# 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 Champernowne’s constant $C_{10}$ $i^{i}$ (has no imaginary part) or $e^{\frac{-\pi}{2}}$ http://planetmath.org/CopelandErdsConstantCopeland-Erdős constant Meissel-Mertens constant Lévy’s constant The prime constant $\rho$ Lehmer’s constant ${6\over{\pi^{2}}}$, the probability that a random integer is squarefree Cahen’s constant Landau-Ramanujan constant Gauss’s constant $\Gamma(\frac{3}{2})=\frac{1}{2}\sqrt{\pi}$ Catalan’s constant $K$ Apéry’s constant $\zeta(3)$ $\Gamma(\frac{3}{4})$ Mills’ constant The plastic constant Square root of two $\sqrt{2}$ Ramanujan-Soldner constant Erdős-Borwein constant The golden ratio $\phi$ $\zeta(2)=\frac{\pi^{2}}{6}$, the solution to the Basel problem Square root of three $\sqrt{3}$ Vijayaraghavan’s infinite nested radical $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\ldots}}}}}$ $\Gamma(\frac{1}{2})=\sqrt{\pi}$ Square root of five $\sqrt{5}$ $2^{\sqrt{2}}$ Khinchin’s constant The silver ratio $\delta_{S}$ Sierpiński’s constant The natural log base $e$ The ratio of a circle’s radius to its circumference $\pi$ $\Gamma(\frac{1}{4})$ $\sqrt{2e\pi}$ Feigenbaum’s constant $\delta$ $e^{2}$ The imaginary part of the first nontrivial zero of the Riemann zeta function (the real part is $\frac{1}{2}$) $e^{e}$ $\pi^{\pi}$

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 $\Gamma(x)$
Title index of important irrational constants IndexOfImportantIrrationalConstants 2013-03-22 17:03:10 2013-03-22 17:03:10 PrimeFan (13766) PrimeFan (13766) 17 PrimeFan (13766) Topic msc 00A08