Mathematical constants by continued fraction representation

Mathematical constants by continued fraction representation: This is a list of mathematical constants sorted by their representations as continued fractions.

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.

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.

Symbol Member of decimal Continued fraction Notes

$0$ $\mathbb{Z}$ 0.00000 00000 [0; ]

$ϕ - 1$ $\mathbb{A}\setminus\mathbb{Q}$ 0.61803 39887 [0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] First positive irrational number in the list.

$C$ $\mathbb{R}$ 0.64341 05463 [0; 1, 1, 1, 4, 9, 169, 16641, 639988804, 177227652025317609, 72589906463585427805281295977816196, …] Continued fraction truncated at 10 terms due to large size.

$C 2$ $\mathbb{R}$ 0.66016 18158 [0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …] Hardy–Littlewood's twin prime constant. Presumed irrational, but not proved.

$γ$ $\mathbb{R}$ 0.57721 56649 [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …] Presumed irrational, but not proved.

$Ω$ $\mathbb{R}\setminus\mathbb{A}$ 0.56714 32904 [0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]

$\beta^\star$ $\mathbb{R}$ 0.70258 [0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …] Value only known to 5 decimal places.

$K$ $\mathbb{R} (\setminus\mathbb{Q}?)$ 0.76422 36535 [0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …] May have been proven irrational.

$\frac 1{M(1,\sqrt 2)}$ $\mathbb{R}\setminus\mathbb{A}$ 0.83462 68417 [0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …] Gauss's constant

$B 4$ $\mathbb{R}$ 0.87058 83800 [0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …] Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005.

$C 2$ $\mathbb{R}\setminus\mathbb{A}$ 0.86224 01259 [0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …] Base 2 Champernowne constant. The binary expansion is $C_2 = 0.1 10 11 100 101 110 111 1000\ldots_2$

$G$ $\mathbb{R}$ 0.91596 55942 [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …] Presumed irrational, but not proved.

$\frac 12$ $\mathbb{Q}$ 0.50000 00000 [0; 2]

$β$ $\mathbb{R}$ 0.28016 94990 [0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …] Presumed irrational, but not proved.

$M$ $\mathbb{R}$ 0.26149 72128 [0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …] Presumed irrational, but not proved.

$M R B$ $\mathbb{R}$ 0.18785 96424 [0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]

$C 10$ $\mathbb{R}\setminus\mathbb{A}$ 0.12345 67891 [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, $4.57540\times 10^{165}$ , 6, 1, …] Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C 10$ has 2504 digits.

$1$ $\mathbb{N}$ 1.00000 00000 [1; ]

$ϕ$ $\mathbb{A}\setminus\mathbb{Q}$ 1.61803 39887 [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]

$E$ $\mathbb{R}\setminus\mathbb{Q}$ 1.60669 51524 [1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …] Not known whether algebraic or transcendental.

$B 2$ $\mathbb{R}$ 1.90216 05823 [1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …] Brun's twin prime constant. Estimated value; best bounds $1.8304 < B 2 < 2.347$ .

$B$ $\mathbb{R}$ 1.45607 49485 [1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4,, …]

$ζ(3)$ $\mathbb{R}\setminus\mathbb{Q}$ 1.20205 69032 [1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]

$V$ $\mathbb{R}$ 1.13198 82488 [1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …] Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica.

$\sqrt 2$ $\mathbb{A}\setminus\mathbb{Q}$ 1.41421 35624 [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]

$μ$ $\mathbb{R}$ 1.45136 92349 [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …] Presumed irrational, but not proved.

$ρ$ $\mathbb{A}\setminus\mathbb{Q}$ 1.32471 95724 [1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]

$2$ $\mathbb N$ 2.00000 00000 [2; ]

$2^\sqrt 2$ $\mathbb{R}\setminus\mathbb{A}$ 2.66514 41426 [2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]

$α$ $\mathbb{R}$ 2.50290 78751 [2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]

$e$ $\mathbb{R}\setminus\mathbb{A}$ 2.71828 18285 [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]

$K 0$ $\mathbb{R}$ 2.68545 20011 [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]

$F$ $\mathbb{R}$ 2.80777 02420 [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]

$P 2$ $\mathbb{R}\setminus\mathbb{A}$ 2.29558 71494 [2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]

$3$ $\mathbb{N}$ 3.00000 00000 [3; ]

$π$ $\mathbb{R}\setminus\mathbb{A}$ 3.14159 26536 [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]

$ψ$ $\mathbb{R}\setminus\mathbb{Q}$ 3.35988 56662 [3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]

$4$ $\mathbb N$ 4.00000 00000 [4; ]

$δ$ $\mathbb R$ 4.66920 16091 [4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]

$e π$ $\mathbb R\setminus\mathbb A$ 23.14069 26328 [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …] Gelfond's constant. Can also be expressed as $( - 1) - i$ ; from this form, it is transcendental due to the Gelfond–Schneider theorem.

See also

Physical constant

Categories:
Mathematical constants
Continued fractions

Symbol	Member of	decimal	Continued fraction	Notes
$0$	$\mathbb{Z}$	0.00000 00000	[0; ]
$ϕ - 1$	$\mathbb{A}\setminus\mathbb{Q}$	0.61803 39887	[0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]	First positive irrational number in the list.
$C$	$\mathbb{R}$	0.64341 05463	[0; 1, 1, 1, 4, 9, 169, 16641, 639988804, 177227652025317609, 72589906463585427805281295977816196, …]	Continued fraction truncated at 10 terms due to large size.
$C 2$	$\mathbb{R}$	0.66016 18158	[0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …]	Hardy–Littlewood's twin prime constant. Presumed irrational, but not proved.
$γ$	$\mathbb{R}$	0.57721 56649	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]	Presumed irrational, but not proved.
$Ω$	$\mathbb{R}\setminus\mathbb{A}$	0.56714 32904	[0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]
$\beta^\star$	$\mathbb{R}$	0.70258	[0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …]	Value only known to 5 decimal places.
$K$	$\mathbb{R} (\setminus\mathbb{Q}?)$	0.76422 36535	[0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …]	May have been proven irrational.
$\frac 1{M(1,\sqrt 2)}$	$\mathbb{R}\setminus\mathbb{A}$	0.83462 68417	[0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …]	Gauss's constant
$B 4$	$\mathbb{R}$	0.87058 83800	[0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …]	Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005.
$C 2$	$\mathbb{R}\setminus\mathbb{A}$	0.86224 01259	[0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …]	Base 2 Champernowne constant. The binary expansion is $C_2 = 0.1 10 11 100 101 110 111 1000\ldots_2$
$G$	$\mathbb{R}$	0.91596 55942	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]	Presumed irrational, but not proved.
$\frac 12$	$\mathbb{Q}$	0.50000 00000	[0; 2]
$β$	$\mathbb{R}$	0.28016 94990	[0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …]	Presumed irrational, but not proved.
$M$	$\mathbb{R}$	0.26149 72128	[0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …]	Presumed irrational, but not proved.
$M R B$	$\mathbb{R}$	0.18785 96424	[0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]
$C 10$	$\mathbb{R}\setminus\mathbb{A}$	0.12345 67891	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, $4.57540\times 10^{165}$ , 6, 1, …]	Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C 10$ has 2504 digits.
$1$	$\mathbb{N}$	1.00000 00000	[1; ]
$ϕ$	$\mathbb{A}\setminus\mathbb{Q}$	1.61803 39887	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
$E$	$\mathbb{R}\setminus\mathbb{Q}$	1.60669 51524	[1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …]	Not known whether algebraic or transcendental.
$B 2$	$\mathbb{R}$	1.90216 05823	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]	Brun's twin prime constant. Estimated value; best bounds $1.8304 < B 2 < 2.347$ .
$B$	$\mathbb{R}$	1.45607 49485	[1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4,, …]
$ζ(3)$	$\mathbb{R}\setminus\mathbb{Q}$	1.20205 69032	[1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]
$V$	$\mathbb{R}$	1.13198 82488	[1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …]	Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica.
$\sqrt 2$	$\mathbb{A}\setminus\mathbb{Q}$	1.41421 35624	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]
$μ$	$\mathbb{R}$	1.45136 92349	[1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …]	Presumed irrational, but not proved.
$ρ$	$\mathbb{A}\setminus\mathbb{Q}$	1.32471 95724	[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]
$2$	$\mathbb N$	2.00000 00000	[2; ]
$2^\sqrt 2$	$\mathbb{R}\setminus\mathbb{A}$	2.66514 41426	[2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]
$α$	$\mathbb{R}$	2.50290 78751	[2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]
$e$	$\mathbb{R}\setminus\mathbb{A}$	2.71828 18285	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]
$K 0$	$\mathbb{R}$	2.68545 20011	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]
$F$	$\mathbb{R}$	2.80777 02420	[2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]
$P 2$	$\mathbb{R}\setminus\mathbb{A}$	2.29558 71494	[2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]
$3$	$\mathbb{N}$	3.00000 00000	[3; ]
$π$	$\mathbb{R}\setminus\mathbb{A}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]
$ψ$	$\mathbb{R}\setminus\mathbb{Q}$	3.35988 56662	[3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]
$4$	$\mathbb N$	4.00000 00000	[4; ]
$δ$	$\mathbb R$	4.66920 16091	[4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]
$e π$	$\mathbb R\setminus\mathbb A$	23.14069 26328	[23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …]	Gelfond's constant. Can also be expressed as $( - 1) - i$ ; from this form, it is transcendental due to the Gelfond–Schneider theorem.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

Mathematical constants (sorted by continued fraction representation) — This is a list of mathematical constants sorted by their representations as continued fractions:(Constants known to be irrational have infinite continued fractions: their last term is ....)Symbol NumeralNumber setValue Continued fraction… … Wikipedia
Continued fraction — Finite continued fraction, where a0 is an integer, any other ai are positive integers, and n is a non negative integer. In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the… … Wikipedia
Mathematical coincidence — This article is about numerical curiosities. For the technical mathematical concept of coincidence, see coincidence point. A mathematical coincidence can be said to occur when two expressions show a near equality that lacks direct theoretical… … Wikipedia
Mathematical constant — A mathematical constant is a special number, usually a real number, that is significantly interesting in some way .[1] Constants arise in many different areas of mathematics, with constants such as e and π occurring in such diverse contexts as… … Wikipedia
e (mathematical constant) — Euler s number redirects here. For γ, a constant in number theory, see Euler s constant. For other uses, see List of topics named after Leonhard Euler#Euler numbers. e is the unique number a, such that the derivative of the exponential function… … Wikipedia
E (mathematical constant) — The mathematical constant e is the unique real number such that the function e x has the same value as the slope of the tangent line, for all values of x . [Keisler, H.J. [http://www.vias.org/calculus/08 exp log functions 03 01.html Derivatives… … Wikipedia
Approximations of π — Timeline of approximations for pi … Wikipedia
Square root of 2 — The square root of 2, also known as Pythagoras constant, often denoted by:sqrt{2} or √2but can also be written as:2^{1/2},,is the positive real number that, when multiplied by itself, gives the number 2. Its numerical value approximated to 65… … Wikipedia
Gauss–Kuzmin–Wirsing operator — GKW redirects here. For the Indian engineering firm see Guest Keen Williams.In mathematics, the Gauss–Kuzmin–Wirsing operator occurs in the study of continued fractions; it is also related to the Riemann zeta function. IntroductionThe… … Wikipedia
Padé table — In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants : R m , n to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often be shown to… … Wikipedia

Academic Dictionaries and Encyclopedias

Mathematical constants by continued fraction representation

See also

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Mathematical constants by continued fraction representation

See also

Look at other dictionaries:

Share the article and excerpts

Direct link