Birth, growth and computation of pi to ten trillion digits

Advances in Difference Equations, Sep 2018

The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed.

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Birth, growth and computation of pi to ten trillion digits

Ravi P Agarwal 0 Hans Agarwal Syamal K Sen 0 Department of Mathematics, Texas A&M University-Kingsville , Kingsville, TX , 78363, USA The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed. - pi = distance around a circle distance across and through the center of the circle Since the exact date of birth of is unknown, one could imagine that existed before the universe came into being and will exist after the universe is gone. Its appearance in the disks of the Moon and the Sun, makes it as one of the most ancient numbers known to humanity. It keeps on popping up inside as well as outside the scientific community, for example, in many formulas in geometry and trigonometry, physics, complex analysis, cosmology, number theory, general relativity, navigation, genetic engineering, statistics, fractals, thermodynamics, mechanics, and electromagnetism. Pi hides in the rainbow, and sits in the pupil of the eye, and when a raindrop falls into water emerges in the spreading rings. Pi can be found in waves and ripples and spectra of all kinds and, therefore, occurs in colors and music. The double helix of DNA revolves around . Pi has lately turned up in super-strings, the hypothetical loops of energy vibrating inside subatomic particles. Pi has been used as a symbol for mathematical societies and mathematics in general, and built into calculators and programming languages. Pi is represented in the mosaic outside the mathematics building at the Technische Universitt Berlin. Pi is also engraved on a mosaic at Delft University. Even a movie has been named after it. Pi is the secret code in Alfred Hitchcocks Torn Curtain and in The Net starring Sandra Bullock. Pi day is celebrated on March (which was chosen because it resembles .). The official celebration begins at : p.m., to make an appropriate . when combined with the date. In , the United States House of Representatives supported the designation of Pi Day. Albert Einstein was born on Pi Day ( March ). Throughout the history of , which according to Beckmann () is a quaint little mirror of the history of man, and James Glaisher (-) has engaged the attention of many mathematicians and calculators from the time of Archimedes to the present day, and has been computed from so many different formula, that a complete account of its calculation would almost amount to a history of mathematics, one of the enduring challenges for mathematicians has been to understand the nature of the number (rational/irrational/transcendental), and to find its exact/approximate value. The quest, in fact, started during the pre-historic era and continues to the present day of supercomputers. The constant search by many including the greatest mathematical thinkers that the world produced, continues for new formulas/bounds based on geometry/algebra/analysis, relationship among them, relationship with other numbers such as = cos(/), /, where is the Golden section (ratio), and ei + = , which is due to Euler and contains of the most important mathematical constants, and their merit in terms of computation of digits of . Right from the beginning until modern times, attempts were made to exactly fix the value of , but always failed, although hundreds constructed circle squares and claimed the success. These amateur mathematicians have been called the sufferers of morbus cyclometricus, the circle-squaring disease. Stories of these contributors are amusing and at times almost unbelievable. Many came close, some went to tens, hundreds, thousands, millions, billions, and now up to ten trillion () decimal places, but there is no exact solution. The American philosopher and psychologist William James (-) wrote in the thousandth decimal of Pi sleeps there though no one may ever try to compute it. Thanks to the twentieth and twenty-first century, mathematicians and computer scientists, it sleeps no more. In , Hermann Schubert (-), a Hamburg mathematics professor, said there is no practical or scientific value in knowing more than the decimal places used in the foregoing, already somewhat artificial, application, and according to Arndt and Haenel (), just decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the radius of a hydrogen atom. Further, an expansion of to only decimal places would be sufficiently precise to inscribe a circle around the visible universe that does not deviate from perfect circularity by more than the distance across a single proton. The question has been repeatedly asked why so many digits? Perhaps the primary motivation for these computations is the human desire to break records; the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms (a stress test for a computer, a kind of digital cardiogram), the statistical distribution of the digits, which is expected to be uniform, that is, the frequency with which the digits ( to ) appear in the result will tend to the same limit (/) as the number of decimal places increases beyond all bounds, and in recent years these digits are being used in applied problems as a random sequence. It appears experts in the field of are looking for surprises in the digits of . In fact, the Chudnovsky brothers once said: We are looking for the appearance of some rules that will distinguish the digits of from other numbers. If you see a Russian sentence that extends for a whole page, with hardly a comma, it is definitely Tolstoy. If someone gave you a million digits from somewhere in , could you tell it was from ? Some interesting observations are: The first digits of add up to (which many scholars say is the mark of the Beast); Since there are degrees in a circle, some mathematicians were delighted to discover that the number is at the th digit position of . A mysterious crop circle in Britain shows a coded image representing the first digits of . The Website The Pi-Search Page finds a persons birthday and other well-known numbers in the digits of . Several people have endeavored to memorize the value of with increasing precision, leading to records of over , digits. We believe that the study and discoveries of will never end; there will be books, research articles, new record-setting calculations of the digits, clubs and computer programs dedicated to . In what follows, we shall discuss the growth and the computation of chronologically. For our ready reference, we also give some digits of , . About BC. The meaning of the word sulv is to measure, and geometry in ancient India came to be known by the name sulba or sulva. Sulbasutras means rule of chords, which is another name for geometry. The Sulbasutras are part of the larger corpus of texts called the Shrautasutras, considered to be appendices to the Vedas, which give rules for constructing altars. If the ritual sacrifice was to be successful, then the altar had to conform to very precise measurements, so mathematical accuracy was seen to be of the utmost importance. The sulbas contain a large number of geometric constructions for squares, rectangles, parallelograms and trapezia. Sulbas also contain remarkable approximations which gives = . . . . , and ( ) = , which gives = . . . . . About BC. Aryabhatta was born in BC in Patliputra in Magadha, modern Patna in Bihar (India). He was teaching astronomy and mathematics when he was years of age in BC. His astronomical knowledge was so advanced that he could claim that the Earth rotated on its own axis, the Earth moves round the Sun and the Moon rotates round the Earth; incredibly he believed that the orbits of the planets are ellipses. He talks about the position of the planets in relation to its movement around the Sun. He refers to the light of the planets and the Moon as reflection from the Sun. He explains the eclipse of the Moon and the Sun, day and night, the length of the year exactly as days. He calculated the circumference of the Earth as , miles, which is close to modern day calculation of , miles. In his Aryabhattiyam, which consists of the verses and introductory verses, and is divided into four padas or chapters (written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system), Aryabhatta included verses giving mathematical rules ganita on pure mathematics. He described various original ways to perform different mathematical operations, including square and cube roots and solving quadratic equations. He provided elegant results for the summation of series of squares and cubes. He made use of decimals, the zero (sunya) and the place value system. To find an approximate value of , Aryabhatta gives the following prescription: Add to , multiply by and add to ,. This is approximately the circumference of a circle whose diameter is ,. This means = ,/, = .. It is important to note that Aryabhatta used the word asanna (approaching), to mean that not only is this an approximation of , but that the value is incommensurable or irrational, i.e., it cannot be expressed as a ratio of two integers. About BC. Great pyramid at Gizeh was built around BC in Egypt. It is one of the most massive buildings ever erected. It has at least twice the volume and thirty times the mass (the resistance an object offers to a change in its speed or direction of motion) of the Empire Sate Building in New York, and built from individual stones weighing up to tons each. From the dimensions of the Great Pyramid, it is possible to derive the value of , namely, = half the perimeter of the base of the pyramid, divided by its height = + / . . . . . About BC. In a tablet found in in Susa (Iraq), Babylonians used the value which yields = / = .. They were also satisfied with = . About BC. Ahmes (around - BC) (more accurately Ahmose) was an Egyptian scribe. A surviving work of Ahmes is part of the Rhind Mathematical Papyrus, BC (named after the Scottish Egyptologist Alexander Henry Rhind who went to Thebes for health reasons, became interested in excavating and purchased the papyrus in Egypt in ) located in the British Museum since . When new, this papyrus was about feet long and inches high. Ahmes states that he copied the papyrus from a nowlost Middle Kingdom original, dating around BC. This curious document entitled directions for knowing all dark things, deciphered by Eisenlohr in , is a collection of problems in geometry and arithmetic, algebra, weights and measures, business and recreational diversions. The problems are presented with solutions, but often with no hint as to how the solution was obtained. In problem no. , Ahmes states that a circular field with a diameter of units in area is the same as a square with sides of units, i.e., (/) = , and hence the Egyptian value of is = . . . . , which is only very slightly worse than the Babylonians value, and in contrast to the latter, an overestimation. We have no idea how this very satisfactory result was obtained (probably empirically), although various justifications are available. Maya value of was as good as that of the Egyptians. About BC. The earliest Chinese mathematicians, from the time of Chou-Kong used the approximation = . Some of those who used this approximation were mathematicians of considerable attainments in other respects. According to the Chinese mythology, is used because it is the number of the Heavens and the circle. About BC. In the Old Testament (I Kings vii., and Chronicles iv.), we find the following verse: Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about. Hence the biblical value of is / = . The Jewish Talmud, which is essentially a commentary on the Old Testament, was published about AD. This shows that the Jews did not pay much attention to geometry. However, debates have raged on for centuries about this verse. According to some, it was just a simple approximation, while others say that . . . the diameter perhaps was measured from outside, while the circumference was measured from inside. About BC. Shatapatha Brahmana (Priest manual of paths) is one of the prose texts describing the Vedic ritual. It survives in two recensions, Madhyandina and Kanva, with the former having the eponymous brahmanas in books, and the latter brahmanas in books. In these books, is approximated by / = . . . . . About BC. Anaxagoras of Clazomanae (- BC) came to Athens from near Smyrna, where he taught the results of the Ionian philosophy. He neglected his possessions in order to devote himself to science, and in reply to the question, what was the object of being born, he remarked: The investigation of the Sun, Moon and heaven. He was the first to explain that the Moon shines due to reflected light from the Sun, which explains the Moons phases. He also said that the Moon had mountains and he believed that it was inhabited. Anaxagoras gave some scientific accounts of eclipses, meteors, rainbows, and the Sun, which he asserted was larger than the Peloponnesus: this opinion, and various other physical phenomena, which he tried to explain which were supposed to have been direct action of the Gods, led him to a prosecution for impiety. While in prison he wrote a treatise on the quadrature of the circle. (The general problem of squaring a figure came to be known as the quadrature problem.) Since that time, hundreds of mathematicians tried to find a way to draw a square with equal area to a given circle; some maintained that they have found methods to solve the problem, while others argued that it is impossible. We will see that the problem was finally laid to rest in the nineteenth century. About BC. Hippocrates of Chios was born about BC, and began life as a merchant. About BC he came to Athens from Chios and opened a school of geometry, and began teaching, thus became one of the few individuals ever to enter the teaching profession for its financial rewards. He established the formula r for the area of a circle in terms of its radius. It means that a certain number exists, and is the same for all circles, although his method does not give the actual numerical value of . In trying to square the circle (unsuccessfully), Hippocrates discovered that two moon-shaped figures (lunes, bounded by pair of circular arcs) could be drawn whose areas were together equal to that of a right-angled triangle. Hippocrates gave the first example of constructing a rectilinear area equal to an area bounded by one or more curves. About BC. Antiphon of Rhamnos (around - BC) was a sophist who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. Thus, he provided preliminary concept of infinitesimal calculus. About BC. Bryson of Heraclea was born around BC. He was a student of Socrates. Bryson considered the circle squaring problem by comparing the circle to polygons inscribed within it. He wrongly assumed that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons. About BC. Hippias of Elis was born about BC. He was a Greek Sophist, a younger contemporary of Socrates. He is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix (x = y cot( y/)), by means of which an angle can be trisected, or indeed divided in any given ratio. It is not known whether Hippias realized that by means of his curve the circle could be squared; perhaps he realized but could not prove it. He lectured widely on mathematics and as well on poetry, grammar, history, politics, archeology and astronomy. Hippias was also a prolific writer, producing elegies, tragedies and technical treatises in prose. His work on Homer was considered excellent. BC. Aristophanes (- BC) in his play The Birds makes fun of circle squarers. Around BC. Plato of Athens (around - BC) was one of the greatest Greek philosophers, mathematicians, mechanician, a pupil of Socrates for eight years, and teacher of Aristotle. He is famous for Platos Academy. Let no man ignorant of mathematics enter here is supposed to have been inscribed over the doors of the Academy. He is supposedly obtained for his day a fairly accurate value for = + = . . . . . About BC. Eudoxus of Cnidus (around - BC) was the most celebrated mathematician. He developed the theory of proportion, partly to place the doctrine of incommensurables (irrationals) upon a thoroughly sound basis. Specially, he showed that the area of a circle is proportional to its diameter squared. Eudoxus established fully the method of exhaustions of Antiphon by considering both the inscribed and circumscribed polygons. He also considered certain curves other than the circle. He explained the apparent motions of the planets as seen from the earth. Eudoxus also wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the Sun, Moon and stars were attached, and which by their rotation produced the effects observed. In all, he required spheres. About BC. Dinostratus (around - BC) was a Greek mathematician. He used Hippias quadratrix to square the circle. For this, he proved Dinostratus theorem. Hippias quadratrix later became known as the Dinostratus quadratrix also. However, his demonstration was not accepted by the Greeks as it violated the foundational principles of their mathematics, namely, using only ruler and compass. About BC. Archimedes of Syracuse (- BC) ranks with Newton and Gauss as one of the three greatest mathematicians who ever lived, and he is certainly the greatest mathematician of antiquity. Galileo called him divine Archimedes, superhuman Archimedes; Sir William Rowan Hamilton (-) remarked who would not rather have the fame of Archimedes than that of his conqueror Marcellus?; Alfred North Whitehead (-) commented no Roman ever died in contemplation over a geometrical diagram; Godfrey Harold Hardy (-) said Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not; and Figure 1 Circle with diameter 1. Voltaire remarked there was more imagination in the head of Archimedes than in that of Homer. His mathematical work is so modern in spirit and technique that it is barely distinguishable from that of a seventeenth-century mathematician. Among his mathematical achievements, Archimedes developed a general method of exhaustion for finding areas bounded by parabolas and spirals, and volumes of cylinders, parabolas, segments of spheres, and specially to approximate , which he called as the parameter to diameter. His approach to approximate is based on the following fact: the circumference of a circle lies between the perimeters of the inscribed and circumscribed regular polygons (equilateral and equiangular) of n sides, and as n increases, the deviation of the circumference from the two perimeters becomes smaller. Because of this fact, many mathematicians claim that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless. If an and bn denote the perimeters of the inscribed and circumscribed regular polygons of n sides, and C the circumference of the circle, then it is clear that {an} is an increasing sequence bounded above by C, and {bn} is a decreasing sequence bounded below by C. Both of these sequences converge to the same limit C. To simplify matters, suppose we choose a circle with the diameter , then from Figure it immediately follows that It is clear that limn an = = limn bn. Further, bn is the harmonic mean of an and bn, and an is the geometric mean of an and bn, i.e., an = n sin bn = n tan . n bn = an = From () for the hexagon, i.e., n = it follows that a = , b = . Then Archimedes successively took polygons of sides , , and , used the recursive relations (), and the inequality < < , , . . . . = < < = . . . . . which he probably found by what is now called Herons method, to obtain the bounds It is interesting to note that during Archimedes time algebraic and trigonometric notations, and our present decimal system were not available, and hence he had to derive recurrence relations () geometrically, and certainly for him the computation of a and b must have been a formidable task. The approximation / is often called the Archimedean value of , and it is good for most purposes. If we take the average of the bounds given in (), we obtain = . . . . . The above method of computing by using regular inscribed and circumscribed polygons is known as the classical method of computing . It follows that an inscribed regular polygon of n sides takes up more than /n of the area of a circle. Heron of Alexandria (about AD) in his Metrica, which had been lost for centuries until a fragment was discovered in , followed by a complete copy in , refers to an Archimedes work, where he gives the bounds . . . . = ,, < < ,, = . . . . . Clearly, in the above right inequality, there is a mistake as it is worse than the upper bound / found by Archimedes earlier. Heron adds Since these numbers are inconvenient for measurements, they are reduced to the ratio of the smaller numbers, namely, /. Archimedes polygonal method remained unsurpassed for centuries. Archimedes also showed that a curve discovered by Conon of Samos (around - BC) could, like Hippias quadratrix, be used to square the circle. The curve is today called the Archimedean Spiral. About BC. Daivajna Varahamihira (working BC) was an astronomer, mathematician and astrologer. His picture may be found in the Indian Parliament along with Aryabhata. He was one of the nine jewels (Navaratnas) of the court of legendary king Vikramaditya I (- BC). In BC, Varahamihira wrote Pancha-Siddhanta (The Five Astronomical Canons), in which he codified the five existing Siddhantas, namely, Paulisa Siddhanta, Romaka Siddhanta, Vasishtha Siddhanta, Surya Siddhanta and Paitamaha Siddhanta. He also made some important mathematical discoveries such as giving certain trigonometric formulae; developing new interpolation methods to produce sine tables; constructing a table for the binomial coefficients; and examining the pandiagonal magic square of order four. In his work, he approximated as . BC. Marcus Vitruvius Pollio (about - BC), a Roman writer, architect and engineer, in his multi-volume work De Architectura (On Architecture) used the value = / = ., which is the same as Babylonians had used , years earlier. He was the first to describe direct measurement of distances by the revolution of a wheel. About BC. Liu Xin (Liu Hsin) (about BC- AD) was an astronomer, historian and editor during the Xin Dynasty (- AD). Liu created a new astronomical system, called Triple Concordance. He was the first to give a more accurate calculation of as ., the exact method he used to reach this figure is unknown. This was first mentioned in the Sui shu (-). He also found the approximations ., . and .. Around AD. Liu Xin ( BC-AD ) was a Chinese astronomer, historian and editor during the Xin Dynasty (- AD). He was the son of Confucian scholar Liu Xiang (- BC). Liu created a catalog of , stars, where he used the scale of magnitudes. He was the first in China to give a more accurate calculation of as .. The method he used to reach this figure is unknown. AD. Brahmagupta (born BC) wrote two treatises on mathematics and astronomy: the Brahmasphutasiddhanta (The Correctly Established Doctrine of Brahma) but often translated as (The Opening of the Universe), and the Khandakhadyaka (Edible Bite) which mostly expands the work of Aryabhata. As a mathematician he is considered as the father of arithmetic, algebra, and numerical analysis. Most importantly, in Brahmasphutasiddhanta he treated zero as a number in its own right, stated rules for arithmetic on negative numbers and zero, and attempted to define division by zero, particularly he wrongly believed that / was equal to . He used a geometric construction for squaring the circle, which amounts to = . . Zhang Heng (- AD) was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman and literary scholar. He proposed a theory of the universe that compared it to an egg. The sky is like a hens egg and is as round as a crossbow pellet. The Earth is like the yolk of the egg, lying alone at the center. The sky is large and the Earth is small. According to him the universe originated from chaos. He said that the Sun, Moon and planets were on the inside of the sphere and moved at different rates. He demonstrated that the Moon did not have independent light, but that it merely reflected the light from the sun. He is most famous in the West for his rotating celestial globe, and inventing in the first seismograph for measuring earthquakes. He proposed (about .) for . He also compared the celestial circle to the width (i.e., diameter) of the earth in the proportion of to , which gives as .. . Claudius Ptolemaeus (around - AD) known in English as Ptolemy, was a mathematician, geographer, astrologer, poet of a single epigram in the Greek Anthology, and most importantly astronomer. He made a map of the ancient world in which he employed a coordinate system very similar to the latitude and longitude of today. One of his most important achievements was his geometric calculations of semichords. Ptolemy in his famous Syntaxis mathematica (more popularly known by its Arabian title of the Almagest), the greatest ancient Greek work on astronomy, obtained, using chords of a circle and an inscribed -gon, an approximate value of in sexagesimal notation, as , which is the same as / = . . . . . Eutocius of Ascalon (about -) refers to a book Quick delivery by Apollonius of Perga (around - BC), who earned the title The Great Geometer, in which Apollonius obtained an approximation for , which was better than known to Archimedes, perhaps the same as /. . Wang Fan (-) was a mathematician and astronomer. He calculated the distance from the Sun to the Earth, but his geometric model was not correct. He has been credited with the rational approximation / for , yielding = .. . Liu Hui (around -) wrote two works. The first one was an extremely important commentary on the Jiuzhang suanshu, more commonly called Nine Chapters on the Mathematical Art, which came into being in the Eastern Han Dynasty, and believed to have been originally written around BC. (It should be noted that very little is known about the mathematics of ancient China. In BC, the emperor Shi Huang of the Chin dynasty had all of the manuscript of the kingdom burned.) The other was a much shorter work called Haidao suanjing or Sea Island Mathematical Manual. In Jiuzhang suanshu, Liu Hui used a variation of the Archimedean inscribed regular polygon with sides to approximate as . and suggested / = . as a practical approximation. About . Pappus of Alexandria (around -) was born in Alexandria, Egypt, and either he was a Greek or a Hellenized Egyptian. The written records suggest that, Pappus lived in Alexandria during the reign of Diocletian (-). His major work is Synagoge or the Mathematical Collection, which is a compendium of mathematics of which eight volumes have survived. Pappus Book IV contains various theorems on circles, study of various curves, and an account of the three classical problems of antiquity (the squaring of the circle, the duplication of a cube, and the trisection of an angle). For squaring the circle, he used Dinostratus quadratrix and his proof is a reductio ad absurdum. Pappus is remembered for Pappuss centroid theorem, Pappuss chain, Pappuss harmonic theorem, Pappuss hexagon theorem, Pappuss trisection method, and for the focus and directrix of an ellipse. . He Chengtian (-) gave the approximate value of as ,/, = . . . . . . Tsu Chung-chih (Zu Chongzhi) (-) created various formulas that have been used throughout history. With his son he used a variation of Archimedes method to find . < < .. He also obtained a remarkable rational approximation /, which yields correct to six decimal digits. In Chinese this fraction is known as Mil. To compute this accuracy for , he must have taken an inscribed regular gon and performed lengthy calculations. Note that = / can be obtained from the values of Ptolemy and Archimedes: = . ++ (( )) = . He declared that / is an inaccurate value whereas / is the accurate value of . We also note that = / can be obtained from the values of Liu Hui and Archimedes. In fact, by using the method of averaging, we have . Bhaskara II or Bhaskaracharya (working ) wrote Siddhanta Siromani (crown of treatises), which consists of four parts, namely, Leelavati Bijaganitam, Grahaganitam and Goladhyaya. The first two exclusively deal with mathematics and the last two with astronomy. His popular text Leelavati was written in AD in the name of his daughter. His contributions to mathematics include: a proof of the Pythagorean theorem, solutions of quadratic, cubic, and quartic indeterminate equations, solutions of indeterminate quadratic equations, integer solutions of linear and quadratic indeterminate equations, a cyclic Chakravala method for solving indeterminate equations, solutions of the Pells equation and solutions of Diophantine equations of the second order. He solved quadratic equations with more than one unknown, and found negative and irrational solutions, provided preliminary concept of infinitesimal calculus, along with notable contributions toward integral calculus, conceived differential calculus, after discovering the derivative and differential coefficient, stated Rolles theorem, calculated the derivatives of trigonometric functions and formulae and developed spherical trigonometry. He conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity. He speculated the nature of the number / by stating that it is like the Infinite, Invariable God who suffers no change when old worlds are destroyed or new ones created, when innumerable species of creatures are born or as many perish. He gave several approximations for . According to him ,/, is an accurate value, / is an inaccurate value, and is for ordinary work. The first value may have been taken from Aryabhatta. This approximation has also been credited to Liu Hui and Zu Chongzhi. He also gave the value / = ., which is of uncertain origin; however, it is the same as that by Ptolemy. . Anicius Manlius Severinus Boethius (around -) introduced the public use of sun-dials, water-clocks, etc. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court. King Theodoric sentenced him to death while absent from Rome, seized at Ticinum (now Pavia), and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October , . His Geometry consists of the enunciations (only) of the first book of Euclid, and of a few selected propositions in the third and fourth books, but with numerous practical applications to finding areas, etc. According to him, the circle had been squared in the period since Aristotles time, but noted that the proof was too long. . Abu Jafar Mohammed Ibn Musa al-Khwarizmi (around -) Mohammed the father of Jafar and the son of Musa was a scholar in the academy Bait al-Hikma (House of Wisdom) founded by Caliph al-Mamun (-). His task (along with several other scholars) was to translate the Greek and Sanskrit scientific manuscripts. They also studied, and wrote on algebra, geometry and astronomy. There al-Khwarizmi encountered the Hindu place-value system based on the numerals , , , , , , , , , , including the first use of zero as a place holder in positional base notation, and he wrote a treatise around AD, on what we call Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum (that is, al-Khwarizmi on the Hindu Art of Reckoning), a name given to the work by Baldassarre Boncompagni in , much changed from al-Khwarizmis original text (of which even the title is unknown) is known. The French Minorite friar Alexander de Villa Dei, who taught in Paris around , mentions the name of an Indian king named Algor as the inventor of the new art, which itself is called the algorismus. Thus, the word algorithm was tortuously derived from al-Khwarizmi (Alchwarizmi, al-Karismi, Algoritmi, Algorismi, Algorithm), and has remained in use to this day in the sense of an arithmetic operation. This Latin translation was crucial in the introduction of Hindu-Arabic numerals to medieval Europe. Al-Khwarizmi used = / in algebra, = in geometry, and = ,/, = . in astronomy. . Mahavira (-) in his work Ganita Sara Samgraha summarized and extended the works of Aryabhatta, Bhaskara, Brahmagupta and Bhaskaracharya. This treatise contains: a naming scheme for numbers from up to , formulas for obtaining cubes of sums; techniques for least common denominators (LCM), techniques for combinations nCr = n(n )(n ) (n r + )/r!, techniques for solving linear, quadratic as well higher order equations, arithmetic and geometric series, and techniques for calculating areas and volumes. He was the first person to mention that no real square roots of negative numbers can exist. According to Mahavira whatever is there in all the three worlds, which are possessed of moving and non-moving beings, all that indeed cannot exist without mathematics. He used the approximate value of as . He also mentions that the approximate volume of a sphere with diameter d is (/)(d/), i.e., = ., and exact volume is (/)(/)(d/), i.e., = .. About . Franco von Lttich (around -) claimed to have contributed the only important work in the Christian era on squaring the circle. His works are published in six books, but only preserved in fragments. . Fibonacci (Leonardo of Pisa) (around -) after the Dark Ages is considered the first to revive mathematics in Europe. He wrote Liber Abbaci (Book of the Abacus) in . In this book, he quotes that The nine Indian numerals are. . . with these nine and with the sign which in Arabic is sifr, any desired number can be written. His Practica geometria, a collection of useful theorems from geometry and (what would eventually be named) trigonometry appeared in , which was followed five years later by Liber quadratorum, a work on indeterminate analysis. A problem in Liber Abbaci led to the introduction of the Fibonacci sequence for which he is best remembered today; however, this sequence earlier appeared in the works of Pingala (about BC) and Virahanka (about AD). In Practica geometriae, Fibonacci used a -sided polygon, to obtain the approximate value of as / = . . . . . . Johannes Campanus (around -) was chaplain to three popes, Pope Urban IV, Pope Nicholas IV and Pope Boniface VIII. He was one of the four greatest contemporary mathematicians. Campanus wrote a Latin edition of Euclids Elements in books around . He used the value of as /. About . Zhao Youqin (born ) used a regular polygon of sides to derive = .. About . Albert of Saxony (around -) was a German philosopher known for his contributions to logic and physics. He wrote a long treatise De quadratura circuli (Question on the Squaring of the Circle) consisting mostly philosophy. He said following the statement of many philosophers, the ratio of circumference to diameter is exactly /; of this, there is proof, but a very difficult one. . Madhava of Sangamagrammas (-) work has come to light only very recently. Although there is some evidence of mathematical activities in Kerala (India) prior to Madhava, e.g., the text Sadratnamala (about ), he is considered the founder of the Kerala school of astronomy and mathematics. Madhava was the first to have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series of sine, cosine, tangent and arctangent, which is This series is valid for < x < , and also for x = . He also gave rational approximations of infinite series, tests of convergence of infinite series, estimate of an error term, early forms of differentiation and integration and the analysis of infinite continued fractions. He fully understood the limit nature of the infinite series. Madhava discovered the solutions of transcendental (transcends the power of algebra) equations by iteration, and found the approximation of transcendental numbers by continued fractions. He also gave many methods for calculating the circumference of a circle. The value of correct to decimal places is attributed to Madhava. However, the text Sadratnamala, usually considered as prior to Madhava, while some researchers have claimed that it was compiled by Madhava, gives the astonishingly accurate value of correct to decimal places. . Jemshid al-Kashi (around -), astronomer royal to Ulugh Beg of Samarkand, wrote several important books Sullam al-sama (The Stairway of Heaven), Mukhtasar dar ilm-i hayat (Compendium of the Science of Astronomy), Khaqani Zij on astronomical tables, Risala dar sharh-i alat-i rasd (Treatise on the Explanation of Observational Instruments), Nuzha al-hadaiq fi kayfiyya sana al-ala almusamma bi tabaq almanatiq (The Method of Construction of the Instrument Called Plate of Heavens), Risala al-muhitiyya (Treatise on the Circumference), The Key to Arithmetic, and The Treatise on the Chord and Sine. In these works al-Kashi showed a great venality in numerical work. In , he calculated to decimal places, and later in to decimal places. For this, he used classical polygon method of sides. . George Prbach (-) whose real surname is unknown, was born in Prbach, a town upon the confines of Bavaria and Austria. He studied under Nicholas de Cusa, and one of his most famous pupils is Regiomontanus. Prbach wrote a work on planetary motions which was published in ; an arithmetic, published in ; and a table of eclipses, published in . He calculated tables of sines for every minute of arc for a radius of , units. This table was published in . He approximated by the rational ,/,, which is exactly the same as given by Aryabhatta. . Nicholas of Cusa (-) is often referred to as Nicolaus Cusanus and Nicholas of Kues (Cusa was a Latin place-name for a city on the Mosel). He was a German cardinal of the Roman Catholic Church, a philosopher, jurist, mathematician and an astronomer. Most of his mathematical ideas can be found in his essays, De Docta Ignorantia (Of Learned Ignorance), De Visione Dei (Vision of God) and On Conjectures. He made important contributions to the field of mathematics by developing the concepts of the infinitesimal and of relative motion. He gave the approximations of as (/)( + ) and / = . . . . . Nicholas thought this to be the exact value. Nicholas said, if we can approach the Divine only through symbols, then it is most suitable that we use mathematical symbols, for these have an indestructible certainty. He also said that no perfect circle can exist in the universe. In accordance with his wishes, his heart is within the chapel altar at the Cusanusstift in Kues. . Johann Regiomontanus (Johannes Mller) (-) is considered as one of the most prominent mathematicians of his generation. He was the first to study Greek mathematical works in order to make himself acquainted with the methods of reasoning and results used there. He also well read the works of the Arab mathematicians. In most of this study, he compiled in his De Triangulis, which was completed in , however, was published only in . Regiomontanus used algebra to find solutions of geometrical problems. He criticized Nicholas of Cusas approximations and methods to approximate the value of and gave the approximation .. About . Nilakanthan Somayajis (around -) most notable work Tantrasangraha elaborates and extends the contributions of Madhava. He was also the author of Aryabhatiya-Bhashya, a commentary of the Aryabhatiya. Of great significance in Nilakanthans work includes the inductive mathematical proofs, a derivation and proof of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, and in Sanskrit poetry the series which follows from Madhavas series () when x = . In the literature () is known as Gregory-Leibniz series. He also gave sophisticated explanations of the irrationality of , the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system. If sn denotes the nth partial sum of (), then s = , s = . . . . , s = . . . . , s, = . . . . , s, = . . . . and Roy North showed that s, = . (where underlined digits are incorrect) indicating an annoyingly slow convergence of the partial sums. Since this is an alternating series, the error committed by stopping at the nth term does not exceed /(n + ) in absolute value. Thus, to compute / to eight decimals from () would require n > terms. Hence, although it is only of theoretical interest, the expressions on the right are arithmetical, while arises from geometry. We also note that the series () can be written as This series converges faster than (). Before . Leonardo da Vinci (-) was an Italian painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer. He briefly worked on squaring the circle, or approximating . . Michael Stifel (-) served in several different Churches at different positions; however, every time due to bad circumstances had to resign and flee. He made the error of predicting the end of the world on October , and other time used a clever rearrangement of the letters LEO DECIMVS to prove that Leo X was , the number of the beast given in the Book of Revelation. He was forced to take refuge in a prison after ruining the lives of many believing peasants who had abandoned work and property to accompany him to heaven. In the later part of his life, he lectured on mathematics and theology. He invented logarithms independently of Napier using a totally different approach. His most famous work is Arithmetica integra which was published in . This work contains binomial coefficients, multiplication by juxtaposition, the term exponent, and the notation +, and , and the opinion that the quadrature of is impossible. According to him the quadrature of the circle is obtained when the diagonal of the square contains parts of which the diameter of the circle contains . Thus, /. . Albrecht Drer (-) was a famous artist and mathematician. His book Underweysung der Messung mit dem Zirckel und Richtscheyt provides measurement of lines, areas and solids by means of compass and ruler, particularly there is a discussion of squaring the circle. . Oronce Fie (-) was a prolific author of mathematical books. He was imprisoned in , probably for practicing judicial astrology. He approximated as / = . . . . . Later, he gave / = . . . . and, in , / = . . . . . . Johannes Buteo (-), a French scholar published a book De quadratura circuli, which seems to be the first book that accounts the history of and related problems. . Valentin Otho (around -) was a German mathematician and astronomer. In , he came to Wittenberg and proposed to Johannes Praetorius the Tsu Chung-chih approximate value of as /. . Tycho Brahe was an astronomer and an alchemist and was known for his most accurate astronomical and planetary observations of his time. His data was used by his assistant, Kepler, to derive the laws of planetary motion. He observed a new star in and a comet in . In , when he was just , he lost his nose partially in a duel with another student in Wittenberg and wore throughout his life a metal insert over his nose. His approximation to is / = . . . . . . Simon Duchesne finds = (/) = . . . . . About . Zhu Zaiyu (-), a noted musician, mathematician and astronomercalendarist, Prince of the Ming Dynasty, obtained the twelfth root of two. He also gave the approximate value of as /. = . . . . . Around the same time Xing Yunlu adopted as . and ., while Chen Jinmo and Fang Yizhi, respectively, took as . and /. . Simon van der Eycke (Netherland) published an incorrect proof of the quadrature of the circle. He approximated as ,/ = . . . . . In , he gave the value .. . Adriaen Anthoniszoon (-) was a mathematician and fortification engineer. He rediscovered the Tsu Chung-chih approximation / to . This was apparently lucky incident, since all he showed was that / > > /. He then averaged the numerators and the denominators to obtain the exact value of . . Francois Vite (-) is frequently called by his semi-Latin name of Vieta. In relation to the three famous problems of antiquity, he showed that the trisection of an angle and the duplication of a cube problems depend upon the solution of cubic equations. He has been called the father of modern algebra and the foremost mathematician of the sixteenth century. In his book, Supplementum geometriae, he showed . < < ., i.e., gave the value of correct to places. For this, he used the classical polygon of = , sides. He also represented as an infinite product For this, we note that sin x = cos x sin x = cos x cos x sin x = = The above formula () is one of the milestones in the history of . The convergence of Vietas formula was proved by Ferdinand Rudio (-) in . It is clear that Vietas formula cannot be used for the numerical computation of . In fact, the square roots are much too cumbersome, and the convergence is rather slow. It is clear that if we define a = / and an+ = ( + an)/, then () is the same as aaa = / . . Adrianus van Roomen (-), more commonly referred to as Adrianus Romanus, successively professor of medicine and mathematics in Louvain, professor of mathematics at Wrzburg, and royal mathematician (astrologer) in Poland, proposed a challenge to all contemporary mathematicians, to solve a certain th degree equation. The Dutch ambassador presented van Roomens book to King Henry IV with the comment that at present there is no mathematician in France capable of solving this equation. The King summoned and showed the equation to Vieta, who immediately found one solution to the equation, and then the next day presented more. However, negative roots escaped him. In return, Vieta challenged van Roomen to solve the problem of Apollonius, to construct a circle tangent to three given circles, but he was unable to obtain a solution using Euclidean geometry. When van Roomen was shown proposers elegant solution, he immediately traveled to France to meet Vieta, and a warm friendship developed. The same year Rooman used the classical method with sides, to approximate to correct decimal places. . Joseph Justus Scaliger (-) was a religious leader and scholar. He is known for ancient Greek, Roman, Persian, Babylonian, Jewish and Egyptian history. In his work, Cyclometrica elementa duo he claimed that is equal to . . Ludolph van Ceulen (-) was a German who emigrated to the Netherlands. He taught Fencing and Mathematics in Delft until , when he moved to Leiden and opened a Fencing School. In , he was appointed to the Engineering School at Leiden, where he spent the remainder of his life teaching Mathematics, Surveying and Fortification. He wrote several books, including Van den Circkel (On The Circle, ), in which he published his geometric findings, and the approximate value of correct to decimal places. For this, he reports that he used classical method with , i.e., ,,, sides. This book ends with Whoever wants to, can come closer. . Ludolph van Ceulen (-) in his work De Arithmetische en Geometrische fondamenten, which was published posthumously by his wife in , computed correct to decimal places by using classical method with sides. This computational feat was considered so extraordinary that his widow had all digits of die Ludolphsche Zahl (the Ludolphine number) was engraved on his tombstone in St. Peters churchyard in Leiden. The tombstone was later lost but was restored in . This was one of the last major attempts to evaluate by the classical method; thereafter, the techniques of calculus were employed. . Willebrord Snell (Snellius) (-) was a Dutch astronomer and mathematician. At the age of , he is said to have been acquainted with the standard mathematical works, while at the age of , he succeeded his father as Professor of Mathematics at Leiden. His fame rests mainly on his discovery in of the law of refraction, which played a significant role in the development of both calculus and the wave theory of light. However, it is now known that this law was first discovered by Ibn Sahl (-) in . Snell cleverly combined Archimedean method with trigonometry, and showed that for each pair of bounds on given by the classical method, considerably closer bounds can be obtained. By his method, he was able to approximate to seven places by using just sides, and to van Ceulens decimal places by using polygons having only sides. The classical method with such polygons yields only two and fifteen decimal places. . Yoshida Mitsuyoshi (-) was working during Edo period. His work named as Jinkoki deals with the subject of soroban arithmetic, including square and cube root operations. In this work, he used . for . . Christoph (Christophorus) Grienberger (-) was an Austrian Jesuit astronomer. The crater Gruemberger on the Moon is named after him. He used Snells refinement to compute to decimal places. This was the last major attempt to compute by the Archimedes method. . Celiang quanyi (Complete Explanation of Methods of Planimetry and Stereometry) gives without proof the following bounds . < < ., i.e., correct to digits. . William Oughtred (-), an English mathematician offered free mathematical tuition to pupils, which included even Wallis. His textbook, Clavis Mathematicae (The Key to Mathematics) on arithmetic published in was used by Wallis and Newton amongst others. In this work, he introduced the symbol for multiplication, and the proportion sign (double colon ::). He designated the ratio of the circumference of a circle to its diameter by /. His notation was used by Isaac Barrow (-) a few years later, and David Gregory (-). Before him, mathematicians described in round-about ways such as quantitas, in quam cum multipliectur diameter, proveniet circumferential, which means the quantity which, when the diameter is multiplied by it, yields the circumference. . Grgoire de Saint-Vincent (-), a Jesuit, was a mathematician who discovered that the area under the hyperbola (xy = k) is the same over [a, b] as over [c, d] when a/b = c/d. This discovery played an important role in the development of the theory of logarithms and an eventual recognition of the natural logarithm. In , Nicolaus Mercator (Kauffmann) (-) wrote a treatise entitled Logarithmo-technica, and discovered the series however, the same series was independently discovered earlier by Saint-Vincent. In his book, Opus geometricum quadraturae circuli et sectionum coni he proposed at least four methods of squaring the circle, but none of them were implemented. The fallacy in his quadrature was pointed out by Huygens. . Ren Descartes (-) was a thoughtful child who asked so many questions that his father called him my little philosopher. In , he published his Discourse on Method, which contained important mathematical work, and three essays, Meteors, Dioptrics and Geometry, produced an immense sensation and his name became known throughout Europe. The rectangular coordinate system is credited to Descartes. He is regarded as a genius of the first magnitude. He was one of the most important and influential thinkers in human history and is sometimes called the founder of modern philosophy. After his death, a novel geometric approach to approximate was found in his papers. His method consisted of doubling the number of sides of regular polygons while keeping the perimeter constant. In modern terms, Descartes method can be summarized as xk+(xk+ xk) = k, x = /. The sequence {xk} generated by the above recurrence relation converges to / . . John Wallis (-) in was appointed as Savilian professor of geometry at the University of Oxford, which he continued for over years until his death. He was the most influential English mathematician before Newton. In his most famous work, Arithmetica infinitorum, which he published in , he established the formula This formula is a great milestone in the history of . Like Vites formula (), Wallis had found in the form of an infinite product, but he was the first in history whose infinite sequence involved only rational operations. In his Opera Mathematica I (), Wallis introduced the term continued fraction. He rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity, specially showed that > . He had great ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. On December , he when in bed, occupied himself in finding the integral part of the square root of ; and several hours afterward wrote down the result from memory. Two months later, he was challenged to extract the square root of a number of digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing. Wallis life was embittered by quarrels with his contemporaries including Huygens, Descartes, and the political philosopher Hobbes, which continued for over years, ending only with Hobbes death. Hobbes called Arithmetica infinitorum a scab of symbols, and claimed to have squared the circle. It seems that to some, individuals quarrels give strength, encouragement and mental satisfaction. To derive (), we note that We know that for all x (, /) the inequalities sinm x > sinm x > sinm+ x hold. Thus, an integration from to / gives Im Im Im+, and hence In = / sinn x dx satisfies the recurrence relation In = Im = m m Im+ = m + m . From these relations, a termwise division leads to Now, it suffices to show that = . . Further, from (), we have IImm+ = mm+ , mlim IImm+ = mlim mm+ = . aa an Finally, a combination of () and () immediately gives (). If we define an = /(n), then () is equivalent to aaa = / . We also note that . William Brouncker, nd Viscount Brouncker (-) was one of the founders and the second President of the Royal Society. His mathematical contributions are: reproduction of Brahmaguptas solution of a certain indeterminate equation, calculations of the lengths of the parabola and cycloid, quadrature of the hyperbola which required approximation of the natural logarithm function by infinite series and the study of generalized continued fractions. He undertook some calculations to verify formula (), and showed that . . . . < < . . . . , which is very satisfactory. He also converted Wallis result () into the continued fraction Neither of the expressions (), and (); however, later has served for an extensive calculation of . Another continued fraction representation of which follows from the series () is . Christiaan Huygens (-) is famous for his invention of the pendulum clock, which was a breakthrough in timekeeping. He formulated the second law of motion of Newton in a quadratic form, and derived the now well-known formula for the centripetal force, exerted by an object describing a circular motion. Huygens was the first to derive the formula for the period of an ideal mathematical pendulum (with massless rod or cord), T = /g. For the computation of , he gave the correct proof of Snells refinement, and using an inscribed polygon of only sides obtained the bounds . < < ., for the same accuracy the classical method requires almost , sides. . Muramatsu Shigekiyo (-) published Sanso, or Stack of Mathematics, in which he used classical polygon method of sides to obtain = .. . Sir Isaac Newton (-), hailed as one of the greatest scientist-mathematicians of the English-speaking world, had the following more modest view of his own monumental achievements: . . . to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. As he examined these shells, he discovered to his amazement more and more of the intricacies and beauties that lay in them, which otherwise would remain locked to the outside world. At the age of , he succeeded Barrow as Lucasian professor of mathematics at Cambridge. About him, Aldous Huxley (-) had said If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood and many other desirable things. As a man he was a failure; as a monster he was superb. Newton made some of the greatest discoveries the world ever knew at that time. Newton discovered: . The nature of colors. . The law of gravitation and the laws of mechanics. . The fluxional calculus. Most of the history books say that to compute Newton used the series sin x = x + x Several other such poems not only in English, but almost in every language including Albanian, Bulgarian, Czech, Dutch, French, German, Italian, Latin, Polish, Portuguese, Romanian, Spanish and Swedish are known. However, there is a problem with this type of mnemonic, namely, how to represent the digit zero. Fortunately, a zero does not occur in until the thirty-second place. Several people have come up with ingenious methods of overcoming this, most commonly using a ten-letter word to represent zero. In other cases, a certain piece of punctuation is used to indicate a naught. Michael Keith (with such similar understanding) in his work Circle digits: a self-referential story, Mathematical Intelligencer, vol. (), -, wrote an interesting story which gives first decimals of . . Ernest William Hobson (-) was Sadleirian Professor at the University of Cambridge from to . His work on real analysis was very influential in England. In his book, Squaring the circle: A History of the Problem, he used a geometrical construction to obtain = . . . . . . Srinivasa Ramanujan (-) was a famous mathematical prodigy. He collaborated with Hardy for five years, proving significant theorems about the number of partitions of integers. Ramanujan also made important contributions to number theory and also worked on continued fractions, infinite series and elliptic functions. In , he became the youngest Fellow of the Royal Society. According to Hardy, the limitations of Ramanujans knowledge were as startling as its profundity. Here was a man who could workout modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta-function, and the dominant terms of the many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchys theorem, and had indeed but the vaguest idea of what a function of a complex variable was. Ramanujan considered mathematics and religion to be linked. He said, an equation for me has no meaning unless it expresses a thought of God. He was endowed with an astounding memory and remembered the idiosyncrasies of the first , integers to such an extent that each number became like a personal friend to him. Once Hardy went to see Ramanujan when he was in a nursing home and remarked that he had traveled in a taxi with a rather dull number, viz ,, Ramanujan exclaimed, No, Hardy, , is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes viz , = + = + , and the next such number is very large. His life can be summed up in his own words, I really love my subject. His paper on Modulus functions and approximation to contains several new innovative empirical formulas and geometrical constructions for approximating . One of the remarkable formulas for its elegance and inherent mathematical depth , m= ((mm!))! (,+m,m) . It has been used to compute to a level of accuracy, never attained earlier. Each additional term of the series adds roughly digits. He also developed the series The first series has the property that it can be used to compute the second block of k (binary) digits in the decimal expansion of without calculating the first k digits. The following mysterious approximation which approximates to correct decimal places is also due to Ramanujan ln ( + )( + ) . . T.M.P. Hughes in his work A triangle that gives the area and circumference of any circle, and the diameter of a circle equal in area to any given square, Nature , , doi:./a uses a geometric construction to obtain = . . . . . . In March , the University of Minnesota was notified that Gottfried Lenzer (a native of Germany who lived in St. Paul for many years) had bequeathed to the university a series of drawings from - and explanatory notes concerning the three classical problems of antiquity. He used a geometrical construction for squaring the circle to obtain = . . . . . . Alexander Osipovich Gelfond (-) was a Soviet mathematician. He proved that e (Gelfonds constant) is transcendental, but nothing yet is known about the nature of any of the numbers + e, e or e. . Helen Abbot Merrill (-) earned her Ph.D. from Yale in on the thesis On Solutions of Differential Equations which Possess an Oscillation Theorem. She served as an associate editor of The American Mathematical Monthly during -, and was a vice-president from to of the Mathematical Association of America. Her book Mathematical Excursions: Side Trips Along Paths not Generally Traveled in Elementary Courses in Mathematics, Bruce Humphries, Inc., Boston, was a text for the general public. In this book, a geometric construction is given (perhaps by an earlier author) which leads to = . . . . . . Edmund Georg Hermann (Yehezkel) Landau (-) was a child prodigy. In , he gave a simpler proof of the prime number theorem. His masterpiece of was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory. Landau wrote over papers on number theory, which had a major influence on the development of the subject. Despite his outstanding talents as both a teacher and researcher, Landau annoyed many of his colleagues at Gttingen. He started criticizing privately, and often publicly, their results. Landau in his work defined / as the value of x between and for which cos x vanishes. One cannot believe this definition was used, at least as an excuse, for a racial attack on Landau. This unleashed an academic dispute which was to end in Landaus dismissal from his chair at Gttingen. Ludwig Georg Elias Moses Bieberbach (-) famous for his conjecture, explained the reasons for Landaus dismissal: Thus the valiant rejection by the Gttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture. Hardy replied immediately to Bieberbach about the consequences of this un-German definition of : There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for ones own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbachs reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true. . A Cleveland businessman Carl Theodore Heisel published a book Mathematical and Geometrical Demonstrations in which he announced the grand discovery that was exactly equal to /, a value that the Egyptians had used some , years ago. Substituting this value for calculations of areas and circumferences of circles with diameters , , . . . up to , he obtained numbers which showed consistency of circumference and area, thereby furnishing incontrovertible proof of the exact truth of his ratio. He also rejected decimal fractions as inexact (whereas ratios of integers as exact and scientific), and extracted roots of negative numbers thus: a = a, a = a. He published this book on his own expense and distributed to colleges and public libraries throughout the United States without charge. . Miff Butler claimed discovery of a new relationship between and e. He stated his work to be the first basic mathematical principle ever developed in USA. He convinced his congressman to read it into the Congressional Record on June . . H.S. Uhler used Machins formula () to compute to decimal places. -. D.F. Ferguson of England used the formula to find that his value disagreed with that of William Shanks in the th place. In , he approximated to decimal places, and in January to decimal places. In the same month William Shanks used Machins formula () to compute -place value of , but Ferguson soon found an error in the rd place. For all the calculations, he used desk calculator. . Ivan Morton Niven (-) gave an elementary proof that is irrational. . Ferguson and John William Wrench, Jr. (-) using a desk calculator, computed , decimal digits of . This record was broken only by the electronic computers. September . John Wrench and L.R. Smith (also attributed to George Reitwiesner et al.) were the first to use an electronic computer Electronic Numerical Integrator and Computer (ENIAC) at the Army Ballistic Research Laboratories in Aberdeen, Maryland, to calculate to , decimal places. For this, they programed Machins formula (). It took hours, a pitifully long time by todays standards. In this project, John Louis von Neumann (-), one of the most versatile and smartest mathematicians of the twentieth century, also took part. In , The ENIAC became obsolete, and it was dismembered and moved to the Smithsonian Institution as a museum piece. . Konrad Knopp gave the following two expansions of : k + k + = . Kurt Mahler (-) showed that is not a Liouville number: A real number x is called a Liouville number if for every positive integer n, there exist integers p and q with q > and such that A Liouville number can thus be approximated quite closely by a sequence of rational numbers. In , Liouville showed that all Liouville numbers are transcendental. . S.C. Nicholson and J. Jeenel programmed NORC (Naval Ordnance Research Calculator) at Dahlgren, Virginia to compute to , decimals. For this, they used Machins formula (). The run took only minutes. . John Gurland established that for all positive integers n, . March . G.E. Felton used the Ferranti Pegasus computer to find , decimal places of in hours. The program was based on Klingenstiernas formula However, a subsequent check revealed that a machine error had occurred, so that only , decimal places were correct. The run was therefore repeated in May , but the correction was not published. January . Francois Genuys programmed an IBM at the Paris Data Processing Center. He used Machins type formula (). It yielded , decimal places of in hour and minutes. July . Genuys programmed an IBM at the Commissariat lEnergie Atomique in Paris to compute to , decimal places. He used Machins type formula (). It took hours and minutes. July . Daniel Shanks (-) and William Shanks used Strmers formula () on an IBM (at the IBM Data Processing Center, New York) to compute to , digits, of which the first , digits were published by photographically reproducing the print-out with , digits per page. The time required for this computation was hours and minutes. They also checked the calculations by using Gauss formula (), which required hours and minutes. . Machins formula () was also the basis of a program run on an IBM at the London Data Center in July , which resulted in , decimal places and required only minutes running time. February . Jean Guilloud and J. Filliatre used an IBM at the Commissariat lEnergie Atomique in Paris to obtain an approximation of extending to , decimal places on a STRETCH computer. For this, they used Strmers and Gauss formulas () and (). It took almost hours. February . Guilloud and M. Dichampt used CDC (Control Data Corporation) in Paris to approximate to , decimal places. For this, they used Strmers and Gauss formulas () and (). The computer that churned out half a million digits needed only hours and minutes (plus hour and minutes to convert that final result from binary to decimal notation). . In the Putnam Competition, the first problem was This integral was known to Mahler in the mid-s, and has later appeared in several exams. It is also discussed by Borwein, Bailey, and Girgensohn in their book on p.. . K.Y. Choong, D.E. Daykin and C.R. Rathbone used , digits of Daniel Shanks and William Shanks () to generate the first , partial quotients of the continued fraction expansion of . . Ralph William Gosper, Jr. (born ), known as Bill Gosper, is a mathematician and programmer. He is best known for the symbolic computation, continued fraction representations of real numbers, Gospers algorithm, and Gosper curve. He used a refinement of Euler transform on () to obtain the series = + + + + + . . Guilloud with Martine Bouyer (Paris) used formulas () and () on a CDC to compute to ,, digits. The run time required for this computation was hours and minutes, of which hour minutes was used to convert the final result from binary to decimal. Results of statistical tests, which generally support the conjecture that is simply normal (in , Flix douard Justin mil Borel (-) defined: A real number a is simply normal in base b if in its representation in base b all digits occur, in an asymptotic sense, equally often) were also performed. . Louis Comtet developed the following Eulers type expansion of : = m= m mm . bk = . Richard Brent and Eugene Salamin independently discovered an algorithm which is based on an arithmetic-geometric mean and modifies slightly Gauss-Legendre algorithm. Set a = , b = / and s = /. For k = , , , . . . compute ck = ak bk, sk = sk kck, pk = Then pk converges quadratically to , i.e., each iteration doubles the number of accurate digits. In fact, successive iterations must produce , , , , , , , and correct digits of . The twenty-fifth iteration must produce million correct decimal digits of . . Kazunori Miyoshi and Kazuhika Nakayama of the University of Tsukuba, Japan calculated to ,, significant figures in . hours on a FACOM M- computer. They used Klingenstiernas formula () and checked their result with Machins formula (). . Guilloud computed ,, decimal digits of . . Rajan Srinivasan Mahadevan (born ) recited from memory the first , digits of . This secured him a place in the Guinness Book of World Records, and he has been featured on Larry King Live and Readers Digest. . Kikuo Takano (-) was a Japanese poet and mathematician. He developed the following Machin-like formula for calculating : = tan . Yoshiaki Tamura on MELCOM II computed ,, decimal places of . For this, he used the Salamin-Brent algorithm (). . Yoshiaki Tamura and Yasumasa Kanada (born , life-long pi digit-hunter, set the record of the past times) on HITAC M-H computed ,, decimal places of . For this, they used the Salamin-Brent algorithm (). . Yoshiaki Tamura and Yasumasa Kanada on HITAC M-H computed ,, decimal places of . For this, they used the Salamin-Brent algorithm (). October . Yasumasa Kanada, Yoshiaki Tamura, Sayaka Yoshino and Yasunori Ushiro on HITAC S-/ computed ,, decimal places of . For this, they used the Salamin-Brent algorithm (). In this work to gather evidence that is simply normal, they also performed statistical analysis. It showed expected behavior. In the first ten million digits, the frequencies for each ten digits are ,; ,; ,,; ,; ,,; ,,; ,; ,,; ,; and ,,. Further, the rate at which the relative frequencies approach / agrees with theory. As an example, for the digit relative frequencies in the first i, i = , , , , , , , digits are , ., ., ., ., ., ., which seem to be approaching / at rate predicted by the probability theory for random digits, i.e., a speed approximately proportional to /n. But this is far from a formal proof of simple normalcy perhaps for a proof the current mathematics is not sufficiently developed. In spite of the fact that the digits of pass statistical tests for randomness, contains some sequences of digits that, to some, may appear non-random, such as Feynman point, which is a sequence of six consecutive s that begins at the nd decimal place. A number is said to be normal if all blocks of digits of the same length occur with equal frequency. Mathematicians expect to be normal, so that every pattern possible eventually will occur in the digits of . xk+ = (xk + /xk)/, yk+ = Then k converges to quartically. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits of . . Morris Newman and Daniel Shanks proved the following: Set . Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura on HITAC M-H computed ,, decimal places of . For this, they used the Salamin-Brent algo rithm(.Jo)n.athan Borwein and Peter Borwein gave the following algorithm. Set x = , y = and = + . Iterate c = + + , + ,, d = + + , + ,, ak+ = ak( + yk+) k+yk+ + yk+ + yk+ . . Gosper used Symbolics , and Ramanujans formula () to compute to ,, decimal digits. . Jonathan Borwein and Peter Borwein gave the following algorithm. Set a = and y = . Iterate Then ak converges quartically to / , i.e., each iteration approximately quadruples the number of correct digits. . The following is not an identity, but is correct to over billion digits n= . Carl Sagan in his novel deals with the theme of contact between humanity and a more technologically advanced, extraterrestrial life form. He suggests that the creator of the universe buried a message deep within the digits of . January . David H. Bailey used Borweins algorithms () and () on CRAY- to compute ,, decimal places of . September . Yasumasa Kanada and Yoshiaki Tamura on HITAC S-/ computes ,, decimal places of . For this, they used algorithms () and (). October . Yasumasa Kanada and Yoshiaki Tamura on HITAC S-/ computed ,, decimal places of . For this, they used algorithm (). January . Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo and others on NEC SX- computed ,, decimal places of . For this they used algorithms () and (). . Jonathan Borwein and Peter Borwein gave the following algorithm. Set x = /, y = / and p = + /. Iterate Then pk decreases monotonically to and |pk | k+ for k . . Hideaki Tomoyori (born ) recited from memory to , places taking hours minutes, including breaks totaling hours minutes, at Tsukuba University Club House. January . Yasumasa Kanada on HITAC S-/ computed ,, decimal places of . For this, he used algorithms () and (). . Jonathan Borwein and Peter Borwein developed the series Each additional term of the series adds roughly digits. . Dario Castellanos gave the following approximation: = . . . . . May . David Volfovich Chudnovsky (born ) and Gregory Volfovich Chudnovsky (born ) have published hundreds of research papers and books on number theory and mathematical physics. Gregory solved Hilberts tenth problem at the age of . They on CRAY- and IBM /VF computed ,, decimal places of . June . David and Gregory Chudnovsky on IBM computed ,, decimal places of . July . Yasumasa Kanada and Yoshiaki Tamura on HITAC S-/ computed ,, decimal places of . For this, they used algorithm (). August . David and Gregory Chudnovsky developed the following rapidly convergent generalized hypergeometric series: (n!()n()!n)! ,(,,+)n,+/,n . Each additional term of the series adds roughly digits. This series is an improved version to that of Ramanujans (). It was used by the Chudnovsky brothers to calculate more than one billion (to be exact ,,,) digits on IBM . November . Yasumasa Kanada and Yoshiaki Tamura on HITAC S-/ computed ,,, decimal places of . For this, they used algorithms () and (). August . David and Gregory Chudnovsky used a home made parallel computer (they called it m zero, where m stands for machine, and zero for the success) to obtain ,,, decimal places of . For this they used series (). . David Boll discovered an occurrence of in the Mandelbrot set fractal. . Jonathan Borwein and Peter Borwein improved on the Salamin-Brent algorithm (). Set a = / and s = ( )/. Iterate rk+ = + ( sk)/ , ak+ = rk+ak k rk+ . Then /ak converges cubically to , i.e., each iteration approximately triples the number of correct digits. Among the several other known iterative schemes, we list the following two which are easy to implement on a computer: Set a = / and s = ( ). Iterate xn+ = /sn , yn+ = (xn+ ) + , zn+ = Then ak converges quintically to / , i.e., each iteration approximately quintuples the number of correct digits, and < an / < n en . Then ak converges nonically to / , i.e., each iteration approximately multiplies the number of correct digits by nine. . Jonathan Borwein and Peter Borwein developed the series + ,,,,,,,, + (,,,,,,,,,,,,,,, + ,,,,,,,,,,,,,,,) / , + ,,,,,,,,, + ,,,(,,,,,,,,,,,,, + ,,,,,,,,,,,,,) / C = ,,,,, ,,,,, ,(,,,,,,,,, + ,,,,,,,,,) / . Each additional term of the series adds approximately digits. However, computation of this series on a computer does not seem to be easy. May . David and Gregory Chudnovsky used a home made parallel computer m zero to obtain ,,, decimal places of . For this they used series (). June . Yasumasa Kanada and Daisuke Takahashi on HITAC S-/ (dual CPU) computed ,,, decimal places of . For this, they used algorithms () and (). August . Yasumasa Kanada and Daisuke Takahashi on HITAC S-/ (dual CPU) computed ,,, decimal places of . For this, they used algorithms () and (). October . Yasumasa Kanada and Daisuke Takahashi on HITAC S-/ (dual CPU) computed ,,, decimal places of . For this, they used algorithms () and (). . David Bailey, Peter Borwein and Simon Plouffe developed the following formula (known as BBP formula) to compute the nth hexadecimal digit (base ) of without having the previous n digits m= m To show the validity of (), for any k < , we have / xk+m dx = k/ The discovery of this formula came as a surprise. For centuries, it had been assumed that there was no way to compute the nth digit of without calculating all of the preceding n digits. Since this discovery, many such formulas for other irrational numbers have been discovered. Such formulas have been called as spigot algorithms because, like water dripping from a spigot, they produce digits that are not reused after they are calculated. . Simon Plouffe discovered an algorithm for the computation of in any base. Later he expressed regrets for having shared credit for his discovery of this formula with Bailey and Borwein. March . David and Gregory Chudnovsky used a home made parallel computer m zero to obtain ,,, decimal places of . For this, they used series (). They said we are looking for the appearance of some rules that will distinguish the digits of from other numbers, i.e., if someone were to give you a million digits from somewhere in , could you tell it was from ? The digits of form the most nearly perfect random sequence of digits that has ever been discovered. However, each digit appears to be orderly. If a single digit in were to be changed anywhere between here and infinity, the resulting number would no longer be , it would be garbage. Around the three-hundred-millionth decimal place of , the digits go -eight eights pop up in a row. Does this mean anything? It appears to be random noise. Later, ten sixes erupt: . What does this mean? Apparently nothing, only more noise. Somewhere past the half-million mark appears the string . It is an accident, as it were. We do not have a good, clear, crystallized idea of randomness. It cannot be that is truly random. Actually, a truly random sequence of numbers has not yet been discovered. . Gosper posted the following fascinating formula April . Yasumasa Kanada and Daisuke Takahashi on HITACHI SR (, CPU) computed ,,, decimal places of . For this, they used algorithms () and (). July . Yasumasa Kanada and Daisuke Takahashi on HITACHI SR (, CPU) computed ,,, decimal places of . The computation tool just over hours, at an average rate of nearly , digits per second. For this, they used algorithms () and (). . Fabrice Bellard developed the following formula: m+ m+ + m+ m+ m+ m+ + m + , which can used to compute the nth digit of in base . It is about % faster then (). The following exotic formula is also due to him: P(m) = ,,m + ,,,m ,,,m + ,,,m ,,m + ,,. April . Yasumasa Kanada and Daisuke Takahashi on HITACHI SR ( of nodes) computed ,,, decimal places of . For this, they used algorithms () and (). September . Yasumasa Kanada and Daisuke Takahashi on HITACHI SR/MPP ( nodes) computed ,,,, i.e., billion decimal places of . For this, they used algorithms () and (). . J. Munkhammar gave the following formula which is related to Vites (): + + an = Another closely related formula is which as a recurrence relation can be written as = limn n+an, where a = , and m n . Robert Palais believes that the notation is wrongly used right from the beginning. According to him, some suitable symbol (now popular as tau ) must have been used for . He justifies his claim by giving several formulas where appears naturally rather than just . For some people, June , is Taus Day and they celebrate. November . Yasumasa Kanada used Machin-like formulas () and () to compute the value of to ,,,, decimal places. The calculation took more than hours on nodes of a HITACHI SR/MPP supercomputer. The work was done at the Department of Information Science at the University of Tokyo. For this, he used arctangent formulas () and (). . Daniel Tammet, at age , recited , decimal places of , scoring the European record. For an audience at the Museum of the History of Science in Oxford, he said these numbers aloud for hours and minutes. Unfortunately, he made his first mistake at position , and did not correct this error immediately and without outside help, but only after he was told that there was a mistake. . Stephen K. Lucas found that Several other integral formulas of this type are known, here we give the following: then it follows that < .. November . Chao Lu, a chemistry student, at age , broke the Guiness record by reciting from memory to , places. For this, he practiced for years. The attempt lasted hrs min and was recorded on video tapes. The attempt was witnessed by officials, math professors and students. . Kate Bush in the song (in her album Aerial) sings the number to its th decimal place (though she omits the th to th decimal places). October . Akira Haraguchi a retired engineer from Chiba recited from memory to , digits in September , , digits in December , , digits in July , and , digits in October . He accomplished the last recitation in /hours in Tokyo. He says memorization of the digits of is the religion of the universe. . Simon Plouffe found the following curious formula: k= k(ek ) k= k(ek ) . In Midnight (tenth episode of the fourth series of British science fiction television series Doctor Who), the character, the businesswoman, Sky Silvestry mimics the speech of The Doctor by repeating the square root of to decimal places .. . Syamal K. Sen and Ravi P. Agarwal suggested four Matlab based procedures, viz, (i) Exhaustive search, (ii) Principal convergents of continued fraction based procedure, (iii) Best rounding procedure for decimal (rational) approximation, and (iv) Continued fraction based algorithm with intermediate convergents. While the first procedure is exponential-time, the remaining three are polynomial-time. Roughly speaking, they have demonstrated that the absolute best k-digit rational approximation of will be as good as k-digit decimal approximation of . The absolute best k-digit rational approximation is most desired for error-free computation involving /any other irrational number. . Syamal K. Sen, Ravi P. Agarwal and Ghoolam A. Shaykhian have demonstrated through numerical experiment using Matlab that has always scored over (golden ratio), as a source of uniformly distributed random numbers, statistically in one-dimensional Monte Carlo (M.C.) integration; whether fares better than for double, triple and higher dimensional M.C. integration or not deserves exploration. . Syamal K. Sen, Ravi P. Agarwal and Ghoolam A. Shaykhian compared the four procedures they proposed in () for computing best k-digit rational approximations of irrational numbers in terms of quality (error) and cost (complexity). They have stressed on the fact that ultra-high-speed computing along with abundance of unused computing power allows employing an exponential-time algorithm for most real-world problems. This obviates the need for acquiring and employing the mathematical knowledge involving principal and intermediate convergents computed using a polynomial-time algorithm for practical problems. Since is the most used irrational number in the physical world, the simple concise Matlab program would do the job wherever /any other irrational number is involved. . Syamal K. Sen, Ravi P. Agarwal and Raffela Pavani have provided, using Matlab, the best possible rational bounds bracketing /any irrational number with absolute error and the time complexity involved. Any better bounds are impossible. In these rational bounds, either the lower bound or the upper bound will always be the absolute best rational approximation. The absolute error computed provides the overall error bounds in an error-free computational environment involving /any other irrational number. . Tue N. Vu has given Machin-type formula (http://seriesmathstudy.com/sms/ machintypetv): For each positive integer n, . Cetin Hakimoglu-Brown developed the following expansion: which can be written as , k= (k)!(,k + ,,k + ,k + ,) ()k+(/)k(/)k(/)k(/)k August . Daisuke Takahashi et al. used a massive parallel computer called the TK Tsukuba System to compute to ,,,, decimal places in hours minutes. For this, they used algorithms () and (). December . Fabrice Bellard used Chudnovsky brothers series () to compute ,,,,, i.e., . trillion decimal places of in days. For this, he used a single desktop PC, costing less than $,. August . Shigeru Kondo and Alexander J. Yee used Chudnovsky brothers series () to compute ,,,,, i.e., trillion decimal places of in days. For this, they used a server-class machine running dual Intel Xeons, equipped with GB of RAM. . Michael Keith used , digits of to establish a new form of constrained writing, where the word lengths are required to represent the digits of . His book contains a collection of poetry, short stories, a play, a movie script, crossword puzzles and other surprises. . Syamal K. Sen and Ravi P. Agarwal in their monograph systematically organized their work of and on and other irrational numbers. They also included several examples to illustrate the importance of their findings. . During the auction for Nortels portfolio of valuable technology patents, Google made a series of strange bids based on mathematical and scientific constants, including . October . Shigeru Kondo and Alexander J. Yee used Chudnovsky brothers series () to compute ,,,,, i.e., trillion decimal places of in days. . Cristinel Mortici improved Gurlands bounds () to n < < n, n where n = + n n + n + ,n ,n n . n . n . n = + n n + n + ,n n+ ((nn)!!)!! . n = n exp n + n n + ,n ,n n = n exp n + n n + ,n Conclusions No number system can capture exactly. We are deeply and almost completely involved in the conventional decimal number system in representing any real quantity. This is not the only number system for the representation. There are other number systems such as binary, octal, hexadecimal, binary-coded decimal, negative radix, p-adic and modular number systems. If the circumference of a circle is exactly represented, then its diameter will not have exact representation and vice versa. Reading the mathematicians in pre-computer days. An important focus of this paper is that the reader besides, however, knowing the usual chronology of the events in the life of , could get a feel and also read how the mind of a mathematician has been working when he ponders over either independently without much knowledge/concern of what has been done in the past or with considerable knowledge of the work done by his predecessors. Hyper-computers ( flops) of were completely non-existence and even beyond the imagination of all the mathematicians/scientists until almost the mid-twentieth century. Also, publication machinery was too poor until the beginning of the twentieth century. Consequently, all the work on that has been carried out during thousands of years prior to the twentieth century was not a monotonic improvement in the value as well as in the exploration of its wonderful character. Many have worked on standalone while others have contributed with some prior knowledge of the earlier work. All of them were severely handicapped due to the non-existence of todays ultra-high speed computers. They entirely depended on their ingenuity and on whatever negligible computing device they had. It is really interesting under this environment to read these scientists/mathematicians and realize how fortunate we are in the gigantic computer age. All that has been done during the last years (-) amounts to much more than what has been achieved during the past several millennia. Matlab is well-suited to check/evaluate merits of all past formulas. Widely used userfriendly Matlab that needs no formal programming knowledge along with the vpa (variable precision arithmetic) and format long g commands can be used to easily and readily check all that has been done during the past several thousand years and possibly appreciate the inherent intellectual import of the bygone scientists (having practically no computing device) and their expected pitfalls, bias and incorrect beliefs. Checking exactness of billions of digits of is difficult. Are all the billions of digit of computed % error-free? We are familiar with the age-old proverb that To err is human (living being). Maybe a new proverb Not to err is computer (non-living being) can be taken as true in the modern computer age. Here, err means mistake. The arithmetic operations, particularly subtraction operations of two nearly equal numbers, involved in a formula could be sometimes error introducer. However, different computers with different formulas used to compute would help verification and obviate possible error in computation. Computing nth decimal digit exactly always without preceding digits seems yet an open computational problem. While probabilistically one may determine the nth digit of without computing the preceding n digits, obtaining nth digit exactly (correctly) always for any n does not seem to be possible without a large precision. It seems yet an open computational problem that needs exploration. Thus, formulas such as () seem more of theoretical/academic interest than of practical usage as of now. PI for testing performance and stability of a computer. Super PI is a computer program of million digits. It uses the Gauss-Legendre algorithm and is a Windows port of the many overclockers to test the performance and stability of their computers. Overclocking is the process of making a computer run faster than the clock frequency specified by the manufacturer by modifying system parameters. Acknowledgements Dedicated to V Lakshmikantham (1924-2012).


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Ravi P Agarwal, Hans Agarwal, Syamal K Sen. Birth, growth and computation of pi to ten trillion digits, Advances in Difference Equations, 2013, 100, DOI: 10.1186/1687-1847-2013-100