How Can Someone Find π?
Throughout the centuries that men have tried to discover the extent of π, they have come up with a cornucopia of ways to figure out as many digits as possible.
The earliest attempts at unraveling the mysteries of π were really just "guess and check" figures. They included everything from 22/7 to 211875/67441. This was sufficient to satisfy the needs of the time; however, mathematicians continued to endeavor to find more and more about the ratio.
The next step was a leap to what would be the next, and latest, phase in the calculation of pi: infinite products and sums. This trend began with Francois Viete and his formula:
This form of equation allows one to compute one term at a time, thus allowing one mathematician to work on one term and another to pick up where the other one left off. Although Viete's method was extremely slow and clumsy, it created a base for almost all advancements in π that followed.
The next great advancement was that of James Gregory's arctangent formula. The arctangent, or inverse tangent, is the angle that has a tangent equal to a certain number. His equation stated that π/4=1-(1/3)+(1/5)-(1/7)+... This was especially useful in the serch for π because of the fact that tan(pi) = 1, and therefore arctan(1) = π. Gregory plugged 1 into his equation and had a form that would become the base of many formulas to follow.
Gregory's early methods proved to be very slow, however. In fact, in order to calculate the first one hundred digits of π using this method, one would have to calculate more terms than there are particles in the universe! In order to fix this huge drawback, many mathematicians were able to find π by using combination of arctangents. Some examples include pi/4 = 4 arctan(1/5) - arctan(1/239) and pi/4 = arctan(1/2) + arctan(1/3). These proved to be much faster than Gregory's original formula.
Then came the computer age, and formulas for finding and verifying π flooded the academic world. These provided easy platforms for computers to calculate millions and billions of digits in just days. Finally, one more wave of calculation slammed into the mathematic world. David Bailey, Peter Borwein, and Simon Plouffe jointly discovered an algorithm to find an individual digit of π without knowing the previous digits. This allowed individual digits to be calculated. (This is the algorithm upon which Pi Client is based).
We have seen the evolution of different algorithms through time. Now see how the speed of each algorithm changes over time. Visit the Finding Pi Applet.