Pi Approximations

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Convergents of the pi continued fractions are the simplest approximants to pi. The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.

Two approximations follow from the near-identity function 3sinx/(2+cosx) evaluated at x=pi/4 and pi/8, giving

pi approx (12)/7(2sqrt(2)-1)

(1)

approx [画像:(24sqrt(2-sqrt(2)))/(4+sqrt(2+sqrt(2))),]

(2)

which are good to 2 and 3 digits, respectively. Kochanski's approximation is the root of

9x^4-240x^2+1492

(3)

given by

pi approx sqrt((40)/3-sqrt(12)) approx 3.141533,

(4)

which is good to 4 digits.

Another curious fact is the almost integer

e^pi-pi=19.999099979...,

(5)

which can also be written as

(pi+20)^i=-0.9999999992-0.0000388927i approx -1

(6)

cos(ln(pi+20)) approx -0.9999999992.

(7)

Here, e^pi is Gelfond's constant. Applying cosine a few more times gives

cos(picos(picos(ln(pi+20)))) approx -1+3.9321609261×10^(-35).

(8)

Another approximation involving e is given by

pi approx sqrt(4e-1),

(9)

which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).

An apparently interesting near-identity is given by

sin(1/(555555) degrees) approx pi×10^(-8),

(10)

which becomes less surprising when it is noted that 555555 is a repdigit, so the above is just a special case of the near-identity

[画像:sin((pi/(180))/(d(10^n-1)/9))] approx [画像:sin((pi/(180))/(d(10^n)/9))]

(11)

approx pi/(2d)10^(-(n+1))

(12)

with d=5 and n=6.

An approximation involving the golden ratio phi arises from the volume of the canonical tetragonal antiwedge with unit midradius, namely

pi approx 8/3phi(2-sqrt(phi))

(13)

= 8/3(1+sqrt(5)-sqrt(2+sqrt(5)))

(14)

= 3.14105...

(15)

(cf. Pegg 2018), which is good to 3 digits. Anopther approximation involving phi is

pi approx 6/5phi^2

(16)

= [画像:6/5((sqrt(5)+1)/2)^2]

(17)

= 3/5(3+sqrt(5))

(18)

= 3.14164...,

(19)

which is good to 4 digits. A similar approximation due to S. Mircea-Mugurel (pers. comm., Oct. 30, 2002) is given by

pi approx 4phi^(-1/2)=3.1446...,

(20)

which however is only good to two decimal places. Another approximation involving the golden ratio phi is given by

[画像: pi approx ((802phi-801)/(602phi-601))^4, ]

(21)

which is good to 7 digits (K. Rashid, pers. comm.).

Some approximations due to Ramanujan include

pi approx [画像:(19sqrt(7))/(16)]

(22)

approx 7/3(1+1/5sqrt(3))

(23)

approx 9/5+sqrt(9/5)

(24)

approx [画像:((2143)/(22))^(1/4)=(9^2+(19^2)/(22))^(1/4)=(102-(2222)/(22^2))^(1/4)=(97+1/2-1/(11))^(1/4)=(97+9/(22))^(1/4)]

(25)

approx [画像:(63)/(25)((17+15sqrt(5))/(7+15sqrt(5)))]

(26)

approx [画像:(355)/(113)(1-(0.0003)/(3533))]

(27)

approx [画像:(12)/(sqrt(130))ln[((3+sqrt(13))(sqrt(8)+sqrt(10)))/2]]

(28)

approx [画像:(24)/(sqrt(142))ln[(sqrt(10+11sqrt(2))+sqrt(10+7sqrt(2)))/2]]

(29)

approx [画像:(12)/(sqrt(190))ln[(3+sqrt(10))(sqrt(8)+sqrt(10))]]

(30)

approx [画像:(12)/(sqrt(310))ln[1/4(3+sqrt(5))(2+sqrt(2))(5+2sqrt(10)+sqrt(61+20sqrt(10)))]]

(31)

approx [画像:4/(sqrt(522))ln[((5+sqrt(29))/(sqrt(2)))^3(5sqrt(29)+11sqrt(6))(sqrt((9+3sqrt(6))/4)+sqrt((5+3sqrt(6))/4))^6],]

(32)

which are accurate to 3, 4, 4, 8, 8, 9, 14, 15, 15, 18, 23, 31 digits, respectively (Ramanujan 1913-1914; Hardy 1952, p. 70; Wells 1986, p. 54; Berndt 1994, pp. 48-49 and 88-89). Equation (◇) and the similar

[画像: pi approx (66sqrt(2))/(33sqrt(29)-148) ]

(33)

are also given by Borwein and Bailey (2003, p. 135). Ramanujan also gave

[画像: pi approx (99^2)/(2206sqrt(2)) ]

(34)

(Wells 1986, p. 54).

It is not hard to find rational approximations to pi using two pandigital numbers (A. Povolotsky, pers. comm., Aug.29, 2022). The best such approximation is

[画像: pi approx (8405139762)/(2675439081)=3.141592653591... ]

(35)

which approximates pi to 10 decimal digits (E. Weisstein, Sep. 7, 2022). S. Irvine (pers. comm.) noted that (◇), giving an approximation to pi good to 8 digits, can be written in a pandigital form as

pi approx [画像:0+sqrt(sqrt(3^4+(19^2)/(78-56)))]

(36)

= [画像:(9^2+(19^2)/(22))^(1/4)]

(37)

= [画像:((2143)/(22))^(1/4)]

(38)

(S. Plouffe, pers. comm.; cf. Wells 1986, p. 54). E. Pegg (pers. comm.) found the pandigital approximation

0+3+(1-(9-8^(-5))^(-6))/(7+2^(-4))=(233546921420225777694970883318153571)/(74340293968115785654927455866388593)

(39)

which approximates pi to 9 digits. Another pandigital formula is given by

[画像: pi approx 3+4/(28)-1/(790+5/6)=3.14159265392... ]

(40)

(B. Astle, pers. comm., Jan. 9, 2004), which approximates pi to 9 digits. Surpassing both of these is the pandigital approximation

[画像: 2^(5^(.4))-.6-((.3^9)/7)^(.8^(.1)). ]

(41)

which gives 10 correct digits (B. Ziv, pers. comm., Jul. 7, 2004). A further pandigital approximation is given by

[画像: (ln{[2×5!+(8-1)!]^(sqrt(9))+4!+(3!)!})/(sqrt(67)), ]

(42)

which is good to 17 digits (G. W. Barbosa, pers. comm.).

M. Schneider (pers. comm., May 6, 2008) found the approximation

pi approx sqrt(7+sqrt(6+sqrt(5))),

(43)

which is good to 3 decimal digits. P. Lindborg (pers. comm.) noted that the convergent 104348/33125 can be written in the slightly curious form

[画像: (314+142)/(2·3·5·7)(1373)/(13·73), ]

(44)

which is good to 9 digits.

Other approximations due to E. Pegg include

pi approx 4-((105)/(166))^(1/3),

(45)

which is good to 6 digits (pers. comm., March 2, 2002) and

pi approx (22)/(17)+(37)/(47)+(88)/(83),

(46)

which is good to 9 digits (pers. comm., Dec. 30, 2002).

A simple approximation involving the cube root is

pi approx 31^(1/3),

(47)

which is good to 3 digits (M. Joseph, pers. comm., May 3, 2006). A more exotic one is given by

pi approx (ln6)^((ln5)^((ln4)^((ln3)^(ln2)))),

(48)

which is good to 4 digits (M. Joseph, pers. comm., May 3, 2006).

Castellanos (1988ab) gives a slew of curious formulas:

pi approx (2e^3+e^8)^(1/7)

(49)

approx [画像:((553)/(311+1))^2]

(50)

approx (3/(14))^4((193)/5)^2

(51)

approx ((296)/(167))^2

(52)

approx [画像:((66^3+86^2)/(55^3))^2]

(53)

approx 1.09999901·1.19999911·1.39999931·1.69999961

(54)

approx [画像:(47^3+20^3)/(30^3)-1]

(55)

approx 2+sqrt(1+((413)/(750))^2)

(56)

approx ((77729)/(254))^(1/5)

(57)

approx [画像:(31+(62^2+14)/(28^4))^(1/3)]

(58)

approx [画像:(1700^3+82^3-10^3-9^3-6^3-3^3)/(69^5)]

(59)

approx [画像:(95+(93^4+34^4+17^4+88)/(75^4))^(1/4)]

(60)

approx [画像:(100-(2125^3+214^3+30^3+37^2)/(82^5))^(1/4),]

(61)

which are accurate to 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, and 13 digits, respectively. An extremely accurate approximation due to Shanks (1982) is

[画像: pi approx 6/(sqrt(3502))ln(2u)+7.37×10^(-82), ]

(62)

where u is the product of four simple quartic units.

David W. Hoffman (pers. comm.) gave

[画像: pi approx ((10^(100))/(11222.11122))^(1/193), ]

(63)

where the numerator is one googol, which is good to 9 digits. The approximations

pi approx e^(e^(e^(-2)))

(64)

approx 2+e^(e^(-2))

(65)

give 2 digits (G. von Hippel, pers. comm.).

A sequence of approximations due to Plouffe and Borwein and Bailey (2003, pp. 115 and 134-135) includes

pi approx 43^(7/23)

(66)

approx [画像:(ln2198)/(sqrt(6))]

(67)

approx [画像:((13)/4)^(1181/1216)]

(68)

approx [画像:(689)/(396ln((689)/(396)))]

(69)

approx [画像:ln5280sqrt(9/(67))]

(70)

approx ((63023)/(30510))^(1/3)+1/4+1/2(sqrt(5)+1)

(71)

approx [画像:(48)/(23)ln((60318)/(13387))]

(72)

approx (228+(16)/(1329))^(1/41)+2

(73)

approx [画像:(125)/(123)ln((28102)/(1277))]

(74)

approx [画像:3/(sqrt(163))ln(640320)]

(75)

approx [画像:((276694819753963)/(226588))^(1/158)+2]

(76)

approx [画像:(ln(640320^3+744))/(sqrt(163)),]

(77)

which are accurate to 4, 5, 7, 7, 9, 10, 11, 11, 11, 15, 23, and 30 digits, respectively.

The last expression, which follows from the series expansion of the j-function. Carrying this one step further gives

-e^(pisqrt(163))+744-196886e^(-pisqrt(163))+...=-640320^3

(78)

e^(pisqrt(163))(1+196884e^(-2pisqrt(163))) approx 640320^3+744

(79)

e^(2pisqrt(163))(1+196884e^(-2pisqrt(163)))^2 approx (640320^3+744)^2

(80)

e^(2pisqrt(163))+2·196884 approx (640320^3+744)^2

(81)

giving

[画像: pi approx (ln[(640320^3+744)^2-2·196884])/(2sqrt(163)), ]

(82)

which is good to 46 decimal digits (Warda, pers. comm., Nov. 15, 2004).

PiApproximationsSqrt

Interestingly, ln(nint(exp(pisqrt(163n))))/sqrt(163n) gives successively good approximations to pi for larger and larger n (Warda, pers. comm., Nov. 22, 2004). In particular, the number of correct digits for n=1, 2, ... are given by 30, 28, 31, 46, 40, 44, 48, 51, 61, 57, 59, 62, 65 (OEIS A100935).

An approximation due to Stoschek using powers of two and the special number 163 (the largest Heegner number) is given by

[画像: pi approx (2^9)/(163)=(512)/(163) approx 3.1411043, ]

(83)

which is good to 3 digits. A fraction with small numerator and denominator which gives a close approximation to pi is

[画像: (311)/(99)=3.14141414.... ]

(84)

Some approximations involving the ninth roots of rational numbers include

pi approx [画像:((4297607660)/(144171))^(1/9)]

(85)

approx [画像:((34041350274878)/(1141978491))^(1/9),]

(86)

which are good to 12 and 15 digits, respectively (P. Galliani, pers. comm.).

de Jerphanion (pers. comm.) found

pi approx ln(23+1/(22)+2/(21))=ln(23+1/6-2/(77))=ln((10691)/(462)),

(87)

which is good to 9 digits, and J. Iuliano found

[画像: pi approx ((19)/(60)+1/(sqrt(3·123449)))^(-1), ]

(88)

which is good to 11 digits.

Definite integrals giving approximations to pi were considered by Backhouse (1995) and Lucas (2005).

F. Voormanns (pers. comm., Dec. 12, 2003) found the curious astronomical approximation

[画像: pi approx 1/( week)((13 years-6 weeks)/(13 years)+3 weeks), ]

(89)

which is accurate to 8 digits if the year is taken as exactly 365 days, or 6 digits if the average Gregorian year (365.2425 days) or tropical year (365.242190 days) is used.

Rivera gives other approximation formulas.

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References

Backhouse, N. "Note 79.36. Pancake Functions and Approximations to pi." Math. Gaz. 79, 371-374, 1995.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988a.Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag. 61, 148-163, 1988b.Contest Center. "Pi Competition." http://www.contestcen.com/pi.htm.Friedman, E. "Problem of the Month (August 2004)." https://erich-friedman.github.io/mathmagic/0804.html.Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1952.Lucas, S. K. "Integral Proofs that 355/113>pi." Gaz. Austral. Math. Soc. 32, 263-266, 2005.Pegg, E. Jr. "For Pi Day: Volume = 3.141--The Canonical Tetragonal Antiwedge Hexahedron." Mar. 14, 2018. https://community.wolfram.com/groups/-/m/t/1301599.Plouffe, S. "A Few Approximations of Pi." http://pi.lacim.uqam.ca/eng/approximations_en.html.Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.Rivera, C. "Problems & Puzzles: Puzzle 050-The Best Approximation to Pi with Primes." http://www.primepuzzles.net/puzzles/puzz_050.htm.Shanks, D. "Dihedral Quartic Approximations and Series for pi." J. Number. Th. 14, 397-423, 1982.Sloane, N. J. A. Sequences A002485/M3097, A002486/M4456, A100935, and A114526 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.

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Weisstein, Eric W. "Pi Approximations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PiApproximations.html

Pi Approximations

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