What is Pi really?

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This is a part of presentation by @Eli during the Pi Day on Saturday March 14, 2015 at Mnazi MMoja in Dar Es Salaam, Tanzania.

Today is the Pi Day, so I find it worth sharing with you a brief but comprehensive history of pi.

Introduction

The mathematical constant \(\pi\) is the ratio of a circle's circumference to its diameter and is commonly used in mathematics, physics and engineering. The name of the Greek letter \(\pi\) is pi (pronounced pie), and this spelling can be used in typographical contexts where the Greek letter is not available. \(\pi\) is also known as Archimedes' constant. It is so called because Archimedes devoted a lot of energy and time and made considerable contribution during its development. The symbol for \(\pi\) was introduced by the English mathematician William Jones in 1706, who wrote

\(3.14159 = \pi.\)

In Euclidean plane geometry, \(\pi\) may be defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Advanced textbooks define \(\pi\) analytically using trigonometric functions, for example as the smallest positive \(x\) for which sin(\(x\)) = \(0\), or as twice the smallest positive \(x\) for which cos(\(x = 0\)).

The numerical value of \(\pi\) rounded to 100 decimal places is:

\(3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679\).

Although this precision is more than sufficient for use in engineering and science, much effort over the last few centuries has been put in computing more digits and investigating the number's properties, testing algorithms, for the purpose of record breaking and in testing performance and precision of supercomputers. Despite much analytical work, in addition to supercomputers calculations that have determined over 10 trillion digits of pi, no pattern in the digits, such as infinitely repeating patterns of digits, has ever been found! The sequence of digits in pi has passed all known tests for statistical randomness.

Like all other irrational numbers, \(\pi\) cannot be represented as common (simple) fraction. But \(\pi\) can be represented by infinite series of nested fractions, called continued fractions and mathematicians have discovered several which exhibit certain patterns, for example

\begin{align}\nonumber
\Large
\frac{4}{1+\frac{1^{2}}{3+ \frac{2^{2}}{5+ \frac{3^{2}}{7+\frac{4^{2}}{9 + \frac{5^{2}}{11+\frac{6^{2}}{\dots}}}}}}}\end{align}

Truncation of the continued fraction at any point generates a value that is an approximation for \(\pi\).

Evolution of Pi

\(\pi\) has been known for almost 4000 years. The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave the value of \(\pi = 3\).

The ratio of the circumference to the diameter of a circle is constant, namely pi, and this has been recognized for as long as we have written records. A ratio of 3:1 appears in the following little famous Biblical verse which 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)."

The same verse appears in II Chronicles 4: 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and it is interesting here it gives \(\pi = 3\). Although this value of course and not even very accurate in its days, the fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. Rabbi Nehemiah (Israelite mathematician, 150 AD) maintained that the discrepancy in the value of pi was due to the thickness of the vessel.

One Babylonian tablet (ca. 1900 - 1680 BC) indicates a value of 3.1250 for \(\pi\). In the Egyptian, Rhind Papyrus (ca. 1650 BC), there is evidence that Egyptians and Mesopotamians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi. Early successful attempts to calculate \(\pi\) were made by Archimedes who used up to 96-sided polygons to inscribe and circumscribe a circle in calculating \(\pi\), Liu Hiu used up to 3,072-sided polygon to calculate pi to 3.14159 and Tsu Ch'ung and his son inscribed polygons with as many as 24,576 sides to calculate \(\pi\).

Properties of Pi

\(\pi\) is an irrational number; that is, it cannot be written as the ratio of two integers, as was proved in 1761 by Johann Heinrich Lambert; \(\pi\) is also transcendental, as was proved by Ferdinand Von Lindemann in 1882. This suggests that, there is no non-constant polynomial with rational coefficients of which \(\pi\) is a root. An important consequence of the transcendence of \(\pi\) is the fact that it is not constructible! That is, it is impossible to construct, using rule and compass alone, a square whose area is equal to the area of a given circle. However, \(\pi\) can be represented as infinite series.

Pi Development

The first theoretical calculation of pi seems to have been carried out by Archimedes of Syracuse (287 - 212 BC).

In one of his famous work Archimedes considered a circle of unit radius inscribing a regular polygon of \(2\times 2^{n-1}\) sides, with semiperimeter \(b_n\), and superscribing a regular polygon of \(3\times 2^{n-1}\) sides, with semiperimeter \(a_n\). Archimedes used up to the polygon of 96 sides, and went through very sophisticated arguments and ideas that he centred out about pi. His conclusion was that

\(\frac{223}{71}<\pi<\frac{22}{7}\).

If we take his best estimate as the average of his two bounds we obtain 3.1418, with an error of about 0.0002 compared to real value.

Archimedes did not has the advantage of an algebraic and trigonometrical notation and had to make all the mathematical derivations by purely geometrical means. So it was pretty stupendous feat both of imagination and of calculation, and the wonder is not that he stopped with polygons of 96 sides, but that he went so far!

Several people were committed enough to devote vast amount of time and effort to this tedious and seemingly wholly useless pursuit. With the available formula like the one developed by Archimedes, the only difficulty in computing \(\pi\) is the sheer boredom of continuing the calculations. On the other side, there is no reason in principle why one should not go on. Various people did, including: Ptolemy (ca. 150 AD), Zu Chongzhi (430 - 501 AD) who approximated pi to \(\frac{355}{113}\), Al-Khwarizmi (ca. 800), Al-Kashi (ca. 1430), Vite (1540 - 1603), Roomen (1561 - 1615), Van Ceulen (ca. 1600), Wallebrord Snellius (1616 - 1703) and James Gregory (1638 - 1675). In \(18^{\rm th}\) century, a French mathematician named
Georges Buffon devised a way to calculate \(\pi\) based on probability.

Although mathematicians began using the Greek letter \(\pi\) in the 1700s, use of the symbol was popularized by Euler who adapted it in 1737.

An improvement continued, and in 1699, Sharp used Gregory's result to get 71 correct digits of \(\pi\); in 1701, Machin made an improvement to get 100 digits; and the following used his method: 1719: De Lagny calculated correctly 112 digits, 1789: Vega got 126 decimal places and in 1794 he got 136, 1841: Rutherford calculated 152 digits and in 1853 he calculated 440 digits, 1873: Shanks calculated 707 decimal places of which 527 were correct.

Pi Day

Many scholars consider pi to be the most intriguing number, and its fascination has even popularly been carried over into non-mathematical community. The world is just crazy about pi! There are pi poems, pieces of music based on the digits of pi, people who have memorized up to 67,980 digits (Guinness-recognized record, by Luo Chao, a 24-year old graduate student from China) of pi. There are people claimed to have memorized up to 100,000 digits (Akira Hagaruchi, a retired Japanese engineer, 2006), Andriy Tychonovych Slyusarchuk, A Ukrainian neurosurgeon, nicknamed Dr. Pi who could recite random sequences of pi in a range of 30 million digits. There is even a website where you can find your birthday in pi. However, record settings in calculation of \(\pi\) digits have often resulted in news headlines!

The history of \(\pi\) has paralleled and escalated the development of mathematics from ancient period to classical era to digital age. \(\pi\) is now tremendously used in sciences, mathematics, physics, engineering and other fields.

The history of \(\pi\) is very amazing and has so many twists and turns. Nowadays, there are two days held in honor of the mathematical
constant \(\pi\): Pi Day and Pi Approximation Day.

March 14, written 3-14 in the US date format, is an unofficial celebration for Pi Day derived from the common 3-digit approximation for the number \(\pi\): 3.14. It is usually celebrated at 1: 59 PM in recognition of six-digit approximation: 3.14159.

This day has been celebrated in a variety of ways, groups of people typically pi clubs, exchange thoughts about the role that the number \(\pi\) has played in their lives and imagine the world without \(\pi\)! Enthusiasts also note that the day happens to be Albert Einstein's birthday (1789). The "ultimate" first Pi Day occurred on March \(14^{th}\), 1592, at 6:53 AM and 59 seconds, which corresponds to the first 12 digits 3.14159265359.

Pi Approximation Day is one of two days: either July 22 or April 26 (April 25 on leap year), the day on which planet earth completes two astronomical units worth of its annual orbit. On this day, the total length of the earth's orbit, divided by the length already travelled, equals \(\pi\).

John Von Neumann used ENIAC to compute 2037 digits of \(\pi\) in 1949, a calculation that took him 70 hours. With increasingly computer technology today, the value of pi can be computed to more finer precision in a very short time.
Disclaimer: Most of these images are from Wikipedia.
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Today is PI Day

Pi Approximation Day is one of two days: either July 22 or April 26 (April 25 on leap year), the day on which planet earth completes two astronomical units worth of its annual orbit. On this day, the total length of the earth's orbit, divided by the length already travelled, equals .
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