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Expert comment: National Pi Day

Dr Anna Kirpichnikova Tuesday 14 March 2017

Senior Lecturer in Mathematics Dr Anna Kirpichnikova joins the National Pi Day celebrations.

Pi is believed to be the most well known mathematical constant; it is equal to the ratio of the circle’s circumference to its diameter (in the flat Euclidean geometry). Easily visualised and extremely practical due to its immediate geometrical origin, it has become a challenging problem to calculate it with necessary precision. It gave rise to the development of science in many areas, such as geometry, trigonometry, number theory, statistics, cosmology, thermodynamics, mechanics, and electromagnetism.

The confirmed history of Pi dates back to the first millennium BC in China, then followed by Egypt, Babylon and Ancient Greece. Some mathematicians split the history of Pi into three major eras. The first one was the search for the approximate value due to its needs in geometry; the second one gave a push to the development of the mathematical analysis and resulted in a pool of infinite series identities to calculate an approximate value of Pi (the more terms we use, the closer we get to Pi). The third and current stage of the interest in Pi is related mostly to computer science (cybersecurity).

In the XVIII-th century the story of Pi has deviated from getting the best possible approximation to looking at Pi’s special properties. An ancient Greek letter  has been adopted and became popular after Leonard Euler’s works on the connection between Pi and prime numbers, which advanced in to the development and study of Riemann zeta function.

In 1761, Pi was proved to be irrational, namely it is not a rational number (the quotient of two whole numbers), which concluded thousands years of attempts to find such two whole numbers. Later, in 1882, Pi was also proved to be transcendental (genius Euler had suspected that more than a century before). A transcendental number means that it cannot be a solution of any polynomial equation with rational coefficients. The latter property finished attempts to represent Pi as any finite combination of rational numbers and surds, as this representation is impossible for transcendental numbers.

The race for the number of digits in Pi’s decimal approximation resumed again with the start of computers. John von Neumann calculated 2037 digits in 1949 (it took 70 hours for the computer) and 1 million digits was reached in 1973. With the growth of computer power, calculating more digits initiated development of numerical schemes that converge much better, i.e., get closer to Pi faster, comparing to infinite series used before. Since the existing numerical algorithms work pretty well, the success of getting more digits of Pi just follows the advances in computer power, so it is now more a kind of competition, rather than the art of mathematics. Sadly, sometimes mathematics and mathematical skills are perceived in line with the passion to speed up calculations. Most of the time, however, this is not what mathematics is about.

The sequence of Pi’s digits passes statistical tests for randomness. It also contains some sequences of digits that may appear non-random, say, solutions to a class test (infinite monkey theorem).

Pi left its footprint in most of the areas of science and it is still being investigated mathematically, for example, not much is known about the relationship between Pi and another famous mathematical constant e (the base of natural logarithm). We still do not know even if  is rational.

Having said all that, Pi appears to be quite simple in terms of Kolmogorov’s complexity. This is defined as the length of the shortest text that describes the object. Indeed, the complexity of Pi is then equal to the length of the sentence “the ratio of the circle’s circumference to its diameter”. Not too long, not too complex.

Pi Day is celebrated twice a year: on the 14th March (in American date format this day looks like the decimal approximation of Pi: 3.14) and as Pi Approximation Day on the 22nd of July (in European date format this day is 22.7, which recalls the rational approximation of Pi, 22/7). On the 14th March one can go even further: the next few digits in Pi’s decimal approximation are 3.14159, and one can start celebrating at 1:59. 

Have a Happy Pi’s Day!

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