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Archimedes writes:
"It has come to my attention that various and sundry folk have done far better approximations of the ratio of a circle's circumference to its diameter than had been possible with my excellent 96-sided regular polygon method.
"μπÏάβο," I say (or would if this calculating engine of yours could properly handle "Unicode"). It pleases me to no end to know that the great work has continued, lo these many years since that obstreperous Italic fellow with the red cloak so rudely interrupted my research.
However, with an eternity in Hades on my hands, I have endeavored to stay busy by continuing to produce more accurate approximations. (That rake Sisyphus tells me it's a waste of time, but he is not one to talk in that regard.)
What follows is the closest approximation I have made in my posthumous calcutatory diversions. "Pi," to use your modern shorthand, is about...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868741326047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give or take."
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(Score: 4, Interesting) by martyb on Friday March 14 2014, @12:20PM
tl;dr: theoretically, in a physical application: approximately 61 digits.
Thought experiment: how many digits of Pi can we use? In this discussion, I explicitly ignore any number-theoretical application of pi being a transcendental number [wikipedia.org].
Let's imagine calculating the circumference of something really big with as precise a unit of length as we can.
What is the biggest "thing" that has a circumference that we can measure? Let's use the diameter of the observable Universe [wikipedia.org]. This is approximately: 8.8 x 10^26 m.
What is the smallest unit of linear measure? Let's use the Planck Length [wikipedia.org] or approximately: 1.6 x 10^-35 m.
According to Wolfram Alpha [wolframalpha.com] this is 5.4 x 10^61 Planck lengths.
Granted, there is no current means by which we could actually measure something with anywhere near the precision of a Planck length, never mind on the scale of the observable universe. But, assuming we could, approximately 61 significant digits of Pi would suffice to compute its circumference.
Footnote: As a young teen, a challenge by my geometry teacher led to my memorizing Pi to 10 decimal places. A year later, I found a book enumerating Pi to 40 places, which I also took to memorizing. When I got to college, I discovered a guy across the hall from me in my dorm (Hi Rob!) had memorized it to 50 places. The battle was on. We'd one-up the other 5-10 digits at a time. After about 4 months, we called a truce at 200 decimal places. That was quite a while ago, yet I can still recall it to 120 places.
(Score: 3, Insightful) by mrcoolbp on Friday March 14 2014, @12:59PM
Oh 61 eh? I thought it was 42...
Joking aside, it's an elegant thought. Thank you for this! This is why I come here.
(Score:1^½, Radical)
(Score: 2) by LaminatorX on Friday March 14 2014, @01:51PM
Universal expansion has outpaced the change in the dynamics underlying Planck Length. At some point in the past, 42 digits would have been the required precision. ;)
This is our news site. There are others like it, but this one is ours.
(Score: 1) by egcagrac0 on Friday March 14 2014, @02:42PM
... which sadly means that sometime in the future - probably on or about next Tuesday - we will need to go to 62 digits for "good enough" calculation.
(Score: 0) by Anonymous Coward on Friday March 14 2014, @05:20PM
I guess until then we can, for objects a tad smaller than the entire universe, go with just 6 digits of precision. If you ever get stuck having to use a vanilla calculator that does not have Pi built-in, just remember "113355". Split that in the middle into 113 and 355. 355 / 113 = 3.1415292 .. close enough for many applications.
Practically though, if you memorize or use Pi to 31 places where the first zero appears - past that it really cannot mater all that much. We aren't all Daniel Tammet's who can recite Pi to 20,000 places.
(Score: 2) by Foobar Bazbot on Saturday March 15 2014, @01:22AM
Could you explain why it makes sense to remember six digits in order to get an approximation of pi that's only valid to 4 digits?
The 5-digit approximation 3.1416 is more accurate and easier to remember; or you can memorize it to six digits as 3.14159. (The latter is both a correct rounding and a correct truncation, which is nice so if you ever want to learn more digits, you don't have to unlearn the final one.)
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(Score: 2) by maxwell demon on Saturday March 15 2014, @09:40AM
You really don't see the structure in "113355"? If you have to remember that as an arbitrary sequence of six digits, I honestly feel sorry for you.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by Foobar Bazbot on Saturday March 15 2014, @01:20PM
*facepalm*
Don't know how, but I totally missed that. Thanks for pointing it out.
The incredible bit was that I realized it was a non-strictly monotonous sequence, but didn't think the implication of that (w/r/t recovery from transposing adjacent digits) was worth mentioning, because that doesn't make it much easier to remember. (I assumed this property was why the AC concatenated it in denominator:numerator order instead of the usual.) And yet I missed the two elements (odd numbers in sequence, and each number repeated) that were obviously the real point.
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(Score: 1) by tombstone on Saturday March 15 2014, @10:12AM
I agree with you that it seems easier to just remember the digits.
However, I believe the AC made a mistake in the arithmetic. The given fraction agrees with pi to more than 4 decimal places. I suggest you check it yourself, but I believe the first few digits of 355/113 are 3.1415929. This is a better approximation than the AC indicated.
(Score: 2) by egcagrac0 on Wednesday March 19 2014, @02:13PM
355/113 is 7 digits of accuracy in only 6 digits. It's also more convenient if your calculator doesn't do floating point.
It also seems to get you accuracy within 1/1000th of a millimeter per meter - which scales to about 1 millimeter of inaccuracy per kilometer.