84:
74:
53:
166:
1220:
1754:
22:
1852:
real difficulty in finding where I had gone wrong. But at this point I not only found my figures differing completely from those of Shanks, but all my efforts to find my mistake failed." He spent another four months checking his work by means of a different series summation before venturing the opinion that Shanks might have erred.
939:
1482:
910:
I think it would be better to just say that these cultures had formulas for circumference and area which are equivalent to the formulas C=(25/8)D and A=(256/81)r^2 so it is like they had values for pi, but it wasn't like they were using the formulas C=pi*D and A=pi*r^2 and they were trying to use the
1851:
A fuller account of
Ferguson's confrontation with Shanks. Working with a desk calculator, it took him a year to get up to 530 decimal places. "Up to this point, whenever I had disagreed with Shanks' figures (and this has occurred from time to time, owing to copying errors, etc.), I had never had any
1817:
note was from 1946 and didn't say what in year the calculation was supposedly done (it just says "recently"). 1944 is hard to believe because it was the height of WW2 and it's implausible that anyone with the necessary skills would be spending their time calculating pi, as well as tying up a scarce
622:
Currently the article mentions 125648/39995 as a fraction that produces 8 correct digits. This is not wrong, but it's not useful. There are at least 45 better fractions that do the same, and use smaller denominators. And half of then are more accurate. So I am replacing 125648/39995 with 99733/31746
619:
Another use is integer math. For example, if you use integer math with 32 bit numbers to calculate the circumference of an object, and the maximum diameter of that object is 130000 units, then the max denominator would be, 2^32/130000 = 33038. Then the best approximate fraction you can use, would be
906:
But for Egypt, this is much more of a stretch. They have a formula for the area of a circle which is A=(D-D/9)^2. It is a great formula, but to say "treats pi as 256/81" is really not accurate. While it is true that this formula could be written as A=(2r-2r/9)^2=(16r/9)^2=256/81*r^2 it is not
902:
For the babylonians, they have a tablet that basically says that the circumference of a circle is 25/24 multiplied by the perimeter of the inscribed regular hexagon. So if the circle has diameter=1, the side of the hexagon is 0.5 and the perimeter of the hexagon is 3 so the circumference of the
1392:
1215:{\displaystyle \pi \approx \pi _{1}(n)={\frac {3}{(1-R){\sqrt {n}}}},\quad R={\frac {1-{\frac {3}{\pi {\sqrt {n}}}}-24\sum _{r=1}^{\infty }{\frac {r}{\exp(2\pi r{\sqrt {n}})-1}}}{1-24\sum _{r=1}^{\infty }{\frac {2r-1}{\exp(\pi (2r-1){\sqrt {n}})+1}}}}}
836:
2016:
1749:{\displaystyle \pi _{4}(n)={\frac {6}{(1-S){\sqrt {n}}}},\quad S={\frac {1-{\frac {6}{\pi {\sqrt {n}}}}-24\sum _{r=1}^{\infty }{\frac {r}{\exp(\pi r{\sqrt {n}})-1}}}{1+24\sum _{r=1}^{\infty }{\frac {2r-1}{\exp(\pi (2r-1){\sqrt {n}})-1}}}}}
1471:
1818:
desk calculator for however long it was. By 1946 there would have been plenty of people with time on their hands, plus surplus calculators. I have not looked at the
Penguin book about curious numbers cited in the
1865:
and indicates the calculation really was done in 1944-1945, wow. I'll change my earlier edit in a minute, and expand the article a little bit when I get a chance. I haven't yet looked Hayes' other references.
1296:
617:
Aside from some mathematical trivia, generally a good use of approximation of pi would be for the memorization of a smaller number of digits than the approximation can give. For this, only 355/113 is useful.
1894:
In 1853 there appeared, in a paper by W. Rutherford, the value of the constant π to 530 decimals, calculated by W. Shanks. This was eventually extended by Shanks to 607, and in 1873 to 707 decimals.
914:
Might there be a simple way to edit this so that it is more accurate and does not claim that these cultures were aware there there was this constant pi, but not to make it too complicated to explain?
2064:
It is tempting to look at the version of the BBP formula given and think that the 16 factor at the end implies that the rest returns the value of one digit but this is obviously not the case, as the
2078:
Could someone better at this stuff than me please add a formula that actually returns arbitrary digits of π? Or, if this is actually not possible, the article text needs to be updated to say so.
1307:
140:
903:
circle would be 25/24*3=25/8=3 1/8. So this is a formula for circumference of a circle, basically 25/8 * diameter, so it is not totally wrong to say 'by implication treats pi as 25/8".
1784:
This section has become a magnet for the insertion of ad hoc approximations of a few decimal places that anyone can dream up. I recommend removing the section altogether per
2039:
1827:
746:
722:
754:
2059:
the formula can compute any particular digit of π—returning the hexadecimal value of the digit—without having to compute the intervening digits (digit extraction).
1905:
2057:
It's quite frustrating that the section titled "digit extraction methods" doesn't actually state a way of extracting a digit of π, in any base. It states that
847:
2106:
2031:
It is of interest to note that the discrepancy occurs at about the point to which Shanks's first published value extends, that is, in the 530th decimal.
130:
229:
2020:
The value so obtained agrees with Shanks's value only to the 527th decimal place; from that last point it seems that Shanks's value is incorrect.
106:
2101:
1403:
179:
2061:
But then, bizarrely, doesn't give a formula for extracting any particular digit of π, instead giving a series sum to return the value of π.
1857:
This and a few others from Hayes' list look like better references about the
Ferguson calculation. The mathematical Gazette article is at
686:
238:
2075:
Likewise, Plouffe's formula for digits of π in base 10 is only given as a sum with no obvious way to factor it into individual digits.
1237:
1897:
For more than seventy years this has been accepted as the value of π, apparently without any doubts having been expressed in print.
615:
The entries with the '<' signs are particularly interesting because of the ratio of added precision over increase of denominator.
97:
58:
1810:
899:
The problem is that neither of those cultures had yet a concept of pi as either circumference/diameter or as area/(radius^2).
2043:
1831:
1387:{\displaystyle \pi _{1}(58)={\frac {1037785473+70101072{\sqrt {2}}+192518946{\sqrt {29}}+311451846{\sqrt {58}}}{1446914567}}}
2068:
factor only appears in denominators and so all the terms in the first part of the sum will become infinitesimally small as
33:
209:
230:
http://math.stackexchange.com/questions/1097633/how-to-show-frac-pi4-frac2-cdot4-cdot4-cdot6-cdot6-cdot8-dotsm3-cdot
877:
873:
188:
228:
be mentioned on this page or is it too obscure? It does seem that some books use it at leas as an example; see
273:
Here is a list of fractions giving approximations of pi with increasing denominators and increasing precision:
919:
699:
690:
667:
881:
242:
1849:
Ferguson, D. F. 1946. Evaluation of π. Are Shanks' figures correct? The
Mathematical Gazette 30(289):89–90.
915:
663:
39:
83:
1770:
2083:
831:{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\frac {{\sqrt {2}}-{\sqrt {3}}-18}{3921}}=3.141592644\ 0^{+}}
234:
1844:
1774:
21:
1793:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1785:
1766:
194:
89:
892:≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the
727:
703:
73:
52:
2011:{\displaystyle \pi =12\tan ^{-1}{1 \over 4}+4\tan ^{-1}{1 \over 20}+4\tan ^{-1}{1 \over 1985}}
1879:
1858:
843:
628:
190:
165:
1900:
Recently I decided to test numerically a series found by a colleague, R. W. Morris, namely
861:
in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of
2079:
1397:
is complicated. The
Borwein's brothers mention the following approximation in their book.
1789:
258:
225:
2095:
2087:
2047:
1835:
1797:
923:
694:
671:
649:
which yields Pi accurately to 10 decimal places..if you need that much accuracy, but
632:
262:
246:
638:...if I may..I find it difficult to memorize any of those fractions after 355/311.
839:
624:
102:
192:
641:
however, I'd like to go the other way and suggest the following approximation:
79:
838:
is accurate to 8 digits. But I don't think it's worth adding to the article.
254:
1809:
I changed the claimed date of this calculation from 1944 to 1946, to match
1878:(Ferguson, D. F. (16 March 1946). "Value of π". Nature. 157 (3985): 342.
1466:{\displaystyle \pi _{4}(58)={\frac {66{\sqrt {2}}}{33{\sqrt {29}}-148}}}
613:
5419351 / 1725033 = 3.14159265358981 (0.0000000000007068% err) <<
893:
682:"Approximations" of pi are mostly best geometry, rational exhaustion.
1883:
1862:
1291:{\displaystyle \pi _{1}(25)={\frac {9}{5}}+{\sqrt {\frac {9}{5}}}}
685:
Here that's "at least as mucn precision, ..., as 7 digits of pi".
303:
355 / 113 = 3.141592920 (0.0000084914% err) <<<<<
1874:
Regarding the above, here is the contents of
Ferguson's note in
858:
607:
1146408 / 364913 = 3.1415926535914 (0.00000000005127% err) <
195:
159:
15:
2023:
The values from the 521st to 540th decimals are given below:
611:
4272943 / 1360120 = 3.1415926535894 (0.000000000012863% err)
609:
3126535 / 995207 = 3.1415926535886 (0.00000000003637% err)
603:
312689 / 99532 = 3.14159265362 (0.0000000009276% err) <
880:, c. 1600 BCE, although stated to be a copy of an older,
599:
104348 / 33215 = 3.14159265392 (0.00000001055% err) <
605:
833719 / 265381 = 3.141592653581 (0.0000000002774% err)
1807:
857:"one Old Babylonian mathematical tablet excavated near
1908:
1485:
1406:
1310:
1240:
942:
757:
730:
706:
601:
208341 / 66317 = 3.14159265347 (0.000000003894% err)
1765:
is even. I added this approximation to the article.
597:
103993 / 33102 = 3.14159265301 (0.00000001839% err)
595:
103638 / 32989 = 3.14159265210 (0.00000004754% err)
593:
103283 / 32876 = 3.14159265117 (0.00000007689% err)
591:
102928 / 32763 = 3.14159265025 (0.00000010644% err)
101:, a collaborative effort to improve the coverage of
2010:
1748:
1465:
1386:
1290:
1214:
830:
740:
716:
589:102573 / 32650 = 3.1415926493 (0.0000001362% err)
587:102218 / 32537 = 3.1415926483 (0.0000001661% err)
585:101863 / 32424 = 3.1415926474 (0.0000001963% err)
583:101508 / 32311 = 3.1415926464 (0.0000002267% err)
581:101153 / 32198 = 3.1415926455 (0.0000002573% err)
579:100798 / 32085 = 3.1415926445 (0.0000002881% err)
577:100443 / 31972 = 3.1415926435 (0.0000003192% err)
575:100088 / 31859 = 3.1415926425 (0.0000003504% err)
573:99733 / 31746 = 3.1415926415 (0.0000003819% err)
571:99378 / 31633 = 3.1415926406 (0.0000004136% err)
569:99023 / 31520 = 3.1415926395 (0.0000004455% err)
567:98668 / 31407 = 3.1415926385 (0.0000004777% err)
565:98313 / 31294 = 3.1415926375 (0.0000005100% err)
563:97958 / 31181 = 3.1415926365 (0.0000005427% err)
561:97603 / 31068 = 3.1415926355 (0.0000005755% err)
559:97248 / 30955 = 3.1415926344 (0.0000006086% err)
557:96893 / 30842 = 3.1415926334 (0.0000006420% err)
555:96538 / 30729 = 3.1415926323 (0.0000006755% err)
553:96183 / 30616 = 3.1415926313 (0.0000007094% err)
551:95828 / 30503 = 3.1415926302 (0.0000007435% err)
549:95473 / 30390 = 3.1415926291 (0.0000007778% err)
547:95118 / 30277 = 3.1415926280 (0.0000008124% err)
545:94763 / 30164 = 3.1415926269 (0.0000008473% err)
543:94408 / 30051 = 3.1415926258 (0.0000008824% err)
541:94053 / 29938 = 3.1415926247 (0.0000009177% err)
539:93698 / 29825 = 3.1415926236 (0.0000009534% err)
537:93343 / 29712 = 3.1415926225 (0.0000009893% err)
535:92988 / 29599 = 3.1415926213 (0.0000010255% err)
533:92633 / 29486 = 3.1415926202 (0.0000010620% err)
531:92278 / 29373 = 3.1415926190 (0.0000010987% err)
529:91923 / 29260 = 3.1415926179 (0.0000011358% err)
527:91568 / 29147 = 3.1415926167 (0.0000011731% err)
525:91213 / 29034 = 3.1415926155 (0.0000012107% err)
523:90858 / 28921 = 3.1415926143 (0.0000012486% err)
521:90503 / 28808 = 3.1415926131 (0.0000012868% err)
519:90148 / 28695 = 3.1415926119 (0.0000013253% err)
517:89793 / 28582 = 3.1415926107 (0.0000013641% err)
515:89438 / 28469 = 3.1415926095 (0.0000014033% err)
513:89083 / 28356 = 3.1415926082 (0.0000014427% err)
511:88728 / 28243 = 3.1415926070 (0.0000014824% err)
509:88373 / 28130 = 3.1415926057 (0.0000015225% err)
507:88018 / 28017 = 3.1415926044 (0.0000015629% err)
505:87663 / 27904 = 3.1415926032 (0.0000016036% err)
503:87308 / 27791 = 3.1415926019 (0.0000016447% err)
501:86953 / 27678 = 3.1415926006 (0.0000016860% err)
751:Based on a recent addition, this approximation:
499:86598 / 27565 = 3.141592599 (0.0000017278% err)
497:86243 / 27452 = 3.141592597 (0.0000017698% err)
495:85888 / 27339 = 3.141592596 (0.0000018122% err)
493:85533 / 27226 = 3.141592595 (0.0000018550% err)
491:85178 / 27113 = 3.141592593 (0.0000018981% err)
489:84823 / 27000 = 3.141592592 (0.0000019416% err)
487:84468 / 26887 = 3.141592591 (0.0000019854% err)
485:84113 / 26774 = 3.141592589 (0.0000020297% err)
483:83758 / 26661 = 3.141592588 (0.0000020743% err)
481:83403 / 26548 = 3.141592587 (0.0000021192% err)
479:83048 / 26435 = 3.141592585 (0.0000021646% err)
477:82693 / 26322 = 3.141592584 (0.0000022103% err)
475:82338 / 26209 = 3.141592582 (0.0000022565% err)
473:81983 / 26096 = 3.141592581 (0.0000023030% err)
471:81628 / 25983 = 3.141592579 (0.0000023500% err)
469:81273 / 25870 = 3.141592578 (0.0000023973% err)
467:80918 / 25757 = 3.141592576 (0.0000024451% err)
465:80563 / 25644 = 3.141592575 (0.0000024933% err)
463:80208 / 25531 = 3.141592573 (0.0000025419% err)
461:79853 / 25418 = 3.141592572 (0.0000025909% err)
459:79498 / 25305 = 3.141592570 (0.0000026404% err)
457:79143 / 25192 = 3.141592569 (0.0000026904% err)
455:78788 / 25079 = 3.141592567 (0.0000027407% err)
453:78433 / 24966 = 3.141592565 (0.0000027916% err)
451:78078 / 24853 = 3.141592564 (0.0000028429% err)
449:77723 / 24740 = 3.141592562 (0.0000028947% err)
447:77368 / 24627 = 3.141592561 (0.0000029469% err)
445:77013 / 24514 = 3.141592559 (0.0000029996% err)
443:76658 / 24401 = 3.141592557 (0.0000030528% err)
441:76303 / 24288 = 3.141592555 (0.0000031065% err)
439:75948 / 24175 = 3.141592554 (0.0000031608% err)
437:75593 / 24062 = 3.141592552 (0.0000032155% err)
435:75238 / 23949 = 3.141592550 (0.0000032707% err)
433:74883 / 23836 = 3.141592549 (0.0000033265% err)
431:74528 / 23723 = 3.141592547 (0.0000033828% err)
429:74173 / 23610 = 3.141592545 (0.0000034396% err)
427:73818 / 23497 = 3.141592543 (0.0000034970% err)
425:73463 / 23384 = 3.141592541 (0.0000035549% err)
423:73108 / 23271 = 3.141592540 (0.0000036134% err)
421:72753 / 23158 = 3.141592538 (0.0000036725% err)
419:72398 / 23045 = 3.141592536 (0.0000037321% err)
417:72043 / 22932 = 3.141592534 (0.0000037923% err)
415:71688 / 22819 = 3.141592532 (0.0000038532% err)
413:71333 / 22706 = 3.141592530 (0.0000039146% err)
411:70978 / 22593 = 3.141592528 (0.0000039767% err)
409:70623 / 22480 = 3.141592526 (0.0000040393% err)
407:70268 / 22367 = 3.141592524 (0.0000041026% err)
405:69913 / 22254 = 3.141592522 (0.0000041666% err)
403:69558 / 22141 = 3.141592520 (0.0000042312% err)
401:69203 / 22028 = 3.141592518 (0.0000042965% err)
399:68848 / 21915 = 3.141592516 (0.0000043624% err)
397:68493 / 21802 = 3.141592514 (0.0000044290% err)
395:68138 / 21689 = 3.141592512 (0.0000044963% err)
393:67783 / 21576 = 3.141592510 (0.0000045643% err)
391:67428 / 21463 = 3.141592508 (0.0000046331% err)
389:67073 / 21350 = 3.141592505 (0.0000047026% err)
387:66718 / 21237 = 3.141592503 (0.0000047728% err)
385:66363 / 21124 = 3.141592501 (0.0000048437% err)
383:66008 / 21011 = 3.141592499 (0.0000049154% err)
381:65653 / 20898 = 3.141592496 (0.0000049879% err)
379:65298 / 20785 = 3.141592494 (0.0000050612% err)
377:64943 / 20672 = 3.141592492 (0.0000051353% err)
375:64588 / 20559 = 3.141592489 (0.0000052102% err)
373:64233 / 20446 = 3.141592487 (0.0000052859% err)
371:63878 / 20333 = 3.141592485 (0.0000053625% err)
369:63523 / 20220 = 3.141592482 (0.0000054399% err)
367:63168 / 20107 = 3.141592480 (0.0000055182% err)
365:62813 / 19994 = 3.141592477 (0.0000055974% err)
363:62458 / 19881 = 3.141592475 (0.0000056774% err)
361:62103 / 19768 = 3.141592472 (0.0000057584% err)
359:61748 / 19655 = 3.141592470 (0.0000058404% err)
357:61393 / 19542 = 3.141592467 (0.0000059232% err)
355:61038 / 19429 = 3.141592464 (0.0000060071% err)
353:60683 / 19316 = 3.141592462 (0.0000060919% err)
351:60328 / 19203 = 3.141592459 (0.0000061777% err)
349:59973 / 19090 = 3.141592456 (0.0000062645% err)
347:59618 / 18977 = 3.141592454 (0.0000063524% err)
345:59263 / 18864 = 3.141592451 (0.0000064413% err)
343:58908 / 18751 = 3.141592448 (0.0000065313% err)
341:58553 / 18638 = 3.141592445 (0.0000066224% err)
339:58198 / 18525 = 3.141592442 (0.0000067146% err)
337:57843 / 18412 = 3.141592439 (0.0000068079% err)
335:57488 / 18299 = 3.141592436 (0.0000069024% err)
333:57133 / 18186 = 3.141592433 (0.0000069980% err)
331:56778 / 18073 = 3.141592430 (0.0000070949% err)
329:56423 / 17960 = 3.141592427 (0.0000071929% err)
327:56068 / 17847 = 3.141592424 (0.0000072922% err)
325:55713 / 17734 = 3.141592421 (0.0000073928% err)
323:55358 / 17621 = 3.141592418 (0.0000074947% err)
321:55003 / 17508 = 3.141592414 (0.0000075979% err)
319:54648 / 17395 = 3.141592411 (0.0000077024% err)
317:54293 / 17282 = 3.141592408 (0.0000078083% err)
315:53938 / 17169 = 3.141592404 (0.0000079155% err)
313:53583 / 17056 = 3.141592401 (0.0000080242% err)
311:53228 / 16943 = 3.141592398 (0.0000081344% err)
309:52873 / 16830 = 3.141592394 (0.0000082460% err)
307:52518 / 16717 = 3.141592390 (0.0000083592% err)
305:52163 / 16604 = 3.141592387 (0.0000084738% err)
911:best approximation of pi they could think of.
907:accurate to say that it treated pi as 256/81.
623:which is more accurate and needs less digits.
933:Ramanujan's approximation in his 1914 paper:
869:= 3.125, about 0.528% below the exact value.
203:This page has archives. Sections older than
8:
285:22 / 7 = 3.142857 (0.04025% err) <<
232:
47:
1998:
1986:
1966:
1954:
1934:
1922:
1907:
1724:
1677:
1671:
1660:
1626:
1605:
1599:
1588:
1568:
1559:
1550:
1529:
1508:
1490:
1484:
1447:
1435:
1429:
1411:
1405:
1371:
1358:
1345:
1333:
1315:
1309:
1276:
1263:
1245:
1239:
1190:
1143:
1137:
1126:
1092:
1068:
1062:
1051:
1031:
1022:
1013:
992:
971:
953:
941:
822:
791:
781:
778:
768:
758:
756:
731:
729:
707:
705:
277:3 / 1 = 3.000 (4.507% err) <<<
213:when more than 10 sections are present.
49:
19:
2058:
655:my calculator shows the result to be
2036:Royal Naval College, Eaton, Chester.
872:At about the same time, the Egyptian
7:
2028:86021 39494 63952 24737 (D. F. F.).
301:333 / 106 = 3.141509 (0.00265% err)
95:This article is within the scope of
299:311 / 99 = 3.141414 (0.00568% err)
297:289 / 92 = 3.141304 (0.00918% err)
295:267 / 85 = 3.141176 (0.01325% err)
293:245 / 78 = 3.141025 (0.01805% err)
291:223 / 71 = 3.140845 (0.02380% err)
289:201 / 64 = 3.140625 (0.03080% err)
287:179 / 57 = 3.140350 (0.03953% err)
38:It is of interest to the following
1672:
1600:
1138:
1063:
884:text) implies an approximation of
14:
2107:Mid-priority mathematics articles
2026:86021 39501 60924 48077 (Shanks).
275:fraction = approximation (error)
207:may be automatically archived by
115:Knowledge:WikiProject Mathematics
1301:is a simple approximation, but
652:trying to find it is always fun.
164:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
1843:from Bryan Hayes' bibliography:
283:19 / 6 = 3.166667 (0.798% err)
135:This article has been rated as
1811:Chronology of computation of π
1731:
1721:
1706:
1700:
1633:
1617:
1526:
1514:
1502:
1496:
1423:
1417:
1327:
1321:
1257:
1251:
1197:
1187:
1172:
1166:
1099:
1080:
989:
977:
965:
959:
1:
2040:2601:644:8501:AAF0:0:0:0:98EB
1828:2601:644:8501:AAF0:0:0:0:2EE5
1542:
1005:
928:
672:02:58, 15 February 2023 (UTC)
109:and see a list of open tasks.
2102:C-Class mathematics articles
1780:Miscellaneous approximations
1775:21:59, 21 October 2023 (UTC)
695:05:24, 14 October 2022 (UTC)
281:16 / 5 = 3.200 (1.859% err)
279:13 / 4 = 3.250 (3.451% err)
263:20:37, 10 January 2024 (UTC)
2088:12:24, 21 August 2024 (UTC)
924:20:59, 22 August 2023 (UTC)
741:{\displaystyle {\sqrt {3}}}
717:{\displaystyle {\sqrt {2}}}
633:16:15, 18 August 2022 (UTC)
247:15:50, 9 January 2015 (UTC)
2123:
1798:22:14, 21 March 2024 (UTC)
878:Second Intermediate Period
874:Rhind Mathematical Papyrus
678:22/7 is definitely ancient
269:Fractional approximations
134:
67:
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2053:Digit extraction methods
2048:18:06, 21 May 2024 (UTC)
1836:10:32, 21 May 2024 (UTC)
853:Babylonian and Egypt Pi?
848:11:12, 30 May 2023 (UTC)
700:approximations based on
141:project's priority scale
929:Borwein's approximation
98:WikiProject Mathematics
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62:Mid‑priority
40:WikiProjects
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224:Should the
112:Mathematics
103:mathematics
59:Mathematics
2096:Categories
2080:GoldenRing
1890:Value of π
1820:Chronology
1380:1446914567
1337:1037785473
1790:Anita5192
1369:311451846
1356:192518946
1343:70101072
235:unsigned
205:365 days
171:Archives
894:octagon
659:(39623)
139:on the
30:C-class
1876:Nature
1841:Added:
1824:Nature
1815:Nature
1767:Nei.jp
1476:where
840:Dhrm77
625:Dhrm77
253:Done.
36:scale.
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2004:1985
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1771:talk
920:talk
888:as ⁄
865:as ⁄
859:Susa
844:talk
806:3921
724:and
691:talk
668:talk
662:qed
629:talk
259:talk
255:Wqwt
243:talk
2070:n→∞
1984:tan
1952:tan
1920:tan
1886:):
1880:doi
1859:doi
1695:exp
1612:exp
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896:."
131:Mid
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180:1
143:.
42::
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