List of mathematical constants

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery.
The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.

Mathematical constants sorted by their representations as continued fractions

The following list includes the continued fractions of some constants and is sorted by their representations. 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 rounded or padded to 10 places if the values are known.

Name	Symbol	Set	Decimal expansion	Continued fraction	Notes
Zero	0	[Integer\|]	0.00000 00000
Golomb–Dickman constant			0.62432 99885		E. Weisstein noted that the continued fraction has an unusually large number of 1s.
Cahen's constant		[Transcendental number\|]	0.64341 05463		All terms are squares and truncated at 10 terms due to large size. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant			0.57721 56649		Using the continued fraction expansion, it was shown that if is rational, its denominator must exceed 10²⁴⁴⁶⁶³.
First continued fraction constant		[Transcendental number\|]	0.69777 46579		Equal to the ratio of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant			0.91596 55942		Computed up to terms by E. Weisstein.
One half		[Rational number\|]	0.50000 00000
Prouhet–Thue–Morse constant		[Transcendental number\|]	0.41245 40336		Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
Copeland–Erdős constant		[Irrational number\|]	0.23571 11317		Computed up to terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.
Base 10 Champernowne constant		[Transcendental number\|]	0.12345 67891		Champernowne constants in any base exhibit sporadic large numbers; the 40th term in has 2504 digits.
One	1	[Natural number\|]	1.00000 00000
Phi, Golden ratio		[Algebraic number\|]	1.61803 39887		The convergents are ratios of successive Fibonacci numbers.
Brun's constant			1.90216 05831		The n^th roots of the denominators of the n^th convergents are close to Khinchin's constant, suggesting that is irrational. If true, this will prove the twin prime conjecture.
Square root of 2		[Algebraic number\|]	1.41421 35624		The convergents are ratios of successive Pell numbers.
Two	2	[Natural number\|]	2.00000 00000
Euler's number		[Transcendental number\|]	2.71828 18285		The continued fraction expansion has the pattern .
Khinchin's constant			2.68545 20011		For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three	3	[Natural number\|]	3.00000 00000
Pi		[Transcendental number\|]	3.14159 26536		The first few convergents are among the best-known and most widely used historical approximations of.

Sequences of constants

Name	Symbol	Formula	Year	Set
Harmonic number			Antiquity	[Rational number\|]
Gregory coefficients			1670	[Rational number\|]
Bernoulli number			1689	[Rational number\|]
Hermite constants		For a lattice L in Euclidean space Rⁿ with unit covolume, i.e. vol = 1, let λ denote the least length of a nonzero element of L. Then √γn is the maximum of λ over all such lattices L.	1822 to 1901	[Real number\|]
Hafner–Sarnak–McCurley constant			1883	[Real number\|]
Stieltjes constants			before 1894	[Real number\|]
Favard constants			1902 to 1965	[Real number\|]
Generalized Brun's Constant		where the sum ranges over all primes p such that p + n is also a prime	1919	[Real number\|]
Champernowne constants		Defined by concatenating representations of successive integers in base b.	1933	[Transcendental number\|]
Lagrange number		where is the nth smallest number such that has positive.	before 1957	[Algebraic number\|]
Feller's coin-tossing constants		is the smallest positive real root of	1968	[Algebraic number\|]
Stoneham number		where b,c are coprime integers.	1973	[Transcendental number\|]
Beraha constants			1974	[Real number\|]
Chvátal–Sankoff constants			1975	[Real number\|]
Hyperharmonic number		and	1995	[Rational number\|]
Gregory number		for rational x greater than or equal to one.	before 1996	[Transcendental number\|]
Metallic mean			before 1998	[Algebraic number\|]