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 UniversityKingsville
, 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 nonmathematicians 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
superstrings, 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 prehistoric 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 circlesquaring 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 twentyfirst 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 highprecision 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 PiSearch Page finds a persons birthday and other
wellknown 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 recordsetting 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 ChouKong
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 moonshaped figures (lunes,
bounded by pair of circular arcs) could be drawn whose areas were together equal to that
of a rightangled 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 seventeenthcentury 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 PanchaSiddhanta
(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 multivolume 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 ( BCAD ) 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 Chungchih (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 sundials, waterclocks, 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 alKhwarizmi (around ) Mohammed the
father of Jafar and the son of Musa was a scholar in the academy Bait alHikma (House
of Wisdom) founded by Caliph alMamun (). 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 alKhwarizmi encountered the
Hindu placevalue 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 HinduArabic numerals. The Arabic text is lost but a Latin
translation, Algoritmi de numero Indorum (that is, alKhwarizmi on the Hindu Art of
Reckoning), a name given to the work by Baldassarre Boncompagni in , much changed from
alKhwarizmis 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 alKhwarizmi
(Alchwarizmi, alKarismi, 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 HinduArabic numerals to medieval Europe. AlKhwarizmi 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 nonmoving 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 alKashi (around ), astronomer royal to Ulugh Beg of
Samarkand, wrote several important books Sullam alsama (The Stairway of Heaven),
Mukhtasar dar ilmi hayat (Compendium of the Science of Astronomy), Khaqani Zij on
astronomical tables, Risala dar sharhi alati rasd (Treatise on the Explanation of
Observational Instruments), Nuzha alhadaiq fi kayfiyya sana alala almusamma bi tabaq
almanatiq (The Method of Construction of the Instrument Called Plate of Heavens), Risala
almuhitiyya (Treatise on the Circumference), The Key to Arithmetic, and The Treatise on
the Chord and Sine. In these works alKashi 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 placename 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 AryabhatiyaBhashya, 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
GregoryLeibniz 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 Chungchih
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 Chungchih 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 semiLatin 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 roundabout
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 SaintVincent (), 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 Logarithmotechnica, and discovered
the series
however, the same series was independently discovered earlier by SaintVincent. 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 wellknown 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
scientistmathematicians of the Englishspeaking 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 thirtysecond place. Several people have come up with ingenious methods
of overcoming this, most commonly using a tenletter 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 selfreferential 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 zetafunction, 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
vicepresident 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 unGerman 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
unGerman 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 printout 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 mids, 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 arithmeticgeometric mean and modifies slightly GaussLegendre
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 twentyfifth 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 Machinlike formula for calculating :
= tan
. Yoshiaki Tamura on MELCOM II computed ,, decimal places of .
For this, he used the SalaminBrent algorithm ().
. Yoshiaki Tamura and Yasumasa Kanada (born , lifelong pi digithunter, set
the record of the past times) on HITAC MH computed ,, decimal places
of . For this, they used the SalaminBrent algorithm ().
. Yoshiaki Tamura and Yasumasa Kanada on HITAC MH computed ,,
decimal places of . For this, they used the SalaminBrent algorithm ().
October . Yasumasa Kanada, Yoshiaki Tamura, Sayaka Yoshino and Yasunori
Ushiro on HITAC S/ computed ,, decimal places of . For this, they used
the SalaminBrent 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 nonrandom, 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 selfcorrecting; 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 MH
computed ,, decimal places of . For this, they used the SalaminBrent
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 SalaminBrent 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 threehundredmillionth 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 halfmillion 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 Machinlike 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
exponentialtime, the remaining three are polynomialtime. Roughly speaking, they have
demonstrated that the absolute best kdigit rational approximation of will be as good as
kdigit decimal approximation of . The absolute best kdigit rational approximation is
most desired for errorfree 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 onedimensional
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 kdigit rational approximations
of irrational numbers in terms of quality (error) and cost (complexity). They have stressed
on the fact that ultrahighspeed computing along with abundance of unused computing
power allows employing an exponentialtime algorithm for most realworld problems.
This obviates the need for acquiring and employing the mathematical knowledge involving
principal and intermediate convergents computed using a polynomialtime 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
errorfree computational environment involving /any other irrational number.
. Tue N. Vu has given Machintype formula (http://seriesmathstudy.com/sms/
machintypetv): For each positive integer n,
. Cetin HakimogluBrown 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 serverclass 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, binarycoded decimal, negative radix, padic 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 precomputer 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. Hypercomputers ( flops) of were completely nonexistence and even
beyond the imagination of all the mathematicians/scientists until almost the midtwentieth
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 nonexistence of todays ultrahigh 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 wellsuited 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 % errorfree? We are familiar with the ageold proverb that To err is
human (living being). Maybe a new proverb Not to err is computer (nonliving 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 GaussLegendre 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 (19242012).