A brief history of Pi

What is circumference over diameter? Today we refer to it simply as pi. This somewhat mysterious ratio interested people of all ages for centuries. This ratio is ancient and goes back to the beginning of civilization.

Although some historians believe that pi was discovered earlier, the first real traces of pi go as far back as the Bible itself. Of course, the value of pi back then was calculated by hand, using crude tools. As a result, pi was thought to equal 3. This might seem very rough to a modern person. In fact, the great temple of Solomon, around 950BC, used 3 as the value for pi. Surprisingly, the Egyptian Rhind Papyrus, around 1650 BC, shows a good evidence taht the value of pi was thought to be 4(8/9)², which is 3.16.

At the time around 200 BC, a great mathematician named Archimedes, of Syracuse, came along. He was also mystified by pi. However, unlike all the rest, Archimedes did not claim to have found the exact value of pi. He, on the other hand, obtained an approximate value of 223/71 < pi < 22/7. Archimedes derived that astonishing value without the advantages of algebraic or trigonometric notations, and he did not know of decimals. To calculate this value, he resorted to polygons, and went all the way up to 96-sided figures!

There were many more people involved in the process of determining pi. Most of them did not make any new theories, but merely used Archimedes' method and went to higher polygons. Below is a chart of most of these people:

Rhind Papyrus - 2000 BC - 1 decimal place correct
Archimedes - 250 BC - 3 decimal places correct
Vitruvius - 20 BC - 1 decimal place correct
Chang Hong - 130 - 1 decimal place correct
Ptolemy - 150 - 3 decimal places correct
Wang Fan - 250 - 1 decimal place correct
Liu Hui - 263 - 5 decimal places correct
Tsu Ch'ung Chi - 480 - 7 decimal places correct
Aryabhata - 499 - 4 decimal places correct
Brahmagupta - 640 - 1 decimal place correct
Al-Khwarizmi - 600 - 4 decimal places correct
Fibonacci - 1220 - 3 decimal places correct
Madhava - 1400 - 11 decimal places correct
Al-Kashi - 1430 - 14 decimal places correct
Otho - 1573 - 6 decimal places correct
Viète - 1593 - 9 decimal places correct
Romanus - 1593 - 15 decimal places correct
Van Ceulen - 1596 - 35 decimal places correct
Newton - 1665 - 16 decimal places correct
Sharp - 1699 - 71 decimal places correct
Seki Kowa - 1700 - 10 decimal places correct
Kamata - 1730 - 25 decimal places correct
Machin - 1706 - 100 decimal places correct
De Lagny - 1719 - 112 decimal places correct
Takebe - 1723 - 41 decimal places correct
Matsunaga - 1739 - 50 decimal places correct
von Vega - 1794 - 136 decimal places correct
Rutherford - 1824 - 152 decimal places correct
Dase Strassnitzky - 1844 - 200 decimal places correct
Clausen - 1847 - 248 decimal places correct
Lehmann - 1853 - 261 decimal places correct
Rutherford - 1853 - 440 decimal places correct
Shanks - 1874 - 527 decimal places correct
Ferguson - 1946 - 620 decimal places correct

In the year 1761, Lambert proved that pi was irrational. In other words, it could not be written as a ratio of integer values. In about a hundred years, in the year 1882, Lindeman proved that pi was also transcendental. That is, pi is not a root of any algebraic equation with rational coefficients. This proved that a circle does not have a square root, a question that has been troubling mathematicians for generations.

Listed below are the first 1001(including the 3) digits of Pi:

3.1415926535897932384626433832795028841971693993751
05820974944592307816406286208998628034825342117067982148
08651328230664709384460955058223172535940812848111745028
41027019385211055596446229489549303819644288109756659334
46128475648233786783165271201909145648566923460348610454
32664821339360726024914127372458700660631558817488152092
09628292540917153643678925903600113305305488204665213841
46951941511609433057270365759591953092186117381932611793
10511854807446237996274956735188575272489122793818301194
91298336733624406566430860213949463952247371907021798609
43702770539217176293176752384674818467669405132000568127
14526356082778577134275778960917363717872146844090122495
34301465495853710507922796892589235420199561121290219608
64034418159813629774771309960518707211349999998372978049
95105973173281609631859502445945534690830264252230825334
46850352619311881710100031378387528865875332083814206171
77669147303598253490428755468731159562863882353787593751
9577818577805321712268066130019278766111959092164201989

As you can see, the legacy of pi was a very long one. It took centuries to get the pi to what it is now. Luckily, with the invention of a computer, this task has been simplified down to a simple algorithm. Now, practically every calculator and every single computer can calculate pi. Below is a link to a website that allows you to see the 200 millionth decimal place of pi. Warning: this crashed my computer!

http://www.math.com/
http://www.hepl.phys.nagoya-u.ac.jp/~mitsuru/pi-e.html
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html

The modern history of Pi is no less exciting than the ancient history.

In 1999 Yasumasa Kanada calculated 206,158,430,000 decimal digits of Pi, setting a world record, only to top it himself three years later.

In 2002 his team at the University of Tokyo calculated 1,241,100,000,000 digits. This calculation took more than 602 hours on a Hitachi SR8000 computer and required more than 1 terabyte of memory (duh!). Those of you who do not have access to SR8000 may like to know that the 1,241,100,000,000th digit after the decimal point is 5. The results were used to calculate the frequency of all ten digits in Pi and they all are quite close to 10%. Of course, this is not enough to prove that the Pi is normal (that is that every sequence of numbers of certain length has the same probability of appearing in Pi).

But another approach is likely to prove more fruitful.

In 1995 David Bailey, Peter Borwein, and Simon Plouffe made an exciting discovery. They found a simple formula for calculation of Pi that incidentally allowed independent calculation of any single hexadecimal digit of Pi. This formula (named BBP formula after the scientists who discovered it) is provided below:

Pi = SUM_{k=0 to infinity} 16^-k [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ]

This formula still doesn't prove that Pi is normal, but it now links the distribution of digits in Pi to the field of chaotic dynamics. There is an unproved but plausible conjecture that certain sequences (just like the one described by the above formula) "uniformly dance in the limit between 0 and 1". This explains why the digits in Pi (and other constants, like log(2) or square root of 2) appear to be random, but doesn't prove that they actually are. However, if it is proved, the normality (in base 2) of Pi and many other mathematical constants will follow.