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# index of important irrational constants

The following table lists some of the most important irrational constants in mathematics.

Of course importance is sometimes debatable. Hardly anyone disputes the importance of $\pi$ or $e$ (in fact, these are the only two constants in the OEIS to have the keyword “core” attached to them), but for other constants it is not quite clear cut. In general, if a given constant has a name (especially a name hyphenating two famous mathematicians’ last names) I consider it important.

Irrationality is not always clear cut either, e.g., it might be a mistake to exclude the Euler-Mascheroni constant $\gamma$ from this list.

The constants are given to 20 decimal places.

0.1149420448532962007 | Kepler-Bouwkamp constant or polygon-inscribing constant |
---|---|

0.1234567891011121314 | Champernowne’s constant $C_{{10}}$ |

0.2078795763507619085 | $i^{i}$ (has no imaginary part) or $e^{{\frac{-\pi}{2}}}$ |

0.2357111317192329313 | Copeland-Erdős constant |

0.2614972128476427837 | Meissel-Mertens constant |

0.3275822918721811159 | Lévy’s constant |

0.4146825098511116602 | The prime constant $\rho$ |

0.5926327182016361971 | Lehmer’s constant |

0.6079271018540266286 | ${6\over{\pi^{2}}}$, the probability that a random integer is squarefree |

0.6434105462883380261 | Cahen’s constant |

0.7642236535892206629 | Landau-Ramanujan constant |

0.8346268416740731862 | Gauss’s constant |

0.8862269254527580136 | $\Gamma(\frac{3}{2})=\frac{1}{2}\sqrt{\pi}$ |

0.9159655941772190150 | Catalan’s constant $K$ |

1.2020569031595942853 | Apéry’s constant $\zeta(3)$ |

1.2254167024651776451 | $\Gamma(\frac{3}{4})$ |

1.3063778838630806904 | Mills’ constant |

1.3247179572447460260 | The plastic constant |

1.4142135623730950488 | Square root of two $\sqrt{2}$ |

1.4513692348833810502 | Ramanujan-Soldner constant |

1.6066951524152917637 | Erdős-Borwein constant |

1.6180339887498948482 | The golden ratio $\phi$ |

1.6449340668482264364 | $\zeta(2)=\frac{\pi^{2}}{6}$, the solution to the Basel problem |

1.7320508075688772935 | Square root of three $\sqrt{3}$ |

1.7579327566180045327 | Vijayaraghavan’s infinite nested radical $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\ldots}}}}}$ |

1.7724538509055160273 | $\Gamma(\frac{1}{2})=\sqrt{\pi}$ |

2.2360679774997896964 | Square root of five $\sqrt{5}$ |

2.6651441426902251887 | $2^{{\sqrt{2}}}$ |

2.6854520010653064453 | Khinchin’s constant |

2.4142135623730950488 | The silver ratio $\delta_{{S}}$ |

2.5849817595792532170 | Sierpiński’s constant |

2.7182818284590452354 | The natural log base $e$ |

3.1415926535897932385 | The ratio of a circle’s radius to its circumference $\pi$ |

3.6256099082219083119 | $\Gamma(\frac{1}{4})$ |

4.1327313541224929385 | $\sqrt{2e\pi}$ |

4.6692116609102990671 | Feigenbaum’s constant $\delta$ |

7.3890560989306502272 | $e^{2}$ |

14.1347251417346937904 | The imaginary part of the first nontrivial zero of the Riemann zeta function (the real part is $\frac{1}{2}$) |

15.1542622414792641898 | $e^{e}$ |

36.4621596072079117710 | $\pi^{\pi}$ |

In looking these up in the OEIS, you can simply type them with a decimal point and no commas between the digits. If you get no results, try chopping off a couple of the least significant digits.

# References

- 1 Alan Jeffrey, Handbook of Mathematical Formulas and Integrals, 3rd Edition. New York: Elsevier Academic Press (2004): 223, Section 11.1.4 Special values of $\Gamma(x)$

## Mathematics Subject Classification

00A08*no label found*

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