representation)

This is a list of mathematical constants sorted by their representations as continued fractions: A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

(Constants known to be irrational have infinite continued fractions: their last term is ....) In mathematics, an irrational number is any real number that is not a rational number, i. ...

Symbol
|Numeral
Number set Value Continued fraction representation
[ Base 10 ]
Λ   > – 2.7 · 10-9

0 A numeral is a symbol or group of symbols that represents a number. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A number is an abstract entity that represents a count or measurement. ... A numeral is a symbol or group of symbols that represents a number. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

= 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]
B4   ≈ 0.87058 83800 [0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23]
K   ≈ 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]
M1   ≈ 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 One half is the irreducible fraction resulting from dividing one by two (Â½), or any number by its double; multiplication by one half is equivalent to division by two. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ... The Omega constant is the value of W(1) where W is Lamberts W function. ... In mathematics, the Embree-Trefethen constant is a threshold value in number theory labelled Î²*. For a fixed real Î², consider the recurrence xn+1=xnÂ±Î²xn-1 where the sign in the sum is chosen at random for each n independently with equal probabilities for + and -. In can be proven that... In 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 Bruns constant for twin primes and usually denoted by B2 (sequence A065421 in OEIS): in stark contrast to the... Catalans constant K, which occasionally appears in estimates in combinatorics, is defined by or equivalently along with where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. ... One half is the irreducible fraction resulting from dividing one by two (Â½), or any number by its double; multiplication by one half is equivalent to division by two. ... 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 value is approximately M &#8776; 0. ...

= 1 [1;]
ρ ≈ 1.32471 95724 47460 25960 90885 44780 97340 [1; 1, 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, 3, 1, 8, 2, 1, 1, 14, 1, 1, 2, 1, 1, 1, 3, 1, 10, 4 ...]
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, ...]
EB   ≈ 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, ...]
B2   ≈ 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]
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, ...]
μ   ≈ 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 The golden section is a line segment sectioned into two according to the golden ratio. ... The ErdÅ‘sâ€“Borwein constant is the sum of the reciprocals of the Mersenne numbers. ... In 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 Bruns constant for twin primes and usually denoted by B2 (sequence A065421 in OEIS): in stark contrast to the... The Ramanujan-Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function. ...

= 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]
e   ≈ 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, ...]
Kh   ≈ 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 e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the infinitely many terms ai of the continued fraction expansion of x have an astonishing property: their geometric mean is a constant, known as Khinchins constant, which is independent of the value of x. ...

= 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 When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ...

= 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]

The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ...

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