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# The History of Pi

Before I talk about the history of Pi I want to explain what Pi is. Webster’s Collegiate Dictionary defines Pi as “1: the 16th letter of the Greek alphabet… 2 a: the symbol pi denoting the ratio of the circumference of a circle to its diameter b: the ratio itself: a transcendental number having a value to eight decimal places of 3. 14159265” A number can be placed into several categories based on its properties. Is it prime or composite? Is it imaginary or real? Is it transcendental or algebraic? These questions help define a number’s behavior in different situations.

In order to understand where Pi fits in to the world of mathematics, one must understand several of its properties pi is irrational and pi is transcendental. A rational number is one that can be expressed as the fraction of two integers. Rational numbers converted into decimal notation always repeat themselves somewhere in their digits. For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal notation is 0. 142857142857…, a repetition of six digits.

However, the square root of 2 cannot be written as the fraction of two integers and is therefore irrational. For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number. The first attempt at a proof was by Johaan Heinrich Lambert in 1761. Through a complex method he proved that if x is rational, tan(x) must be irrational. It follows that if tan(x) is rational, x must be irrational. Since tan(pi/4)=1, pi/4 must be irrational; therefore, pi must be irrational. Many people saw Lambert’s proof as too simplified an answer for such a complex and long-lived problem.

In 1794, however, A. M. Legendre found another proof which backed Lambert up. This new proof also went as far as to prove that Pi^2 was also irrational. In the long history of the number Pi, there have been many twists and turns, many inconsistencies that reflect the condition of the human race as a whole. Through each major period of world history and in each regional area, the state of intellectual thought, the state of mathematics, and hence the state of Pi, has been dictated by the same socio-economic and geographic forces as every other aspect of civilization.

The following is a brief history, organized by period and region, of the development of our understanding of the number Pi. A transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+… +cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic. It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental.

This feat was finally accomplished for Pi by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler. In 1873, Hermite proved that the constant e was transcendental. Combining this with Euler’s famous equation e^(i*Pi)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, Pi had to be transcendental in order to make i*Pi transcendental. Now that I have explained what Pi is and several of its properties, lets look at its history.

In ancient times, Pi was discovered independently by the first civilizations to begin agriculture. Their new sedentary life style first freed up time for mathematical pondering, and the need for permanent shelter necessitated the development of basic engineering skills, which in many instances required a knowledge of the relationship between the square and the circle (usually satisfied by finding a reasonable approximation of Pi). Although there are no surviving records of individual mathematicians from this period, historians today know the values used by some ancient cultures.

Here is a sampling of some cultures and the values that they used: Babylonians – 3 1/8, Egyptians – (16/9)^2, Chinese – 3, Hebrews – 3 (implied in the Bible, I Kings vii, 23). The first record of an individual mathematician taking on the problem of Pi (often called “squaring the circle,” and involving the search for a way to cleanly relate either the area or the circumference of a circle to that of a square) occurred in ancient Greece in the 400’s B. C. (this attempt was made by Anaxagoras).

Based on this fact, it is not surprising that the Greek culture was the first to truly delve into the possibilities of abstract mathematics. The part of the Greek culture centered in Athens made great leaps in the area of geometry, the first branch of mathematics to be thoroughly explored. Antiphon, an Athenian philosopher, first stated the principle of exhaustion (click on Antiphon for more info). Hippias of Elis created a curve called the quadratrix, which actually allowed the theoretical squaring of the circle, though it was not practical. In the late Greek period (300’s-200’s B.

C. ), after Alexander the Great had spread Greek culture from the western borders of India to the Nile Valley of Egypt, Alexandria, Egypt became the intellectual center of the world. Among the many scholars who worked at the University there, by far the most influential to the history of Pi was Euclid. Through the publishing of Elements, he provided countless future mathematicians with the tools with which to attack the Pi problem. The other great thinker of this time, Archimedes, studied in Alexandria but lived his life on the island of Sicily.

It was Archimedes who approximated his value of Pi to about 22/7, which is still a common value today. Archimedes was killed in 212 B. C. in the Roman conquest of Syracuse. In the years after his death, the Roman Empire gradually gained control of the known world. Despite their other achievements, the Romans are not known for their mathematical achievements. The dark period after the fall of Rome was even worse for Pi. Little new was discovered about Pi until well into the decline of the Middle Ages, more than a thousand years after Archimedes’ death.

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Home » History » The History of Pi

# The History of Pi

A little known verse in the bible reads And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits; and a line of thirty cubits did compass it about(I Kings 7, 23). This passage from the bible demonstrates the ancient nature of the irrational number pi. Pi in fact is mentioned in a number of verses throughout the bible. In II Chronicles 4,2, in the passage describing the building of the great temple of Solomon which was built around 950BC, pi is given as equal to three.

This value is not very accurate at all and should not even be considered accurate for its time, however it should be noted that precision was not needed for the task that was being performed and we should let the general concept of pi that the biblical characters posses impress us. Present knowledge suggests that the concept of pi first developed in 2000 BC in two separate cultures. The Babylonians used pi at a value of 25/8 while an entirely different culture, the ancient Egyptians used pi at a value of 256/81.

While the biblical calculation of pi=3 most likely came from crude measurement, there is strong reason to believe, because of the relative accuracy of the values, that the Babylonians and Egyptians found pi by means of mathematical equations. In the Egyptian Rhind Papyrus, which is dated around 1650 BC, there is strong evidence supporting that the Egyptians used 4(8/9)2 =3. 16 for their value of pi. At that point in history, and for the majority of modern history, pi was not seen as an irrational number as it is today. The next culture that investigated pi was the ancient Greeks.

Starting in 434 BC Greeks were unraveling the mysteries of pi. The mathematician Anaxagoras made an unsuccessful attempt at finding pi, which he called squaring the circle and in 414 BC, 20 years after Anaxagoras failed in his attempt to square the circle, Aristophanes refers to the work of Anaxagoras in his comedy The Birds. It took over 100 years for the Greeks to finally find a value for pi. In 240 BC Archimedes of Syracuse showed that 223/71*pi*22/7. Archimedes knew, what so many people today do not, that pi does not equal 22/7 and he made no claim to have discovered the exact value of pi.

However if we take the average of his two bounds we obtain pi=3. 1418, which was an error of about 0. 0002. Archimedes found the most accurate value of pi up to that time and his value would be used exclusively until the next discovery in the world of pi. The next major finding concerning pi did not occur in the western world, but in China by Tsu Chung-chih who approximated pi at 355/113 in 480 AD. Next to nothing except for this work is known about Tsu Chung-chihs life but it is very unlikely that he had any awareness of Archimedes work.

We shall now notice how during the dark ages of Europe, the lead in the research of pi is passed to the East. Aryabhata, working on his own in Persia without any outside information in 515 AD was able to approximate pi to 3 decimal places. A mathematician from Baghdad named AlKhwarizimi worked with pi however the most accurate finding of pi to date was found even more east in Samarkand by Al-Khashi. In 1430 AD he approximated pi to 16 decimal places, the most to date. His work however, would be the last of note from the east as the European Renaissance brought about a whole new mathematical world.

The first notable discovery in the approximation of pi from the European Renaissance was by Viete in 1593 AD. He expressed pi as an infinite product by using only 2s and square roots. In 1610 Ludolph van Ceulen demonstrated the new thought coming out of the Renaissance by calculating pi to 35 decimal places. Around the same time, Snell refined Archimedess method of calculating pi, and Snells work was used by Grienberger to calculate pi to 39 decimal places in 1630.

The 18th centuary brought about great achievements in the calculating of pi. In 1706, Machin found pi to 100 decimal places, the first time that feat was ever achieved and in the same year, a British mathematician, William Jones first used pi for the circle ratio. In 1737, Euler first used the Greek letter pi to represent the mysterious number therefore giving it its present day name. Up until the 18th centuary, pi was seen as a rational number, however in 1761, Lambert showed that pi was irrational, therefore opening up a whole new world for the research of pi.

Pi became seen as a boundless number, open for limitless exploration. Soon after Lamberts discovery, Legendre showed that pi2 is irrational. The 19th centuary presented two mathematicians, who, without computers were able to find pi to huge amounts of decimal places. In 1844, Johann Dase, who was described by his contemporaries as the lightning calculator found pi to 200 decimal places. Shanks soon overshadowed Dases findings however by finding by to an astounding 707 decimal places in 1873.

While the 19th centuary showed great strides in the calculation of pi, the 20th centuary, with the advent of computers, broke great barriers in finding the most exact value of pi. In 1945, two scientists, Ferguson and Wrench worked on a computer system for calculating pi, however before this system was perfected, they did some manual calculations. In 1945 Ferguson found that the number occupying the 528th place for Shanks value of pi was incorrect. Soon after in 1948, Ferguson and Wrench published the correct value of pi to 808 decimal places.

However in 1949, with their computer up and running, Ferguson and Wrech were able to find pi to its most exact value ever. Their ENIAC system performed the first electronic computation of pi to 2,037 decimal places. It is interested to not that this computer occupied a warehouse the size of a high school fieldhouse and its only purpose was to calculate pi, however the computer represented a huge jump in the research of pi. It opened doors to the intricate calculations of pi we see in our modern day. From this point on, all new calculations of pi would be done electronically.

In 1958, Genuys found pi to 10,000 decimal places, and in 1962 David Shanks, a relative of the 19th centuary mathematician William Shanks, along with Wrench found pi to 100,000 decimal places. In 1973 Guillard and Bouyer were the first to find pi to one million places. The research in pi in the 1980s to the present has pretty much moved across the pacific to Japan. In 1982, Y Tamura and Y Kanada found pi to 8 million places and in 1986 Kanada found it to 33,554,000 places;in 1987 134,217,728 places and in 1988 he found pi to 201,326,000 places.

In November of 1989 Kanada brought the one billion mark by finding pi to 1,073,741,799 places. The great year of 1995 however made the most progress in the calculation of pi. Kanada found pi to 4 billion places, and soon after Borwien, a German mathematician found pi to 10 billion places, a great leap from the biblical approximation of 3. Today you can download files off the internet of values of pi to 2. 5 million places. On the next page you can examine pi to 50,000 places, a relatively low number for todays standards, however still impressive in its own way.

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