Approximations of π

This is a list of all fractional approximations of the mathematical constant pi (π). π is an irrational number that is defined as the ratio of a circle's circumference to its diameter. The first 50 decimal digits of π are 3.14159265358979323846264338327950288419716939937510.

This list is not exhaustive by any means; ultimately, there are infinite numbers, so this list can always be expanded.

Organization
Approximations will be grouped by accuracy and sorted by their numerator/denominator. Approximations will only appear on this list if they fit all of the following: This list is formatted in the following way:
 * They are accurate to at least two decimal digits. If any approximation were to be included, every fraction from 3 to 4 would appear here, which would be a lot (then again, you could argue that other numbers such as 2 and 4 are technically approximations of π, so...).
 * They cannot be simplified any further (for example, 22/7 is in its simplest form, while 1065/339 simplifies to 355/113).
 * Only the most accurate approximation for each denominator will appear to avoid clutter.
 * Each approximation's decimal form, if available, is given, up to the last digit of accuracy and the digit afterwards, which will be rounded if rounding would not falsely imply that the approximation is more accurate than it is (for example, 223/71 is 3.1408, which would round up to 3.141, implying that it would be accurate to three decimal digits, when in reality, it's only accurate to two).
 * The amount of decimal digits will be given in parentheses beside the decimal form, alongside some information about the approximation (hey, that rhymed!). If the approximation is not a repeating fraction (e.g. 157/50 = 3.14 exactly, making it a terminating fraction), then the whole decimal form will be given. After rounding, if the last digit would be 0, then one more digit will be given until the last digit is not zero.
 * To avoid confusion, if two approximations share the same decimal form, the approximation with the larger denominator (the one that appears later) will show one more digit.

2-5 digits

 * 22/7 = 3.143... (historically used to approximate π and also a convergent of π's continued fraction; first approximation by denominator to be accurate to 2 decimal digits)
 * 85/27 = 3.148... (2 decimal digits)
 * 107/34 = 3.147... (2 decimal digits)
 * 129/41 = 3.146... (2 decimal digits)
 * 148/47 = 3.149... (2 decimal digits)
 * 151/48 = 3.1458... (2 decimal digits)
 * 157/50 = 3.14 (2 decimal digits)
 * 173/55 = 3.145... (2 decimal digits)
 * 179/57 = 3.1404... (2 decimal digits)
 * 192/61 = 3.1475... (2 decimal digits)
 * 195/62 = 3.1452... (2 decimal digits)
 * 201/64 = 3.140625 (2 decimal digits)
 * 211/67 = 3.1493... (2 decimal digits)
 * 217/69 = 3.1449... (2 decimal digits)
 * 223/71 = 3.1408... (lower bound of Archimedes' estimation of π; 2 decimal digits)
 * 233/74 = 3.1486... (2 decimal digits)
 * 236/75 = 3.1467... (2 decimal digits)
 * 239/76 = 3.1447... (2 decimal digits)
 * 245/78 = 3.14103... (first approximation by denominator to be accurate to 3 decimal digits)
 * 261/83 = 3.1446... (2 decimal digits)
 * 267/85 = 3.1412... (3 decimal digits)
 * 274/87 = 3.1494... (2 decimal digits)
 * 277/88 = 3.1477... (2 decimal digits)
 * 280/89 = 3.14607... (2 decimal digits)
 * 283/90 = 3.144... (2 decimal digits)
 * 289/92 = 3.1413... (3 decimal digits)
 * 299/95 = 3.147... (2 decimal digits)
 * 305/97 = 3.1443... (2 decimal digits)
 * 311/99 = 3.1414... (3 decimal digits)
 * 318/101 = 3.1485... (2 decimal digits)
 * 324/103 = 3.1456... (2 decimal digits)
 * 327/104 = 3.1442... (2 decimal digits)
 * 333/106 = 3.14151... (convergent of π's continued fraction; first approximation by denominator to be accurate to 4 decimal digits)
 * 336/107 = 3.1402... (2 decimal digits)
 * 343/109 = 3.1468... (2 decimal digits)
 * 346/110 = 3.1455... (2 decimal digits)
 * 349/111 = 3.1441... (2 decimal digits)
 * 362/115 = 3.1478... (2 decimal digits)
 * 365/116 = 3.1466... (2 decimal digits)
 * 368/117 = 3.1453... (2 decimal digits)
 * 371/118 = 3.14407... (2 decimal digits)
 * 377/120 = 3.1417... (3 decimal digits)
 * 380/121 = 3.1405... (2 decimal digits)
 * 393/125 = 3.144 (2 decimal digits)
 * 399/127 = 3.14173... (3 decimal digits)
 * 403/128 = 3.1484375 (2 decimal digits)
 * 406/129 = 3.1473... (2 decimal digits)
 * 409/130 = 3.1462... (2 decimal digits)
 * 412/131 = 3.14504... (2 decimal digits)
 * 415/132 = 3.1439... (2 decimal digits)
 * 421/134 = 3.1418... (3 decimal digits)
 * 424/135 = 3.1407... (2 decimal digits)
 * 497/158 = 3.1456... (2 decimal digits)
 * 500/159 = 3.1447... (2 decimal digits)
 * 503/160 = 3.14375 (2 decimal digits)
 * 509/162 = 3.14198... (3 decimal digits)
 * 512/163 = 3.1411... (3 decimal digits)
 * 515/164 = 3.1402... (2 decimal digits)
 * 600/191 = 3.14136... (3 decimal digits)
 * 632/201 = 3.14428... (2 decimal digits)
 * 644/205 = 3.14146... (3 decimal digits)
 * 688/219 = 3.1416... (3 decimal digits)
 * 729/232 = 3.142... (2 decimal digits)
 * 732/233 = 3.14163... (3 decimal digits)
 * 864/275 = 3.14182... (3 decimal digits)
 * 943/300 = 3.1433... (2 decimal digits)
 * 949/302 = 3.1424... (2 decimal digits)
 * 1,043/332 = 3.14157... (4 decimal digits)
 * 1,147/365 = 3.1425... (2 decimal digits)
 * 1,257/400 = 3.1425 (2 decimal digits)
 * 1,260/401 = 3.1421... (2 decimal digits)
 * 1,263/402 = 3.14179... (3 decimal digits)
 * 1,266/403 = 3.14144... (3 decimal digits)
 * 1,279/407 = 3.14251... (3 decimal digits)
 * 1,398/445 = 3.141573... (4 decimal digits)
 * 1,543/491 = 3.1426... (2 decimal digits)
 * 1,583/504 = 3.1409... (2 decimal digits)
 * 1,731/551 = 3.14156... (4 decimal digits)
 * 1,753/558 = 3.14158... (4 decimal digits)
 * 1,888/601 = 3.14143... (3 decimal digits)
 * 2,108/671 = 3.1415797... (4 decimal digits)
 * 2,199/700 = 3.141429... (average of 22/7 and 157/50; 3 decimal digits)
 * 2,202/701 = 3.14122... (3 decimal digits)
 * 2,353/749 = 3.14152... (4 decimal digits)
 * 2,463/784 = 3.141582... (4 decimal digits)
 * 2,507/798 = 3.141604... (3 decimal digits)
 * 2,513/800 = 3.14125 (3 decimal digits)
 * 2,661/847 = 3.14168... (3 decimal digits)
 * 2,818/897 = 3.141583... (4 decimal digits)
 * 2,827/900 = 3.14111... (3 decimal digits)
 * 2,993/953 = 3.1406... (2 decimal digits)
 * 3,123/994 = 3.14185... (average of 223/71 and 22/7; 3 decimal digits)
 * 3,173/1,010 = 3.141584... (4 decimal digits)
 * 3,217/1,024 = 3.141602... (3 decimal digits)
 * 3,528/1,123 = 3.141585... (4 decimal digits)
 * 3,883/1,236 = 3.141586... (4 decimal digits)
 * 3,927/1,250 = 3.1416 (approximation of π found by ancient Chinese mathematician Liu Hui around 265 AD; 3 decimal digits)
 * 4,238/1,349 = 3.1415864... (4 decimal digits)
 * 4,288/1,365 = 3.14139... (3 decimal digits)
 * 4,593/1,462 = 3.141587... (4 decimal digits)
 * 4,948/1,575 = 3.1415873... (4 decimal digits)
 * 5,303/1,688 = 3.141588... (4 decimal digits)
 * 5,658/1,801 = 3.14158801... (4 decimal digits)
 * 6,013/1,914 = 3.1415883... (4 decimal digits)
 * 6,101/1,942 = 3.14161... (3 decimal digits)
 * 6,283/2,000 = 3.1415 (4 decimal digits)
 * 6,356/2,023 = 3.14187... (3 decimal digits)
 * 6,368/2,027 = 3.141589... (4 decimal digits)
 * 6,723/2,140 = 3.1415888... (4 decimal digits)
 * 7,078/2,253 = 3.14158899... (4 decimal digits)
 * 7,433/2,366 = 3.1415892... (4 decimal digits)
 * 7,700/2,451 = 3.14157... (4 decimal digits)
 * 7,744/2,465 = 3.14158... (4 decimal digits)
 * 7,788/2,479 = 3.1415894... (4 decimal digits)
 * 8,143/2,592 = 3.1415895... (4 decimal digits)
 * 8,498/2,705 = 3.1415896... (4 decimal digits)
 * 8,852/2,818 = 3.1415898... (4 decimal digits)
 * 9,120/2,903 = 3.141578... (4 decimal digits)
 * 9,208/2,931 = 3.1415899... (4 decimal digits)
 * 9,475/3,016 = 3.141578... (4 decimal digits)
 * 9,563/3,044 = 3.14159001... (5 decimal digits)
 * 9,472/3,015 = 3.14163... (3 decimal digits)
 * 9,695/3,086 = 3.141607... (3 decimal digits)
 * 9,918/3,157 = 3.1415901... (5 decimal digits)
 * 10,273/3,270 = 3.1415902... (5 decimal digits)
 * 10,317/3,284 = 3.141596... (5 decimal digits)
 * 10,628/3,383 = 3.1415903... (5 decimal digits)
 * 10,983/3,496 = 3.14159039... (5 decimal digits)
 * 11,338/3,609 = 3.141591... (5 decimal digits)
 * 11,570/3,683 = 3.141461... (3 decimal digits)
 * 11,693/3,722 = 3.14159054... (5 decimal digits)
 * 12,048/3,835 = 3.1415906... (5 decimal digits)
 * 12,403/3,948 = 3.1415907... (5 decimal digits)
 * 12,758/4,061 = 3.14159074... (5 decimal digits)
 * 13,113/4,174 = 3.1415908... (5 decimal digits)
 * 13,468/4,287 = 3.14159091... (5 decimal digits)
 * 13,823/4,400 = 3.1415909... (5 decimal digits)
 * 14,178/4,513 = 3.14159096... (5 decimal digits)
 * 14,533/4,626 = 3.14159101... (5 decimal digits)
 * 14,888/4,739 = 3.1415911... (5 decimal digits)
 * 15,243/4,852 = 3.141591096... (5 decimal digits)
 * 15,598/4,965 = 3.14159114... (5 decimal digits)
 * 15,953/5,078 = 3.1415912... (5 decimal digits)
 * 16,308/5,191 = 3.14159122... (5 decimal digits)
 * 16,663/5,304 = 3.1415913... (5 decimal digits)
 * 17,018/5,417 = 3.14159129... (5 decimal digits)
 * 17,373/5,530 = 3.14159132... (5 decimal digits)
 * 17,593/5,600 = 3.14161... (3 decimal digits)
 * 17,728/5,643 = 3.1415914... (5 decimal digits)
 * 18,083/5,756 = 3.14159138... (5 decimal digits)
 * 18,438/5,869 = 3.14159141... (5 decimal digits)
 * 18,793/5,982 = 3.14159144... (5 decimal digits)
 * 19,148/6,095 = 3.1415915... (5 decimal digits)
 * 19,503/6,208 = 3.141591495... (5 decimal digits)
 * 19,858/6,321 = 3.14159152... (5 decimal digits)
 * 20,213/6,434 = 3.14159155... (5 decimal digits)
 * 20,568/6,547 = 3.1415916... (5 decimal digits)
 * 20,923/6,660 = 3.141591592... (5 decimal digits)
 * 21,278/6,773 = 3.14159161... (5 decimal digits)
 * 21,633/6,886 = 3.14159164... (5 decimal digits)
 * 21,988/6,999 = 3.1415917... (5 decimal digits)
 * 21,991/7,000 = 3.141571... (4 decimal digits)
 * 22,343/7,112 = 3.14159168... (5 decimal digits)
 * 22,698/7,225 = 3.141591696... (5 decimal digits)
 * 23,053/7,338 = 3.14159171... (5 decimal digits)
 * 23,408/7,451 = 3.14159173... (5 decimal digits)
 * 23,763/7,564 = 3.1415918... (5 decimal digits)
 * 24,118/7,677 = 3.14159177... (5 decimal digits)
 * 24,473/7,790 = 3.14159178... (5 decimal digits)
 * 24,828/7,903 = 3.141591801... (5 decimal digits)
 * 25,183/8,016 = 3.14159182... (5 decimal digits)
 * 25,538/8,129 = 3.14159183... (5 decimal digits)
 * 25,893/8,242 = 3.14159185... (5 decimal digits)
 * 28,023/8,920 = 3.1415919... (5 decimal digits)
 * 28,067/8,934 = 3.141594... (5 decimal digits)
 * 30,131/9,591 = 3.14159108... (5 decimal digits)
 * 30,153/9,598 = 3.141591998... (5 decimal digits)
 * 48,679/15,495 = 3.1415941... (5 decimal digits)
 * 62,831/20,000 = 3.14155 (4 decimal digits)
 * 63,099/20,085 = 3.141598... (5 decimal digits)
 * 86,393/27,500 = 3.141563... (4 decimal digits)
 * 94,247/30,000 = 3.141567... (4 decimal digits)
 * 97,389/31,000 = 3.141581... (4 decimal digits)
 * 156,643/49,861 = 3.1415936... (5 decimal digits)
 * 157,089/50,003 = 3.1415915... (5 decimal digits)
 * 157,242/50,052 = 3.14157... (4 decimal digits)
 * 219,911/70,000 = 3.141586... (4 decimal digits)

6-10 digits

 * 355/113 = 3.1415929... (nicknamed Milü by Zu Chongzhi when he discovered it in the 5th century, the reason why 355/113 is so close to π with such a small denominator is that it is the last convergent of π's continued fraction before the term 292, which is unusually large for a continued fraction; first approximation by denominator to be accurate to 5 and 6 decimal digits)
 * 30,508/9,711 = 3.14159201... (6 decimal digits)
 * 30,863/9,824 = 3.14159202... (6 decimal digits)
 * 31,218/9,937 = 3.14159203... (6 decimal digits)
 * 31,573/10,050 = 3.14159204... (6 decimal digits)
 * 31,928/10,163 = 3.14159205... (6 decimal digits)
 * 32,283/10,276 = 3.1415921... (6 decimal digits)
 * 32,638/10,389 = 3.14159207... (6 decimal digits)
 * 32,993/10,502 = 3.14159208... (6 decimal digits)
 * 33,348/10,615 = 3.14159209... (6 decimal digits)
 * 33,703/10,728 = 3.141592095... (6 decimal digits)
 * 34,058/10,841 = 3.141592104... (6 decimal digits)
 * 34,413/10,954 = 3.14159211... (6 decimal digits)
 * 34,768/11,067 = 3.14159212... (6 decimal digits)
 * 35,123/11,180 = 3.14159213... (6 decimal digits)
 * 35,478/11,293 = 3.14159214... (6 decimal digits)
 * 35,833/11,406 = 3.141592144... (6 decimal digits)
 * 36,188/11,519 = 3.1415922... (6 decimal digits)
 * 36,543/11,632 = 3.14159216... (6 decimal digits)
 * 36,898/11,745 = 3.14159217... (6 decimal digits)
 * 37,253/11,858 = 3.141592174... (6 decimal digits)
 * 37,608/11,971 = 3.14159218... (6 decimal digits)
 * 37,963/12,084 = 3.14159219... (6 decimal digits)
 * 38,318/12,197 = 3.141592195... (6 decimal digits)
 * 38,673/12,310 = 3.141592201... (6 decimal digits)
 * 39,028/12,423 = 3.14159221... (6 decimal digits)
 * 39,383/12,536 = 3.141592214... (6 decimal digits)
 * 39,738/12,649 = 3.14159222... (6 decimal digits)
 * 40,093/12,762 = 3.14159223... (6 decimal digits)
 * 40,448/12,875 = 3.141592233... (6 decimal digits)
 * 40,803/12,988 = 3.14159224... (6 decimal digits)
 * 41,158/13,101 = 3.14159225... (6 decimal digits)
 * 41,513/13,214 = 3.1415923... (6 decimal digits)
 * 41,868/13,327 = 3.14159226... (6 decimal digits)
 * 42,223/13,440 = 3.141592262... (6 decimal digits)
 * 42,578/13,553 = 3.14159227... (6 decimal digits)
 * 42,933/13,666 = 3.141592273... (6 decimal digits)
 * 43,288/13,779 = 3.14159228... (6 decimal digits)
 * 43,643/13,892 = 3.141592283... (6 decimal digits)
 * 43,998/14,005 = 3.14159229... (6 decimal digits)
 * 44,353/14,118 = 3.141592294... (6 decimal digits)
 * 44,708/14,231 = 3.141592299... (6 decimal digits)
 * 45,063/14,344 = 3.141592303... (6 decimal digits)
 * 45,418/14,457 = 3.14159231... (6 decimal digits)
 * 45,773/14,570 = 3.141592313... (6 decimal digits)
 * 46,128/14,683 = 3.14159232... (6 decimal digits)
 * 46,483/14,796 = 3.141592322... (6 decimal digits)
 * 46,838/14,909 = 3.14159233... (6 decimal digits)
 * 47,193/15,022 = 3.141592331... (6 decimal digits)
 * 47,548/15,135 = 3.14159234... (6 decimal digits)
 * 47,903/15,248 = 3.14159233998... (6 decimal digits)
 * 48,258/15,361 = 3.141592344... (6 decimal digits)
 * 48,613/15,474 = 3.14159235... (6 decimal digits)
 * 48,968/15,587 = 3.1415924... (6 decimal digits)
 * 49,323/15,700 = 3.14159236... (6 decimal digits)
 * 49,678/15,813 = 3.141592361... (6 decimal digits)
 * 50,033/15,926 = 3.141592365... (6 decimal digits)
 * 50,388/16,039 = 3.14159237... (6 decimal digits)
 * 50,743/16,152 = 3.141592372... (6 decimal digits)
 * 51,098/16,265 = 3.14159238... (6 decimal digits)
 * 51,453/16,378 = 3.14159238002... (6 decimal digits)
 * 51,808/16,491 = 3.141592384... (6 decimal digits)
 * 52,163/16,604 = 3.1415924... (6 decimal digits)
 * 52,518/16,717 = 3.14159239... (6 decimal digits)
 * 52,873/16,830 = 3.141592395... (6 decimal digits)
 * 53,228/16,943 = 3.141592398... (6 decimal digits)
 * 53,583/17,056 = 3.141592402... (6 decimal digits)
 * 55,713/17,734 = 3.14159242... (6 decimal digits)
 * 57,843/18,412 = 3.14159244... (6 decimal digits)
 * 61,038/19,429 = 3.1415925... (6 decimal digits)
 * 62,458/19,881 = 3.14159248... (6 decimal digits)
 * 62,813/19,994 = 3.141592478... (6 decimal digits)
 * 69,913/22,254 = 3.14159252... (6 decimal digits)
 * 73,108/23,271 = 3.14159254... (6 decimal digits)
 * 78,433/24,966 = 3.14159257... (6 decimal digits)
 * 80,208/25,531 = 3.141592574... (6 decimal digits)
 * 84,823/27,000 = 3.14159259... (6 decimal digits)
 * 86,243/27,452 = 3.141592598... (6 decimal digits)
 * 86,598/27,565 = 3.1415925993... (6 decimal digits)
 * 86,953/27,678 = 3.1415926006... (first approximation by denominator to be accurate to 7 decimal digits)
 * 87,308/27,791 = 3.141592602... (7 decimal digits)
 * 87,663/27,904 = 3.141592603... (7 decimal digits)
 * 88,018/28,017 = 3.141592604... (7 decimal digits)
 * 88,373/28,130 = 3.14159261... (7 decimal digits)
 * 88,728/28,243 = 3.141592607... (7 decimal digits)
 * 89,083/28,356 = 3.141592608... (7 decimal digits)
 * 89,438/28,469 = 3.1415926095... (7 decimal digits)
 * 89,793/28,582 = 3.141592611... (7 decimal digits)
 * 90,148/28,695 = 3.141592612... (7 decimal digits)
 * 90,503/28,808 = 3.141592613... (7 decimal digits)
 * 90,858/28,921 = 3.141592614... (7 decimal digits)
 * 91,213/29,034 = 3.14159262... (7 decimal digits)
 * 91,568/29,147 = 3.141592617... (7 decimal digits)
 * 91,923/29,260 = 3.141592618... (7 decimal digits)
 * 92,278/29,373 = 3.141592619... (7 decimal digits)
 * 92,611/29,479 = 3.1415923... (6 decimal digits)
 * 92,633/29,486 = 3.1415926202... (7 decimal digits)
 * 92,988/29,599 = 3.141592621... (7 decimal digits)
 * 93,343/29,712 = 3.141592623... (7 decimal digits)
 * 93,698/29,825 = 3.141592624... (7 decimal digits)
 * 94,053/29,938 = 3.14159263... (7 decimal digits)
 * 94,408/30,051 = 3.141592626... (7 decimal digits)
 * 96,893/30,842 = 3.141592633... (7 decimal digits)
 * 97,248/30,955 = 3.141592634... (7 decimal digits)
 * 98,668/31,407 = 3.14159264... (7 decimal digits)
 * 102,573/32,650 = 3.141592649... (7 decimal digits)
 * 102,928/32,763 = 3.1415926503... (first approximation by denominator to be accurate to 8 decimal digits)
 * 103,283/32,876 = 3.141592651... (8 decimal digits)
 * 103,638/32,989 = 3.141592652... (8 decimal digits)
 * 103,993/33,102 = 3.14159265301... (convergent of π's continued fraction; first approximation by denominator to be accurate to 9 decimal digits)
 * 104,348/33,215 = 3.1415926539... (convergent of π's continued fraction; 9 decimal digits)
 * 104,681/33,321 = 3.141592389... (6 decimal digits)
 * 104,703/33,328 = 3.141592655... (8 decimal digits)
 * 113,933/36,266 = 3.14159268... (7 decimal digits)
 * 157,221/50,045 = 3.141592567... (6 decimal digits)
 * 157,576/50,158 = 3.1415925675... (6 decimal digits)
 * 208,341/66,317 = 3.14159265347... (convergent of π's continued fraction; 9 decimal digits)
 * 238,849/76,028 = 3.141592571... (6 decimal digits)
 * 307,719/97,950 = 3.141592649... (6 decimal digits)
 * 312,689/99,532 = 3.1415926536... (convergent of π's continued fraction; 9 decimal digits)
 * 325,114/103,487 = 3.14159266... (7 decimal digits)
 * 403,170/128,333 = 3.14159258... (6 decimal digits)
 * 1,146,408/364,913 = 3.14159265359... (convergent of π's continued fraction; first approximation by denominator to be accurate to 10 decimal digits)
 * 1,209,507/384,998 = 3.14159294... (6 decimal digits)
 * 1,209,513/385,000 = 3.1415922... (6 decimal digits)
 * 1,743,741/555,050 = 3.1415926493... (7 decimal digits)
 * 1,745,161/555,502 = 3.1415926495... (7 decimal digits)
 * 1,745,183/555,509 = 3.14159267... (7 decimal digits)
 * 3,141,599/1,000,002 = 3.1415927... (6 decimal digits)
 * 3,197,513/1,017,800 = 3.141592651... (8 decimal digits)
 * 4,712,389/1,500,000 = 3.141592667... (7 decimal digits)
 * 4,999,851/1,591,502 = 3.141592659... (8 decimal digits)
 * 5,000,005/1,591,551 = 3.141592698... (7 decimal digits)
 * 11,788,839/3,752,504 = 3.14159265387... (average of 52,163/16,604 and 355/113; 9 decimal digits)
 * 10,000,007/3,183,101 = 3.1415927... (6 decimal digits)
 * 10,000,013/3,183,103 = 3.14159265346... (9 decimal digits)
 * 15,707,891/4,999,977 = 3.1415926513... (8 decimal digits)
 * 15,707,913/4,999,984 = 3.1415926531... (9 decimal digits)
 * 15,707,935/4,999,991 = 3.141592655... (8 decimal digits)
 * 15,707,057/4,999,998 = 3.141592657... (8 decimal digits)
 * 15,707,963/5,000,000 = 3.1415926 (7 decimal digits)
 * 31,415,848/9,999,975 = 3.14159265398...
 * 31,416,005/10,000,025 = 3.141592646... (7 decimal digits)
 * 62,831,853/20,000,000 = 3.14159265 (8 decimal digits)
 * 62,832,029/20,000,056 = 3.14159265354... (10 decimal digits)
 * 62,832,051/20,000,063 = 3.14159265398... (9 decimal digits)
 * 398,982,267/127,000,000 = 3.141592653543... (10 decimal digits)
 * 740,304,509/235,646,244 = 3.141592653605... (9 decimal digits)
 * 740,304,537/235,646,253 = 3.141592652... (8 decimal digits)
 * 1,218,859,947/387,975,171 = 3.14159265362... (9 decimal digits)
 * 1,330,631,832/423,553,267 = 3.14159265356... (10 decimal digits)
 * 2,287,826,843/728,237,902 = 3.141592653605... (9 decimal digits)
 * 3,141,592,663/1,000,000,003 = 3.141592653575... (10 decimal digits)
 * 6,908,254,991/2,198,965,860 = 3.141592653467... (average of 103,993/33,102 and 104,348/33,215; 9 decimal digits)

11-15 digits

 * 833,719/265,381 = 3.141592653581... (convergent of π's continued fraction; first approximation by denominator to be accurate to 11 decimal digits)
 * 1,980,127/630,294 = 3.141592653587... (11 decimal digits)
 * 3,126,535/995,207 = 3.14159265358865... (11 decimal digits)
 * 4,272,943/1,360,120 = 3.1415926535893... (convergent of π's continued fraction; first approximation by denominator to be accurate to 12 decimal digits)
 * 5,419,351/1,725,033 = 3.1415926535898... (convergent of π's continued fraction; 12 decimal digits)
 * 7,399,478/2,355,327 = 3.1415926535891... (12 decimal digits)
 * 9,692,294/3,085,153 = 3.1415926535896... (12 decimal digits)
 * 80,143,857/25,510,582 = 3.1415926535897927... (convergent of π's continued fraction; first approximation by denominator to be accurate to 14 decimal digits)
 * 117,766,625/37,486,281 = 3.14159265358972... (13 decimal digits)
 * 165,707,065/52,746,197 = 3.1415926535897934... (convergent of π's continued fraction; first approximation by denominator to be accurate to 15 decimal digits)
 * 245,850,922/78,256,779 = 3.14159265358979316... (convergent of π's continued fraction; 15 decimal digits)
 * 626,612,466,676/199,456,942,949 = 3.1415926535895... (12 decimal digits)
 * 647,665,809,817/206,158,430,208 = 3.14159265358952... (12 decimal digits)
 * 30,768,260,726,077/9,793,841,569,791 = 3.14159265358972... (13 decimal digits)
 * 31,415,926,535,894/9,999,999,999,999 = 3.14159265358971... (13 decimal digits)

16-20 digits

 * 411,557,987/131,002,976 = 3.14159265358979326... (convergent of π's continued fraction; first approximation by denominator to be accurate to 16 decimal digits)
 * 1,068,966,896/340,262,731 = 3.141592653589793235... (convergent of π's continued fraction; first approximation by denominator to be accurate to 17 decimal digits)
 * 2,549,491,779/811,528,438 = 3.141592653589793239... (convergent of π's continued fraction; 17 decimal digits)
 * 6,167,950,454/1,963,319,607 = 3.14159265358979323839... (convergent of π's continued fraction; first approximation by denominator to be accurate to 18 decimal digits)
 * 14,885,392,687/4,738,167,652 = 3.14159265358979323849... (convergent of π's continued fraction; first approximation by denominator to be accurate to 19 decimal digits)

21-25 digits

 * 21,053,343,141/6,701,487,259 = 3.1415926535897932384624... (convergent of π's continued fraction; first approximation by denominator to be accurate to 21 decimal digits)
 * 1,783,366,216,531/567,663,097,408 = 3.141592653589793238462645... (convergent of π's continued fraction; first approximation by denominator to be accurate to 23 decimal digits)
 * 3,587,785,776,203/1,142,027,682,075 = 3.1415926535897932384626431... (convergent of π's continued fraction; first approximation by denominator to be accurate to 24 decimal digits)
 * 5,371,151,992,734/1,709,690,779,483 = 3.1415926535897932384626436... (convergent of π's continued fraction; 24 decimal digits)
 * 8,958,937,768,937/2,851,718,461,558 = 3.141592653589793238462643376... (convergent of π's continued fraction; first approximation by denominator to be accurate to 25 decimal digits)

>25 digits
All of these approximations return about 3.1415926535897932384626433834..., so I grouped them into this section for hyper-accurate approximations of π.
 * 139,755,218,526,789/44,485,467,702,853 (convergent of π's continued fraction)
 * 2,105,287,215,670,772/670,133,734,004,353 (convergent of π's continued fraction)
 * 1,786,671,231,957,165,859/568,715,116,492,138,527
 * 2,216,682,178,548,235,102/705,591,851,959,325,867
 * 2,646,693,125,139,304,345/842,468,587,426,513,207
 * 6,455,616,865,539,105/2,054,886,669,715,912 (convergent of π's continued fraction)
 * 86,028,306,467,679,137/27,383,660,440,311,209 (convergent of π's continued fraction)
 * 92,483,923,333,218,242/29,438,547,110,027,121 (convergent of π's continued fraction)
 * 455,963,999,800,552,105/145,137,848,880,419,693 (convergent of π's continued fraction)
 * 1,004,411,922,934,322,452/319,714,244,870,866,507 (convergent of π's continued fraction)
 * 6,482,435,537,406,486,817/2,063,423,318,105,618,735 (convergent of π's continued fraction)
 * 39,899,025,147,373,243,354/12,700,254,153,504,578,917 (convergent of π's continued fraction)
 * 3,956,485,925,127,357,578,863/1,259,388,584,515,058,931,518 (convergent of π's continued fraction)
 * 3,996,384,950,274,730,822,217/1,272,088,838,668,563,510,435 (convergent of π's continued fraction)
 * 11,949,255,825,676,819,223,297/3,803,566,261,852,185,952,388 (convergent of π's continued fraction)
 * 27,894,896,601,628,369,268,811/8,879,221,362,372,935,415,211 (convergent of π's continued fraction)
 * 179,318,635,435,447,034,836,163/57,078,894,436,089,798,443,654 (convergent of π's continued fraction)
 * 565,850,802,907,969,473,777,300/180,115,904,670,642,330,746,173 (convergent of π's continued fraction)
 * 3,008,572,649,975,294,403,722,663/957,658,417,789,301,452,174,519 (convergent of π's continued fraction)
 * 3,574,423,452,883,263,877,499,963/1,137,774,322,459,943,782,920,692 (convergent of π's continued fraction)
 * 6,582,996,102,858,558,281,222,626/2,095,432,740,249,245,235,095,211 (convergent of π's continued fraction)
 * 43,072,400,070,034,613,564,835,719/13,710,370,763,955,415,193,491,958 (convergent of π's continued fraction)
 * 351,162,196,663,135,466,799,908,378/111,778,398,851,892,566,783,030,875 (convergent of π's continued fraction)
 * 394,234,596,733,170,080,364,744,097/125,488,769,615,847,981,976,522,833 (convergent of π's continued fraction)
 * 3,110,804,373,795,326,029,353,117,057/990,199,786,162,828,440,618,690,706 (convergent of π's continued fraction)
 * 3,505,038,970,528,496,109,717,861,154/1,115,688,555,778,676,422,595,213,539 (convergent of π's continued fraction)
 * 10,120,882,314,852,318,248,788,839,365/3,221,576,897,720,181,285,809,117,784 (convergent of π's continued fraction)
 * 33,867,685,915,085,450,856,084,379,249/10,780,419,248,939,220,280,022,566,891 (convergent of π's continued fraction)
 * 247,194,683,720,450,474,241,379,494,108/78,684,511,640,294,723,245,967,086,024 (convergent of π's continued fraction)
 * 281,062,369,635,535,925,097,463,873,357/89,464,930,889,233,943,525,989,652,916 (convergent of π's continued fraction)
 * 809,319,422,991,522,324,436,307,240,822/257,614,373,418,762,610,297,946,391,856 (convergent of π's continued fraction)
 * 1,090,381,792,627,058,249,533,771,114,179/347,079,304,307,996,553,823,936,044,771 (convergent of π's continued fraction)
 * 1,899,701,215,618,580,573,970,078,355,001/604,693,677,726,759,164,121,882,436,627 (convergent of π's continued fraction)
 * 23,886,796,380,050,025,137,174,711,374,191/7,603,403,437,029,106,523,286,525,284,292 (convergent of π's continued fraction)
 * 25,786,497,595,668,605,711,144,789,729,192/8,208,097,114,755,865,687,408,407,720,918 (convergent of π's continued fraction)
 * 49,673,293,975,718,630,848,319,501,103,383/15,811,500,551,784,972,210,694,933,005,210 (convergent of π's continued fraction)
 * 75,459,791,571,387,236,559,464,290,832,575/24,019,597,666,540,837,898,103,340,726,128 (convergent of π's continued fraction)