To 15 decimal places of accuracy, the mathematical constant known as "pi", the ratio of a circle's circumference to its diameter, is 3.141592653589793 (the next digit is a "2" so that last "3" is not a rounded-up value).
Thousands of years ago it was noticed that pi was not exactly equal to 3, and that the fraction 22/7 was a reasonable approximation (off by about one part in a thousand).
In the Middle Ages it was discovered that the fraction 355/113 was a much better approximation (off by about three parts in ten million).
I wondered about what other fractions might be decent approximations of pi. Here's a computer program (in QBASIC) I wote to find some:
DIM a AS DOUBLE, b AS DOUBLE, c AS DOUBLE DIM d AS DOUBLE, e AS DOUBLE, f AS LONG CLS a = 3.141592653589793# 'max accuracy QBASIC allows e = 1# 'current level of accuracy-of-computation FOR f = 1 TO 20000000 'twenty million loops b = FIX((f * a) + .01) 'compute numerator if 'f' is denominator c = b / f 'compute the fraction d = c - a 'get difference between this fraction and pi IF (d < 0) THEN 'not trusting the ABS() function here d = -d 'get absolute value of the difference END IF IF (d < e) THEN 'is current fraction a better approximation of pi? e = d 'save that smaller difference for next comparison PRINT b; f, c END IF NEXT f
Here are the results: 3 / 1 -------- 3 22 / 7 -------- 3.142857142857143 179 / 57 ------- 3.140350877192982 201 / 64 ------- 3.140625 223 / 71 ------- 3.140845070422535 245 / 78 ------- 3.141025641025641 267 / 85 ------- 3.141176470588235 289 / 92 ------- 3.141304347826087 311 / 99 ------- 3.141414141414141 333 / 106 ------ 3.141509433962264 355 / 113 ------ 3.141592920353983 52163 / 16604 ---- 3.141592387376536 52518 / 16717 ---- 3.141592390979242 52873 / 16830 ---- 3.141592394533571 53228 / 16943 ---- 3.141592398040489 53583 / 17056 ---- 3.141592401500938 53938 / 17169 ---- 3.141592404915837 54293 / 17282 ---- 3.141592408286078 54648 / 17395 ---- 3.141592411612532 55003 / 17508 ---- 3.141592414896047 55358 / 17621 ---- 3.14159241813745 55713 / 17734 ---- 3.141592421337544 56068 / 17847 ---- 3.141592424497115 56423 / 17960 ---- 3.141592427616926 56778 / 18073 ---- 3.141592430697726 57133 / 18186 ---- 3.14159243374024 57488 / 18299 ---- 3.141592436745178 57843 / 18412 ---- 3.14159243971323 58198 / 18525 ---- 3.141592442645074 58553 / 18638 ---- 3.141592445541367 58908 / 18751 ---- 3.141592448402752 59263 / 18864 ---- 3.141592451229856 59618 / 18977 ---- 3.141592454023292 59973 / 19090 ---- 3.141592456783656 60328 / 19203 ---- 3.141592459511535 60683 / 19316 ---- 3.141592462207496 61038 / 19429 ---- 3.141592464872098 61393 / 19542 ---- 3.141592467505885 61748 / 19655 ---- 3.141592470109387 62103 / 19768 ---- 3.141592472683124 62458 / 19881 ---- 3.141592475227604 62813 / 19994 ---- 3.141592477743323 63168 / 20107 ---- 3.141592480230766 63523 / 20220 ---- 3.141592482690406 63878 / 20333 ---- 3.141592485122707 64233 / 20446 ---- 3.141592487528123 64588 / 20559 ---- 3.141592489907097 64943 / 20672 ---- 3.141592492260062 65298 / 20785 ---- 3.141592494587443 65653 / 20898 ---- 3.141592496889654 66008 / 21011 ---- 3.141592499167103 66363 / 21124 ---- 3.141592501420186 66718 / 21237 ---- 3.141592503649291 67073 / 21350 ---- 3.141592505854801 67428 / 21463 ---- 3.141592508037087 67783 / 21576 ---- 3.141592510196515 68138 / 21689 ---- 3.141592512333441 68493 / 21802 ---- 3.141592514448216 68848 / 21915 ---- 3.141592516541182 69203 / 22028 ---- 3.141592518612675 69558 / 22141 ---- 3.141592520663024 69913 / 22254 ---- 3.14159252269255 70268 / 22367 ---- 3.141592524701569 70623 / 22480 ---- 3.141592526690391 70978 / 22593 ---- 3.141592528659319 71333 / 22706 ---- 3.14159253060865 71688 / 22819 ---- 3.141592532538674 72043 / 22932 ---- 3.141592534449677 72398 / 23045 ---- 3.141592536341939 72753 / 23158 ---- 3.141592538215735 73108 / 23271 ---- 3.141592540071334 73463 / 23384 ---- 3.141592541908997 73818 / 23497 ---- 3.141592543728987 74173 / 23610 ---- 3.141592545531554 74528 / 23723 ---- 3.14159254731695 74883 / 23836 ---- 3.141592549085417 75238 / 23949 ---- 3.141592550837196 75593 / 24062 ---- 3.141592552572521 75948 / 24175 ---- 3.141592554291623 76303 / 24288 ---- 3.14159255599473 76658 / 24401 ---- 3.141592557682062 77013 / 24514 ---- 3.141592559353839 77368 / 24627 ---- 3.141592561010273 77723 / 24740 ---- 3.141592562651577 78078 / 24853 ---- 3.141592564277954 78433 / 24966 ---- 3.14159256588961 78788 / 25079 ---- 3.141592567486742 79143 / 25192 ---- 3.141592569069546 79498 / 25305 ---- 3.141592570638214 79853 / 25418 ---- 3.141592572192934 80208 / 25531 ---- 3.141592573733892 80563 / 25644 ---- 3.14159257526127 80918 / 25757 ---- 3.141592576775246 81273 / 25870 ---- 3.141592578275995 81628 / 25983 ---- 3.141592579763691 81983 / 26096 ---- 3.141592581238504 82338 / 26209 ---- 3.141592582700599 82693 / 26322 ---- 3.14159258415014 83048 / 26435 ---- 3.141592585587289 83403 / 26548 ---- 3.141592587012204 83758 / 26661 ---- 3.14159258842504 84113 / 26774 ---- 3.14159258982595 84468 / 26887 ---- 3.141592591215085 84823 / 27000 ---- 3.141592592592592 85178 / 27113 ---- 3.141592593958618 85533 / 27226 ---- 3.141592595313303 85888 / 27339 ---- 3.141592596656791 86243 / 27452 ---- 3.141592597989217 86598 / 27565 ---- 3.14159259931072 86953 / 27678 ---- 3.141592600621432 87308 / 27791 ---- 3.141592601921485 87663 / 27904 ---- 3.141592603211009 88018 / 28017 ---- 3.141592604490131 88373 / 28130 ---- 3.141592605758976 88728 / 28243 ---- 3.141592607017668 89083 / 28356 ---- 3.141592608266328 89438 / 28469 ---- 3.141592609505076 89793 / 28582 ---- 3.141592610734028 90148 / 28695 ---- 3.141592611953302 90503 / 28808 ---- 3.14159261316301 90858 / 28921 ---- 3.141592614363265 91213 / 29034 ---- 3.141592615554178 91568 / 29147 ---- 3.141592616735856 91923 / 29260 ---- 3.141592617908407 92278 / 29373 ---- 3.141592619071937 92633 / 29486 ---- 3.141592620226548 92988 / 29599 ---- 3.141592621372344 93343 / 29712 ---- 3.141592622509424 93698 / 29825 ---- 3.141592623637888 94053 / 29938 ---- 3.141592624757833 94408 / 30051 ---- 3.141592625869356 94763 / 30164 ---- 3.14159262697255 95118 / 30277 ---- 3.14159262806751 95473 / 30390 ---- 3.141592629154327 95828 / 30503 ---- 3.141592630233092 96183 / 30616 ---- 3.141592631303893 96538 / 30729 ---- 3.14159263236682 96893 / 30842 ---- 3.141592633421957 97248 / 30955 ---- 3.141592634469391 97603 / 31068 ---- 3.141592635509205 97958 / 31181 ---- 3.141592636541484 98313 / 31294 ---- 3.141592637566307 98668 / 31407 ---- 3.141592638583755 99023 / 31520 ---- 3.141592639593909 99378 / 31633 ---- 3.141592640596845 99733 / 31746 ---- 3.141592641592641 100088 / 31859 ---- 3.141592642581374 100443 / 31972 ---- 3.141592643563118 100798 / 32085 ---- 3.141592644537946 101153 / 32198 ---- 3.141592645505932 101508 / 32311 ---- 3.141592646467148 101863 / 32424 ---- 3.141592647421663 102218 / 32537 ---- 3.141592648369548 102573 / 32650 ---- 3.141592649310873 102928 / 32763 ---- 3.141592650245704 103283 / 32876 ---- 3.141592651174109 103638 / 32989 ---- 3.141592652096153 103993 / 33102 ---- 3.141592653011902 104348 / 33215 ---- 3.141592653921421 208341 / 66317 ---- 3.141592653467437 312689 / 99532 ---- 3.141592653618936 833719 / 265381 --- 3.141592653581078 1146408 / 364913 --- 3.141592653591404 3126535 / 995207 --- 3.14159265358865 4272943 / 1360120 -- 3.141592653589389 5419351 / 1725033 -- 3.141592653589815 42208400 / 13435351 - 3.141592653589772 47627751 / 15160384 - 3.141592653589777 53047102 / 16885417 - 3.141592653589781 58466453 / 18610450 - 3.141592653589784
It appears that 355/113 is the best simple fractional approximation of pi to remember. It's also easy to remember, if you think of the sequence 113355, then split the sequence in half and divide the first half into the second half.
For any of those "better" fractions, you would have to memorize almost as many total-number-of digits as if you decided to memorize the actual decimal version itself: 3.14159265358979323846...-- Vernon, Jan 15 2011 Continued fractions http://www.petrospe...kommer/contfrac.htmAnother algorithm for generating rational approximations [Wrongfellow, Jan 15 2011] That's a very brute-force approach. Continued fractions will get you the really good approximations more quickly than that - see the link, which discusses pi towards the end of the page.-- Wrongfellow, Jan 15 2011 //I wondered about what other fractions might be decent approximations of pi//
I'm reasonably sure that others have wondered this too...-- Jinbish, Jan 15 2011 random, halfbakery