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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