Historians estimate that by 2000 B.C. humans had noticed that the ratio of circumference to diameter was the same for all circles. This discovery hinged on the idea of proportion - in this case humans noticed that if you double the distance "across" a circle, then you double the distance "around" it.

 

In today's algebraic notation this implied the formula :

 

 

where Pi was constant. (It wasn't until 1706 that this notation, using the Greek letter seen in the above equation - often written Pi and pronounced like the English 'pie' - was introduced by William Jones).

The significance of this discovery is clear: Circles are everywhere - in the sun, the moon, the pupils of our eyes, the most basic religious rituals and the earliest man-made structures. Achieving a greater mathematical understanding of Pi would lead to scientific and technological advances that would further the development of civilization, as well as creating some very interesting problems in pure mathematics.

But one problem remained - what is the numerical value of Pi?

 

 

The values of Pi through time

Person/People Year Value
Babylonians 2000 B.C. 3+ 1/8
Egyptians 2000 B.C. (16/9)^2= 3.1605
Chinese 1200 B.C. 3
Old Testament ~550 B.C. 3
Archimedes ~300 B.C. proves 3 10/71<Pi<3 1/7
uses 211875/67441=3.14163
Ptolomy ~200 A.D. 377/120=3.14166...
Chung Huing ~300 A.D. sqrt(10)=3.16...
Wang Fau ~263 A.D. 157/50=3.14
Tsu Chung-Chi ~500 A.D. proves 3.1415926<Pi<3.1415929
Aryabhatta ~500 3.1416
Brahmagupta ~600 sqrt(10)
Fibonacci 1220 3.141818
Ludolph van Ceulen 1596 Calculates Pi to 35 decimal places
Machin 1706 100 decimal places
Lambert 1766 Proves Pi is irrational
Richter 1855 500 decimal places
Lindeman 1882 Proves Pi is transcendental
Ferguson 1947 808 decimal places
_Pegasus Computer 1957 7,840 decimal places
IBM 7090 1961 100,000 decimal places
CDC 6600 1967 500,000 decimal places

 

The Babylonians found the first known value for Pi in around 2000 B.C. They used (25/8). The Egyptians used first 3 for Pi improved this to (22 / 7). They also used (256/81). If you imagine a circle in the Great Pyramid at Giza in Egypt like this:

 

Then the circle's circumference is twice the base length of the pyramid, and the circle's area is equal to the pyramid's vertical sectional area through the peak. The ratio of the perimeter of the base of the Great Pyramid to its height is twice Pi. The same ratio for the Pyramid of the Sun in Mexico is four times Pi. Both are built to an accuracy of a few inches.

 

One of the first mathematicians to make a study of the circle was Archimedes.
He wrote a text "Measurement of the Circle", in which he attempted to compute .
The method that Archimedes used to compute leads us to one of our first calculus concepts, the limit.

So what happened between 1220 and 1596? Well, in the late 15th century, European mathematicians (benefiting from the greater intellectual freedom that came with the end of the Middle Ages) figured out how to express Pi exactly as an infinite product. This facilitated the computation of much better approximations. As time passed, mathematicians made the expressions for Pi as an infinite product or sum more concise, and computational methods improved as well.

For example they:

determined that:

showed that:

While derived his famous formula:

  • Today Pi is known to more than 10 billion decimal places.

     

    The first four thousand decimal places of Pi

    3.1415926535897932384626433832795028841971693993751058209749445923078164062 862089986280348253421170679821480865132823066470938446095505822317253594081284 811174502841027019385211055596446229489549303819644288109756659334461284756482 337867831652712019091456485669234603486104543266482133936072602491412737245870 066063155881748815209209628292540917153643678925903600113305305488204665213841 469519415116094330572703657595919530921861173819326117931051185480744623799627 495673518857527248912279381830119491298336733624406566430860213949463952247371 907021798609437027705392171762931767523846748184676694051320005681271452635608 277857713427577896091736371787214684409012249534301465495853710507922796892589 235420199561121290219608640344181598136297747713099605187072113499999983729780 499510597317328160963185950244594553469083026425223082533446850352619311881710 100031378387528865875332083814206171776691473035982534904287554687311595628638 823537875937519577818577805321712268066130019278766111959092164201989380952572 010654858632788659361533818279682303019520353018529689957736225994138912497217 752834791315155748572424541506959508295331168617278558890750983817546374649393 192550604009277016711390098488240128583616035637076601047101819429555961989467 678374494482553797747268471040475346462080466842590694912933136770289891521047 521620569660240580381501935112533824300355876402474964732639141992726042699227 967823547816360093417216412199245863150302861829745557067498385054945885869269 956909272107975093029553211653449872027559602364806654991198818347977535663698 074265425278625518184175746728909777727938000816470600161452491921732172147723 501414419735685481613611573525521334757418494684385233239073941433345477624168 625189835694855620992192221842725502542568876717904946016534668049886272327917 860857843838279679766814541009538837863609506800642251252051173929848960841284 886269456042419652850222106611863067442786220391949450471237137869609563643719 172874677646575739624138908658326459958133904780275900994657640789512694683983 525957098258226205224894077267194782684826014769909026401363944374553050682034 962524517493996514314298091906592509372216964615157098583874105978859597729754 989301617539284681382686838689427741559918559252459539594310499725246808459872 736446958486538367362226260991246080512438843904512441365497627807977156914359 977001296160894416948685558484063534220722258284886481584560285060168427394522 674676788952521385225499546667278239864565961163548862305774564980355936345681 743241125150760694794510965960940252288797108931456691368672287489405601015033 086179286809208747609178249385890097149096759852613655497818931297848216829989 487226588048575640142704775551323796414515237462343645428584447952658678210511 413547357395231134271661021359695362314429524849371871101457654035902799344037 420073105785390621983874478084784896833214457138687519435064302184531910484810 053706146806749192781911979399520614196634287544406437451237181921799983910159 195618146751426912397489409071864942319615679452080951465502252316038819301420 937621378559566389377870830390697920773467221825625996615014215030680384477345 492026054146659252014974428507325186660021324340881907104863317346496514539057 962685610055081066587969981635747363840525714591028970641401109712062804390397 595156771577004203378699360072305587631763594218731251471205329281918261861258 673215791984148488291644706095752706957220917567116722910981690915280173506712 748583222871835209353965725121083579151369882091444210067510334671103141267111 369908658516398315019701651511685171437657618351556508849099898599823873455283 316355076479185358932261854896321329330898570642046752590709154814165498594616 371802709819943099244889575712828905923233260972997120844335732654893823911932 597463667305836041428138830320382490375898524374417029132765618093773444030707 469211201913020330380197621101100449293215160842444859637669838952286847831235 526582131449576857262433441893039686426243410773226978028073189154411010446823 252716201052652272111660396

     

    Our information came from :

    http://www.users.globalnet.co.uk/~nickjh/Pi.htm
    http://skyline.www.cistron.nl/frame.htm?url=pi/pi.htm
    http://users.hol.gr/~xpolalis/Pilinks.html
    netwerk, Maandblad voor multimedia thuis en elders, juni 1999

     

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