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| This is a list of [[mathematical constant]]s sorted by their representations as [[continued fraction]]s.
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| 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 [[rounding|rounded]] or padded to 10 places if the values are known.
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| {| class="wikitable sortable"
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| ! [[List of letters used in mathematics and science|Symbol]]<ref group="lower-greek">Although some of the symbols in the leftmost column are displayed in black due to math markup peculiarities, all are clickable and link to the respective constant's page.</ref>
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| ! [[set (mathematics)|Member of]]
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| ! [[Decimal representation|decimal]]
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| ! [[Continued fraction]]
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| ! Notes
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| | [[0 (number)|<math>0</math>]]
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| || [[integer|<math>\mathbb{Z}</math>]]
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| || 0.00000 00000
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| || [0; ]
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| | [[Golden ratio#Golden ratio conjugate|<math>\phi^{-1}</math>]]
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| || [[algebraic number|<math>\mathbb{A}\setminus\mathbb{Q}</math>]]
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| || 0.61803 39887
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| || [0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
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| || irrational
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| | [[Cahen's constant|<math>C</math>]]
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| || [[real number|<math>\mathbb{R}</math>]]
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| || 0.64341 05463
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| || [0; 1, 1, 1, 4, 9, 169, 16641, 639988804, 177227652025317609, 72589906463585427805281295977816196, …]
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| || Continued fraction truncated at 10 terms due to large size.
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| | [[Twin prime#First Hardy–Littlewood conjecture|<math>C_2</math>]]
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| || [[real number|<math>\mathbb{R}</math>]]
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| || 0.66016 18158
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| || [0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …]
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| || Hardy–Littlewood's twin prime constant. Presumed [[irrational]], but not proved.
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| | [[Euler–Mascheroni constant|<math>\gamma</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.57721 56649
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| || [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]
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| || Presumed irrational, but not proved.
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| | [[Omega constant|<math>\Omega</math>]]
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| || [[transcendental number|<math>\mathbb{R}\setminus\mathbb{A}</math>]]
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| || 0.56714 32904
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| || [0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]
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| | [[Embree–Trefethen constant|<math>\beta^\star</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.70258
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| || [0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …]
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| || Value only known to 5 decimal places.
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| | [[Landau–Ramanujan constant|<math>K</math>]]
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| || <math>\mathbb{R} (\setminus\mathbb{Q}?)</math>
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| || 0.76422 36535
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| || [0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …]
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| || May have been proven irrational.
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| | [[Gauss's constant|<math>\frac 1{M(1,\sqrt 2)}</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 0.83462 68417
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| || [0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …]
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| || Gauss's constant
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| | [[Brun's theorem#Brun's constant|<math>B_4</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.87058 83800
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| || [0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …]
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| || Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005.
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| | [[Champernowne constant|<math>C_2</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 0.86224 01259
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| || [0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …]
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| || Base 2 Champernowne constant. The binary expansion is <math>C_2 = 0.1 10 11 100 101 110 111 1000\ldots_2</math>
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| | [[Catalan's constant|<math>G</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.91596 55942
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| || [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]
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| || Presumed irrational, but not proved.
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| | [[One half|<math>\frac 12</math>]]
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| || [[rational number|<math>\mathbb{Q}</math>]]
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| || 0.50000 00000
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| || [0; 2]
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| | [[Bernstein's constant|<math>\beta</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.28016 94990
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| || [0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …]
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| || Presumed irrational, but not proved.
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| | [[Meissel–Mertens constant|<math>M</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.26149 72128
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| || [0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …]
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| || Presumed irrational, but not proved.
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| | [[MRB constant|<math> C_{{}_{MRB}}</math>]]
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| || <math>\mathbb{R}</math>
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| || 0.18785 96424
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| || [0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]
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| | [[Champernowne constant|<math>C_{10}</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 0.12345 67891
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| || [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, <math>4.57540\times 10^{165}</math>, 6, 1, …]
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| || Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in <math>C_{10}</math> has 2504 digits.
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| | [[1 (number)|<math>1</math>]]
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| || [[natural number|<math>\mathbb{N}</math>]]
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| || 1.00000 00000
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| || [1; ]
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| | [[Golden ratio|<math>\phi</math>]]
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| || <math>\mathbb{A}\setminus\mathbb{Q}</math>
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| || 1.61803 39887
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| || [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
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| | [[Erdős–Borwein constant|<math>E</math>]]
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| || [[irrational number|<math>\mathbb{R}\setminus\mathbb{Q}</math>]]
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| || 1.60669 51524
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| || [1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …]
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| || Not known whether algebraic or transcendental.
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| | [[Brun's theorem#Brun's constant|<math>B_2</math>]]
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| || <math>\mathbb{R}</math>
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| || 1.90216 05823
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| || [1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]
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| || Brun's twin prime constant. Estimated value; best bounds <math>1.8304<B_2<2.347</math>.
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| | [[Backhouse's constant|<math>B</math>]]
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| || <math>\mathbb{R}</math>
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| || 1.45607 49485
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| || [1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, …]
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| | [[Apéry's constant|<math>\zeta(3)</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{Q}</math>
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| || 1.20205 69032
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| || [1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]
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| | [[Random Fibonacci sequence|<math>V</math>]]
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| || <math>\mathbb{R}</math>
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| || 1.13198 82488
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| || [1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …]
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| || Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica.
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| | [[Square root of 2|<math>\sqrt 2</math>]]
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| || <math>\mathbb{A}\setminus\mathbb{Q}</math>
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| || 1.41421 35624
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| || [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]
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| | [[Ramanujan-Soldner constant|<math>\mu</math>]]
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| || <math>\mathbb{R}</math>
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| || 1.45136 92349
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| || [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …]
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| || Presumed irrational, but not proved.
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| | [[Plastic number|<math>\rho</math>]]
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| || <math>\mathbb{A}\setminus\mathbb{Q}</math>
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| || 1.32471 95724
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| || [1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]
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| | [[2 (number)|<math>2</math>]]
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| || <math>\mathbb N</math>
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| || 2.00000 00000
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| || [2; ]
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| | [[Gelfond–Schneider constant|<math>2^\sqrt 2</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 2.66514 41426
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| || [2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]
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| | [[Feigenbaum constants|<math>\alpha</math>]]
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| || <math>\mathbb{R}</math>
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| || 2.50290 78751
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| || [2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]
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| | [[e (mathematical constant)|<math>e</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 2.71828 18285
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| || [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]
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| | [[Khinchin's constant|<math>K_0</math>]]
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| || <math>\mathbb{R}</math>
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| || 2.68545 20011
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| || [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]
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| | [[Fransén–Robinson constant|<math>F</math>]]
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| || <math>\mathbb{R}</math>
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| || 2.80777 02420
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| || [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]
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| | [[Universal parabolic constant|<math>P_2</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 2.29558 71494
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| || [2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]
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| | [[3 (number)|<math>3</math>]]
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| || <math>\mathbb{N}</math>
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| || 3.00000 00000
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| || [3; ]
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| | [[Pi|<math>\pi</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{A}</math>
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| || 3.14159 26536
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| || [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]
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| | [[Reciprocal Fibonacci constant|<math>\psi</math>]]
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| || <math>\mathbb{R}\setminus\mathbb{Q}</math>
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| || 3.35988 56662
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| || [3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]
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| | [[4 (number)|<math>4</math>]]
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| || <math>\mathbb N</math>
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| || 4.00000 00000
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| || [4; ]
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| | [[Feigenbaum constants|<math>\delta</math>]]
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| || <math>\mathbb R</math>
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| || 4.66920 16091
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| || [4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]
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| | [[Gelfond's constant|<math>e^\pi</math>]]
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| || <math>\mathbb R\setminus\mathbb A</math>
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| || 23.14069 26328
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| || [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …]
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| || Gelfond's constant. Can also be expressed as <math>(-1)^{-i}</math>; from this form, it is transcendental due to the [[Gelfond–Schneider theorem]].
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| |}
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| {{Reflist|group=lower-greek}}
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| ==See also==
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| * [[Physical constant]]
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| * [[Mathematical constants and functions]]
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| [[Category:Mathematical constants|*]]
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| [[Category:Continued fractions]]
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