(9) Challenge to Pi (Modern period)Dec 10, 2012
In the 17th century, calculus was applied to the calculation of Pi, and the formula for arctan(x), the Madhava formula, was rediscovered by Gregory and Leibniz respectively. This formula is now referred to as the Madhava series or Gregory-Leibniz series. The Gregory-Leibniz series equals Pi/4 when evaluated with x=1, but it is not practical because of slow convergence. After that, Machin reached 100 digits of Pi with the new arctan formula. The first 50 digits of Pi could be calculated using 35 terms of infinite series, which brought about a change in the calculation algorithm of Pi. From the mid-20th century onwards, all calculations of Pi have been done with the help of calculators or computers. In 1949, the first computer ENIAC reached 2037 digits of Pi in 70 hours using the Machin formula. In 2002, a Japanese team reached 1,241 billion digits of Pi by the similar Takano formula using the supercomputer. As the proof of computer performance, every country's team has been competing to become the world's number one in the computation of Pi.
| Pi (ATAN two terms)|
 Pi (ATAN three terms)