of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, all of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of until the advent of the electronic digital computer three-quarters of a century later. 20th and 21st centuries In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of , including which computes a further eight decimal places of with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . Even using just the first term gives See Ramanujan–Sato series. From the mid-20th century onwards, all calculations of have been done with the help of calculators or computers. In 1944, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. In the early years of the computer, an expansion of to decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of . For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years. In 1989, the Chudnovsky brothers computed to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of : Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at