A competition to find the best approximations for pi
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PI Competition
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       This is an open-ended competition to find the best possible approximations to pi (about 3.14159 26535 89793 23846 26433). A good approximation would be an expression which matches pi to more significant digits than the number of digits that the expression contains. For example, the most common approximation to pi is 22/7 which is about 3.14286. This contains 3 digits, and matches the first 3 digits of pi, so it is a fair approximation. By comparison, the expression

√√√√√√√√√√√√√√√√√√√√√√√√√√√√8 + √√√√√√√√√√√√14 + √√√√√68

contains 5 digits, but matches pi to 9 decimal places, so it is considered an outstanding approximation.
       We will consider 4 types of approximations. The first type uses only the mathematical operations of addition, subtraction, multiplication, division, and square root. It may not use other operations such as decimal fractions, exponentiation, logarithms, or any trig functions. For example, 355/113 (approximately pi+0.00000 0266) would be a valid expression, but 2 arcsin(1) would not. These approximations can be constructed with ruler and compass.
       The second type adds exponentiation, higher roots and nesting of radicals to the list of allowed operators. For example 710/17 would be a valid expression containing 5 digits, and evaluating to about pi-0.000241.
       The third type approximates pi as the root of a polynomial. So far only polynomial equations with integer coefficients have been considered.
       The fourth type approximates pi as the root of a more general equation.
       The 4 tables below show the best approximations that I, and the contributors to this webpage, have found for each of the 4 types.
       Here √19 is the square root of 19, and 3√52 is the cube root of 52. In counting digits, there are two digits in √19 and 3 digits in 3√52.
       In the equations, I count only the digits in the coefficients, and not the digits in the exponents, which are standard. A coefficient of 1 or -1 is counted as 1 digit, even though the 1 is not normally written. This avoids "sparse" polynomials with many terms, most of them having +1 or -1 as the coefficient.
       Anyone who wishes is invited to contribute to this effort, and all worthy results will be posted, with credit. The best approximations, where the number of digits of accuracy exceed the number of digits in the expression by at least 3, have been highlighted in yellow.
       Approximations using only small numbers will be considered superior to those using larger numbers. Please try to avoid any number with more than 5 digits.


Pi Pie (Credit: ??)

Type 1 - Fractions and Square Roots
Digits Expression Value Contributor
1 3 pi-.142 Ancient
2 √10 pi+.0206 Ancient
3 22/7 pi+.00126 Ancient
3 √(79/8) pi+.000859 Brian Sapozhnikov
3 √51 - 4 pi-.000164 ?
4 √14 - 3/5 pi+.00006 47 Frank Rubin
4 √8 + √5/51 pi-.00005 34 Francesco Franco
4 9/5 + √9/5 pi+.00004 81 Brian Astle
4 √√(8+√2) + √√(2+√3) pi-.00003 73 Francesco Franco
4 √√√√62 + √√(9+√7) pi-.00000 494 Francesco Franco
4 √(7+√(9-√(2-√2))) pi+.00000 0103 Jean Jacquelin
5 2/(2+7√3)+3 pi+.00000 672 Cetin Hakimoglu
5 √(227/23) pi-.00000 624 Frank Rubin
5 √21 + (√5)/4 - 2 pi+.00000 00357 Oleg Vlasii
6 355/113 pi+.00000 0266 Tsu Ch'ung Chi 450AD
6 (√(5/(√2+1))+1/3)² pi-.00000 00643 Gregory Koch
6 √2 + √(3-8/495) pi-.00000 00407 N.V. Subramanyan
6 √7 + √6/(6-√91/9) pi+.00000 000297 Oleg Vlasii
6 77 / (49-3/√2) pi-.00000 00009 85 Francesco Franco
7 √73 - √29 + 8/43 pi+.00000 00839 Frank Rubin
7 43/(√425-√48) pi+.00000 00479 Frank Rubin
7 (77+20/3) / 49 pi-.00000 000794 Francesco Franco
7 √(9+√(3/4)) + 7/(3*84) pi-.00000 000530 Cesare G. Ardito
7 (77+√6) / (49-3/√5) pi-.00000 00007 78 Francesco Franco
7 √7 + (√6+√89-7)/√97 pi+.00000 00000 0583 Oleg Vlasii
8 22/7 - 6/4745 pi+.00000 00003 32 Brian Astle
8 (77+√30) / (49-1/√7) pi+.00000 00002 03 Francesco Franco
8 √8 + (2/9-(√57)/80)√6 pi-.00000 00000 0312 Oleg Vlasii
9 (√179-√97)/(√59-√43) pi+.00000 00001 08 Frank Rubin
9 6√94/(√7+√6/3) - 5 - √75 pi-.00000 00000 000347 Oleg Vlasii
9 √8 + (√6 + 2√7)/(√633 - √7/6) pi+.00000 00000 00009 01 Oleg Vlasii
10 √3/(49/√8-8) + √39/2 - 1/6 pi-.00000 00000 00003 21 Oleg Vlasii
10 (6+√2)/(7√7) + (2+9√2)/(5+(√5)/6) pi+.00000 00000 00000 535 Oleg Vlasii
11 √(525665/53261) pi-.00000 00000 192 N.V. Subramanyan
12 77 / (49-(75+73)/7957) pi-.00000 00000 000459 Francesco Franco
11 (√3+√(7-2/41)) / (2-(5-√2)/(√58-√3)) pi-.00000 00000 00000 00940 Oleg Vlasii
12 (√71-6-5/44) ((3+√3)/(5-√5)-1/√8) pi+.00000 00000 00000 000835 Oleg Vlasii
13 355/113 - 1/(444+533) pi+.00000 00000 000543 Gerson Washiski Barbosa
13 357377/(784+√1055) pi-0.00000 00000 00000 369 Gary W. Miller


Type 2 - Higher Roots and Nested Radicals
Digits Expression Value Contributor
2 √(7+√8) pi-.00656 Jean Jacquelin
2 √√97 pi-.00330 Jean Jacquelin
2 2 + √√√√8 pi-.00280 Frank Rubin
3 √(2+√62) pi+.000701 Frank Rubin
3 √(6+√15) pi+.000538 Frank Rubin
3 3√31 pi-.000212 ?
3 √(8+√(7/2)) pi+.000195 Frank Rubin
3 √(7+√(6+√5)) pi+.00003 99 Jean Jacquelin
3 √√√√√√√√2 + √√√√√√√√√√√√√√5 + √√√√8 pi+.00000 549 Frank Rubin
3 √√√7 [√√3+√√√3] pi-.00000 421 Jean Jacquelin
3 √ [√√√√2 / (√√√√5 - √√√√√√√√√√√√√√√√√8)] pi-.00000 143 Jean Jacquelin
4 5√306 pi-.00004 04 Frank Rubin
4 4 - 6√(2/5) pi+.00003 13 Jaume Oliver Lafont
4 2 + 3√√√√24 pi-.00000 621 N.V. Subramanyan
4 8 [√√√(73)-√√8] pi+.00000 0727 Jean Jacquelin
4 √√√√2 + √√√√√7 + √√√√√√√78 pi+.00000 0452 Frank Rubin
4 √√√√√√√√√√√√2 + √√√√√√√√√√√√7 +
√√√√√68 		pi-.00000 00884 Frank Rubin
4 (√√√√√√√√√√2 + √√√√√√√√√3) /
(√√8 - √√√√2) 		pi+.00000 00056 Jean Jacquelin
4 √√√√√√√√√√2 + √√√√√5 +
√√√√√(2+√√√√√√√7) 		pi-.00000 000409 Frank Rubin
4 √√√√√√√√√√√√√√√√√√3 +
√√√√√√√7 + √√√√(8-√√√8) 		pi+.00000 00006 70 Frank Rubin
5 437/23 pi-.00005 28 Frank Rubin
5 2 + √√√√√√4794 pi-.00000 0680 N.V. Subramanyan
5 √√(767 / √62) pi+.00000 00513 Jean Jacquelin
5 √(9+√(5-√6)-√((√7)/5)) pi+.00000 00236 Oleg Vlasii
5 √√√√√√√√√√√√√√6 +
√√√√√√√√√√√√√√√√√√45 + √√√√√69 		pi+.00000 000453 Frank Rubin
5 √√√√√√√√√√√√√986 + √√√√5 + √√√√√3 pi-.00000 000452 Joel Li
5 √√√√√√√√√√√√√√√√436 + √√√√8 +
√√√√√√√√2 		pi-.00000 000202 Joel Li
5 √√(97+9/22) pi-.00000 000101 Randall Munroe
5 √√√√√√√√√√√√√√√√√√√√√√√√√√√√8 +
√√√√√√√√√√√√14 + √√√√√68 		pi-.00000 00002 13 Frank Rubin
5 √√√√√√√√√√√√√√√√5 +
√√√√√√√√√(3+√√√√√√√√√√2) +
√√√√(7+√√√√√√√√7) 		pi-.00000 00000 213 Frank Rubin
6 96 / (32-3√3) pi+.00000 0139 Francesco Franco
6 (77+√√√5) / (49-√3) pi+.00000 000130 Jean Jacquelin
6 √√2143/22 pi-.00000 000101 Srinivasa Ramanujan
6 √√√√√√√√√√√√√√√√√√√√√√96 + √√√√√67 + √√√√√√√√√√√11 pi+.00000 00004 30 Joel Li
6 √√√√√√√√√√√√√√√34 + √√√√√69 +
√√√√√√√√√√√√√√√√√√71 		pi-.00000 00004 01 Frank Rubin
6 4 / √(9/8 + 4/√65) pi+.00000 00000 770 Oleg Vlasii
6 √√√√√√√√√√√√14 + √√√√√65 +
√√√√√√√√√√√√√√√√√√√√√√√√√√√√√72 		pi+.00000 00000 0593 Frank Rubin
6 √√√√√√√√√√(4+√√√√5) +
√√√√√√√√√√(3+√√√√√√√√6) +
√√√√(9-√√√√√√√8) 		pi-.00000 00000 00775 Frank Rubin
6 √√√√√√√√√√(6+√√√√√√√8) + √√√√(7√√√√7) +
√√√√√√√√√√(7-√√√√√√5) 		pi-.00000 00000 00219 Frank Rubin
7 3√318 - 3√50 pi-.00000 00443 Francesco Franco
7 3√(31+3/478) pi+.00000 00179 Frank Rubin and N.V. Subramanyan
7 √√√√√√√√√√√√8 +
√√√√√√√√√√√√√√√√7777 + √√√√√68 		pi+.00000 00000 796 Joel Li
7 (√6-√(√88-6)) * √(1+9√8) pi-.00000 00000 000509 Oleg Vlasii
8 (√41+√35) / (√46-√√67) pi+.00000 00007 20 Francesco Franco
8 3 + (9(2/35 - 7-4)) / 8 pi+.00000 00000 0188 Oleg Vlasii
9 97272/1087 pi+.00000 00006 78 Frank Rubin
9 3√(31+25/3983) pi+.00000 00001 50 Frank Rubin and N.V. Subramanyan
9 √√√√√√√√√√√√√√√√√21777 + √√√√√60 + √√√√√√√√√13 pi+.00000 00000 0119 Joel Li
9 √((7-5/√6)/4 + √58 + √(9/√6-√7)) pi-.00000 00000 00000 386 Oleg Vlasii
10 9√√888582403 pi-.00000 00000 140 Rory Kulz
10 5 - 25√5352083 pi-.00000 00000 111 N.V. Subramanyan
10 88 / (5 √√√(993-√√2688)) pi-.00000 00000 00553 Francesco Franco
10 5√(√580674-456) pi-.00000 00000 00350 Jim Cullen
11 √√(5√(8769956796)) pi-.00000 00000 0148 Nick McGrath
13 √√(6√(854273519914)) pi+.00000 00000 000172 Nick McGrath
14 √√√(199002961/20973) pi-.00000 00000 000952 N.V. Subramanyan
16 √√√√√8105800789910710 pi+.00000 00000 00000 005 Sudipta Das


Type 3 - Roots of Polynomials
Digits Polynomial Value Contributor
3 x3-7x-9 pi+.000669 Frank Rubin
4 x5-99x+5 pi+.00000 519 Frank Rubin
4 x9-31x6-6 pi+.00000 121 Frank Rubin
5 7x3-15x2-69 pi-.00000 114 Frank Rubin
5 10x4-99x2+3 pi+.00000 0121 Frank Rubin
5 9x5-877x+1 pi+.00000 0116 Frank Rubin
5 x12-99x8+5x7-6 pi+.00000 00658 Frank Rubin
6 34x2-4x-323 pi+.00000 0854 Frank Rubin
6 9x3-9x2-44x-52 pi-.00000 0158 Frank Rubin
6 x15-31x12-6x6-x3-2 pi-.00000 00000 265 Oleg Vlasii
7 4x8-8x-37929 pi-.00000 00898 N.V. Subramanyan
7 x8-x6-x-8524 pi-.00000 00106 N.V. Subramanyan
7 8x3+54x2+7x-803 pi-.00000 00005 55 Frank Rubin
7 x17-99x13+5x12-8x5+6x4+4 pi-.00000 00000 0140 Oleg Vlasii
8 x11-x5-293898 pi+.00000 000166 N.V. Subramanyan
8 x8+x5-x4-x-9694 pi-.00000 00005 90 N.V. Subramanyan
8 8x4-37x2+77x-656 pi-.00000 00002 89 Frank Rubin
8 9x5+7x4-15x3-946x+1 pi-.00000 00000 505 Frank Rubin
8 2x9-4x3+9x2-59583 pi-.00000 00000 0128 Frank Rubin
8 3x17-92x14-3x13-2x11+5x6+7x4+9 pi-.00000 00000 000198 Oleg Vlasii
9 49x8-x5-464632 pi-.00000 00000 867 N.V. Subramanyan
9 85x3-69x2-597x-79 pi-.00000 00000 288 Frank Rubin
9 x14-x12-8197902 pi-.00000 00000 0622 N.V. Subramanyan
9 4x5+77x3+81x2-4411 pi+.00000 00000 0263 Frank Rubin
9 2x8-55x3+2x-17278 pi+.00000 00000 006817 Frank Rubin
9 2x16-6x15-3x13+5x11-x10-82x8-4x6-1 pi+.00000 00000 00000 00195 Oleg Vlasii
10 3x12-73x9-x7-593723 pi+.00000 00000 00178 Frank Rubin
10 33x5+x4-74x2-5x-9450 pi-.00000 00000 000172 Frank Rubin
10 6x6-4x5+5x4+2x3-2x2+3x-5083 pi-.00000 00000 00003 16 Jean Jacquelin
11 18x4+39x3-67x2-744x+36 pi+.00000 00000 000254 Frank Rubin
11 6x5-8x4-41x2+85x2-53x-458 pi+.00000 00000 00240 Frank Rubin
11 4x11-22x10-90x6+969966 pi+.00000 00000 000158 Frank Rubin
12 76x4-11x3+51x2+68x-7779 pi+.00000 00000 000580 Frank Rubin
12 14x5+8x4+21x3-7x2-96x-5344 pi+.00000 00000 00001 26 Frank Rubin
13 40x5+x4-4x3+25x2-70x-12241 pi+.00000 00000 00000 497 Frank Rubin
14 66x5-6x4-29x3+57x2-44x-19138 pi+.00000 00000 00000 293 Frank Rubin
15 79x5+70x4-56x3+87x2-60x-29928 pi+.00000 00000 00000 046 Frank Rubin
16 74x5-61x4+49x3-17x2-544x-16346 pi-.00000 00000 00000 118 N.V. Subramanyan
16 87x5+15x4+22x3+13x2-560x-27136 pi-.00000 00000 00000 079 Frank Rubin
24 x22-3x21-x20+6x18-7x16+7x15-10x14+22x13-2x12-13x11
-x9+2x8-x7-x6+3x5-4x4+14x3+7x2+7x+1 		pi-.00000 00000 00000 00000 00000 00000 0711 Paul Cleary
The 13 to 16 digit approximations are pushing the limit of extended precision arithmetic
on my PC, so may not be fully accurate. I could not check the 24 digit approximation.

Type 4 - General Expressions and Roots of General Equations
Digits Equation Value Contributor
2 √(4e-1) pi+.000561 Alexander R. Povolotsky
3 ex=7x+ln(x) pi-.000304 Alexander R. Povolotsky
3 ex-x=20 pi+.00004 06 Frank Rubin
4 ex+e/x7-x=20 pi+.00000 00006 72 Gerson W. Barbosa
6 (x+√2)4-x/4=430 pi+.00000 00074 Gerson W. Barbosa
7 x4+x5=e6 pi+.00000 0029 Jean Jacquelin, based on an approximation by Simon Plouffe
7 ex-ln(x)=22-1/(247+ln(2)) pi-.00000 00000 143 Gerson W. Barbosa
8 2+e709/5354 pi+.00000 000417 N.V. Subramanyan
8 ln(√820-544/99) pi-.00000 00000 508 Jim Cullen
8 ln((7!+5!+5!)3+6!+4!))/√67 pi+.00000 00000 00000 00111 Gerson Washiski Barbosa
11 355/113-1/(eee-65650) pi+.00000 00000 00000 861 Gerson Washiski Barbosa
11 √((88ln(8)+34√√√√6-57√√7)/13) pi+.00000 00000 00000 680 Jim Cullen
11 ln((5!*5336)3+4!+6!)/√163 Accurate to 30 places Gerson Washiski Barbosa
29 2/3 √(2/29) ln(1/(2√(2)) (11√(6) + 5 √(29)) (70 + 13√(29)) (7 + 3√(6) + √(99 + 42√(6)))3) pi-.00000 00000 00000 00000 00000 00000 0706 Paul Cleary


Pandigital: uses all digits once each
Digits Equation Value Contributor
9 3.14 + (7-.9-6+2/8)5 pi-.00000 00000 93 Joel Karnofsky
9 3 + (2^((((5+8)/4)-6)*9))/7.1 pi-.00000 00000 281 Sergey Ioffe
9 3 + (9^(2/(5*7) - (1+6)-4)) / 8 pi+.00000 00000 0188 Oleg Vlasii
9 2^5^.4 - .6 - ((.3^9)/7)^.8^.1 pi+.00000 00000 00660 Richard Hess
9 (2/.98-.3) * (.4+5^(7-.6-.1)) pi-.00000 00000 000411 Oleg Vlasii
10 3 + 4/28 - 1/(790 + 5/6) pi+.00000 00003 32 Brian Astle
10 3 + (5^-8^-((10+2-6-9)/4))/7 pi-.00000 00000 193 Sergey Ioffe
10 (1 + (2/(.8-4^(-.6)))^.5)^(7^(-.03*.9)) pi+.00000 00000 00001 03 Oleg Vlasii


Simon Plouffe uses a measure R2 for the effectiveness of a pi approximation which is the number of digits of accuracy divided by the number of digits in the largest number in the expression. Values over 2 are considered best. Here are some approximations that score well on this scale.
Digits R2 Equation Value Contributor
6 4.574 8/3+3/5-1/8 pi+.00007 40 Frank Rubin
9 5.502 5/8+5/9+8/7+9/11 pi+.00000 186 Frank Rubin
11 4.592 42/25+49/33-1/43 pi+.00000 00173 Frank Rubin
12 4.690 73/39+92/73+1/105 pi+.00000 00003 32 Frank Rubin
12 3.796 427/596+405/167 pi+.00000 00000 291 Frank Rubin
47 11.084 -1/3-4/5-4/7-9/11+9/13+13/17+16/19+16/23
+14/29+26/31+19/37+14/41+2/43+21/47 		pi-.00000 00000 00000 000293 Simon Plouffe


LINKS
Robert Chin's Pi Approximation webpage.
Simon Plouffe's Pi Approximation webpage.
Carlos Rivera's Pi Approximation webpage.
Eric Weisstein's Pi Approximation webpage.


David Blatner's The Joy of Pi webpage.
Boris Gourevitch's L'Univers de Pi webpage.
Ramon Lloren's Pi Webpage.
Martin Rebas's Pi Approximation Day webpage.

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