**The Story of p **

By :

Wrutheran Sinnadurai

Unit Matematik, MPKTBR

The number p is the ratio of the circumference of a circle to its diameter. This ratio is the same for all circles and cannot be expressed exactly as a decimal. A practical approximate value for p that is often used is 3.1416 or 3 6 . Apart from circles, p also appears in problems related to certain surface areas and volumes in solid geometry. However, the use of p is not restricted to geometry alone; many branches of mathematics and physics, such as number theory, statistics, acturial theory, vibrational theory and alternating electric currents, engage the number p .

The use of the number** p **goes as far back as around 2000 B.C. The Egyptians used the value of 3.16 in some of their calculations. However the concept of p among the Egyptians was rather vague. In the Old Babylonian period (c. 1800 - 1600 B.C.) the circumference of a circle was found by taking three times its diameter. This gave p a value of 3. The Hebrews, according to biblical accounts used the same value.

It was the Greeks who showed concerted interest in determining the value of p . Hippocrates (c. 440 B.C.), Dinostratus ( c. 350 B.C.) and Euclid (c. 300 B.C.) all attempted to find values for p . But it was Archimedes who demonstrated ingenuity in arriving at a fairly accurate value. In his work `The Measurement of the Circle’ (c. 240 B.C.), he proved that the area of any circle is equal to the area of a right triangle with one leg equal to the radius and the other, equal to the circumference of the circle. He further showed that p had a value between 3 ^{10}/_{71} and 3 ^{10}/_{70} (or 3 6 ). This he obtained from inscribed and circumscribed polygons, continually doubling the number of sides until the perimeter of the polygon of 96 sides was obtained. As a result of his contribution the value 3 6 has often been called the `Archimedean value of p ’. Archimedes’ method of calculating p is known as the `classical method’. After him Claudius Ptolemy of Alexandria (c. 150 A.D). From his extensive work in astronomy, he developed a table which gave the lengths of the chords of a circle subtended by central angles of each degree and half-degree. From this table he offered the value of p in sexagesimal notation as 3 8’ 30" which translates into a value of ^{377}/_{120} or 3.1416.

Approximations for p have also been offered by mathematicians of other countries. In China, for example, many mathematicians had worked with p . Liu Hsin (c. 23 A.D.) used the value 3.1547, while Chang Heng (c. 78 – 139) used ¶ 10 or the fraction 92/29. Using regular inscribed polygons the third century mathematician used a polygon of 389 sides to derive the value 3.141024 < p < 3.142904. With a 3072-sided polygon he found his best value of 3.14159. In the fifth century, the brilliant Tsu Chung-Chi (430 – 501) refined the method to obtain 3.1415926 < p < 3.1415927. From this he suggested the fraction 22/7 as the ‘inaccurate’ and 355/113 as the ‘accurate’ values of p .

Two outstanding Indian mathematicians known to have done some work with p were Aryabhata (c. 530) and Bhaskara (c. 1150). Aryabhata’s calculation of p translates as follows:

Add four to one hundred, multiply by eight and then add sixty two thousand; the result is approximately the circumference of a circle of diameter twenty thousand. In other words,

p = __circumference__ = __8(100 + 4) + 62000__

diameter 2000

= __62832__ = 3.1416

20000

**Baskara gave several approximations for **p . As his accurate value he gave 3927/1250, 22/7 was his inaccurate value and ¶ 10 was his practical value for ordinary work.

The advent of the Renaissance saw the mushrooming of mathematicians in Europe. The sixteenth and seventeenth centuries witnessed many mathematicians pursuing the value of p . However, the earlier work was mainly confined to the `classical method’ introduced by Archimedes.Using this method the French mathematician Francois Viete (1579) used polygons having 6 (2^{16}) or 393216 sides to compute p to nine decimal places and Adrianus Romanus (1593) of Holland used polygons of 2^{30} sides to obtain p correct to 15 decimal places. Ludolf van Coulen (1610) of Germany spent a large part of his life on this task and using polygons of 2^{62} sides computed p to 35 decimal places. In honour of his work, this number to this day is frequently referred to as the `Ludolphine number’. In 1621 the Dutch physicist, Snell refined the classical method to obtain van Coulen’s 35 decimal places with only a polygon of 2^{30} sides. Grienberger (1630) used Snell’s method to compute p to 39 decimal places making it the last major attempt to compute p by the classical method.

Starting in the mid-seventeenth century mathematicians began to look at infinite series to obtain p . In 1650 the English mathematician John Wallis suggested the expression

p **= **__2 . 2 . 4 . 4 . 6 . 6 . 8 ……..__

2 1 . 3 . 3 . 5 . 5 . 7 . 7 ……

James Gregory (1671) of Scotland obtained the series

Arctan x = x - x^{3} + x^{5} - x^{7} + …….

3 5 7

which Leibniz (1674) modified to

1/4 - 1/3 + 1/5 - 1/7 + .....

**Using Gregory’s series, Abraham Sharp (1699) found **p to 71 correct decimal places with x = ¶ 2 . John Machin (1706) improved that with 100 decimal places using

p /4 = 4 arctan (1/5) - arctan (1/239)

**William Shanks of England used Machin’s equation to compute **p to 707 places in 1873 after 15 years’ work.

With the advent of electronic computers, computation of p became much easier and faster. In 1949 ENIAC computed p to 2037 places in seventy hours. It also detected an error in Shanks’ work in the 528^{th} place.

The most notable contributor of the 20^{th} century to the calculation of p was the Indian mathematical genius, Ramanujam. Although he died at the age of 32, he made many noteworthy contributions to mathematics. He formulated 14 series for the computation of p . Two of these series are:

Both these series have been exploited to calculate p to millions of decimal places using modern day computers.

The adoption of the symbol p for the ratio (circumference / diameter) is essentially due to the usage given it by Leonard Euler from 1736 onwards. However, he was not the first to use it; that credit has been attributed to the English writer, William Jones, in a publication in 1706.

Johann Lambert (1761) was the first to prove that p is an irrational number – a number that is a non-repeating, non-terminating decimal. Ferdinand Lindemann (1882) was the first to prove that p is a transcendental number – an irrational number that cannot be the root of any algebraic equation with rational coefficients. His proof was partly based on Euler’s equation

e^{i}p + 1 = 0

which has been said to contain the five most significant numbers in all of mathematics.

References

Burton, D.M. (1995). *History of mathematics: An introduction. 3 ^{rd} ed. *Dubuque,USA: Wm. C Brown Publishers.

*Encyclopaedia Americana* (1978) Canada: Grolier Ltd. Eves, H. (1964*). An introduction to the history of mathematics*. New York:

Holt,Rinehart & Winston National Council of Teachers of Mathematics (1989*). Historical topics for the mathematics classroom*. USA: N.C.T.M.