Computation of p
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Introduction Long ago men already knew that the circumference of a circle increases as the diameter increases. They were even aware that the circumference of a circle was roughly proportional to the diameter of the circle, with a proportionality constant of approximately 3. There was an old Chinese saying that "a diameter of 1 corresponds to a circumference of 3". In the Bible, the value of p was also taken to be 3. |
Approximation by Areas of Regular Polygons
Measurement of a Circle by Archimedes
Archimedes (287 - 212 BC) was the first one to approximate the value of p by approximating the area of a circle. He noticed that the area of a regular polygon inscribed in a circle provides a lower limit of the area of a circle, and that the area of a regular polygon circumscribing a circle provides an upper limit. Furthermore, as the number of sides of the polygon increases, these estimations become more and more accurate. In this way, he gave in his Measurement of a Circle the approximation 22/7.
Liu Hui's "Principle of Exhaustion for Circles"
In around 220 AD, during the Wei-Jin dynasty in China, the mathematician Liu Hui computed the area of a circle by the so-called "principle of exhaustion for circles". The method was to mark several equally spaced points on the circumference of a circle and then connect them to form a regular polygon inscribed in the circle. When the number of points increases, the area of the inscribed polygon gets closer to the area of the circle. By repeating this process indefinitely, we eventually get the area of the circle. The value of p obtained by Liu Hui was 3.14.
Breakthrough by Zu Chongzi
Another Chinese mathematician Zu Chongzi (430 - 501) approximated p as
van Ceulen computed p up to 35 digits
In 1610, the German mathematician Ludolph van Ceulen (1540 - 1610) computed the area of a polygon with 262 sides to obtain the value of p up to 35 decimal places. In honour of his contribution in calculating the value of p, the following result was engraved on his tombstone:
。
Infinite Product Representation of p
Viète's infinite product representation of p
In 1579, the French mathematician François Viète (1540 - 1603) first represented p as an infinite product. He did that by calculating the areas of regular 4-gons, 8-gons, 16-gons, etc., that are inscribed in a circle with diameter 1.
He found that the area of the inscribed regular 4-gon is given by
.
Next, the area of the regular 8-gon is given by
.
Viète discovered, by continuing to double the number of sides of the inscribed polygon, the formula
.
Therefore,
.
This is known as Viète's Formula. However, there are too many square roots in this formula, which makes it not very practical as a means to compute the value of p.
Breakthrough by Wallis
In 1655, the English mathematician John Wallis (1616 - 1703) established another infinite product representation of p:
.
This is obviously much simpler than Viète's Formula.
Infinite Series Representation of p
Gregory's Series
In 1671, the Scottish mathematician James Gregory (1638 - 1675) discovered a famous formula known as the Gregory's formula:
.
Putting x = 1, we obtain
.
By calculating the first two terms, we have 
; by calculating the first three terms, we get 
; by calculating the first four terms, we get 
. The convergence is so slow that we need to calculate the first 5000 terms before we can get an accuracy of 5 decimal places! Nevertheless, this is the second major breakthrough in the computation of p. The convergence is so slow that we need to calculate the first 5000 terms before we can get an accuracy of 5 decimal places! Nevertheless, this is the second major breakthrough in the computation of p.
Machin's Formula
In 1706, English astronomer John Machin (1680 - 1751) discovered that
.
Using Gregory's formula, this becomes
.
The above expression converges much faster; we get an accuracy of 100 decimal places by calculating only the first six terms.
Other infinite series representations of p
Later on, many mathematicians evaluated p to more and more decimal places. For instance, von Vega (1754 - 1802) got 140 digits in 1789, of which 126 were correct; William Rutherford got 208 digits in 1841, of which 152 were correct; Thomas Clausen (1801 - 1885) got 248 digits in 1847; William Shanks (1812 - 1882) got 707 places in 1873, of which 527 were correct; whereas D. F. Ferguson and J. W. Wrench got 808 places in 1948.
Miracles brought by computers
The latest great breakthrough in the computation of p was due to the invention of computers. We still use our old formulae; only that computers are far more efficient than the human brain in all those lengthy calculations.
With the use of modern computers, the value of p is already computed to over 1.2 trillion digits by 2003.
Reference: Yuan, Xiaoming. Shu Xue Dan Sheng De Gu Shi (Stories about the birth of mathematics). Publisher: Chiu Chang Publishing Company.