Mathematicians are still searching for possible patterns within pi - if its digits should ever begin to repeat. The IBM 7090 broke the 100,000 digit mark in 1961, and the CDC 7600 was able to reach pi’s millionth decimal place in 1973.Įven now, the value of pi continues to be further unravelled up to and beyond its 50 trillionth digit. By 1949, the ENIAC computer could calculate 2,037 digits of pi, and the IBM 704 was able to calculate 16,167 digits a decade later. The subsequent development of more advanced computers meant that pi could be calculated to higher and higher decimal places. Eventually, though, in the early twentieth century, Indian mathematician Srinivasa Ramanujan developed an incredibly efficient formula for calculating pi based on its reciprocal fraction, which was later incorporated into computer algorithms. Progress slowed once again for the next few hundred years, as more calculations were made but with little conceptual development. It was also during this period that the use of the Greek letter pi was introduced by William Jones, in 1706, and that it was later popularized by Leonhard Euler, beginning in 1737. In this same century, two major developments also occurred concerning the nature of pi: in 1761, Johann Lambert proved that pi was irrational - that is, that it cannot be expressed as the ratio of two numbers - and Ferdinand von Lindemann concluded that it was transcendental soon after. By the end of the eighteenth century, over one hundred digits of pi had been calculated using this method and others derived from it. Next came the Gregory-Leibniz series, which made use of both infinite series and trigonometric functions to develop formulae for values of pi divided by four and six. Later that century, Sir Isaac Newton - in one of his many achievements - used his binomial theorem to quickly calculate the value of pi up to 16 decimal places. While calculating an integral in an attempt to find the area of a circle with a radius of one, he established a formula involving the multiplication of an infinite series of fractions that was based on the value of one-half of pi. One of the earliest formulae for calculating pi was proposed by English mathematician John Wallis in 1656. In the seventeenth century, however, new analytical techniques were developed in the field of mathematics which allowed for improved calculations of pi using infinite series. However, based on his results, it is hypothesized that he would have used a method similar to that of Archimedes, starting with a regular polygon with 24,576 sides inscribed inside a circle and performing lengthy calculations involving hundreds of square roots carried out to nine decimal places in order to to arrive at his answer.īy 1600, pi had been calculated up to 35 digits using the method of inscribed polygons, although with little theoretical progress made beyond Archimedes’ work. Unfortunately, very few specifics are known of his work, as his book has been lost to history. One notable contributor to these calculations was Chinese mathematician and astronomer Zu Chongzhi, who approximated pi to be around 355/113. In the centuries that followed, mathematicians around the world were able to extend the number of known decimal places of pi by performing extensive calculations. By doing so, he proved that there was a constant ratio between the area of a circle and the square of its radius, and he was able to obtain an upper bound of 22/7 and a lower bound of 223/71 for the value of pi. He approximated the area of a circle by comparing the areas of a polygon inscribed inside the circle and one that surrounded it. A little further south, in ancient Egypt, the Rhind papyrus, dated circa 1650 BCE, reveals that a value of 256/81 - around 3.16045 - was used to approximate pi.Īround 250 BCE, Archimedes made a major breakthrough in calculating a more accurate value of pi by using the Pythagorean theorem. In 2000 BCE, Babylonian mathematicians performed the first known calculation of the area of a circle using the circumference of an inscribed hexagon, and derived an approximate value for pi of 3.125. The concept of pi is ancient - it’s been known to humanity for at least four thousand years. Irrational, transcendental, and never-ending, it has frustrated and fascinated mathematicians across cultures for centuries, and it has become a universally recognized symbol for the field in popular culture. Pi - the ratio between the circumference and diameter of a circle - is perhaps one of the most famous numbers in all of mathematics.
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