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Introduction
Pi is a mathematical constant that expresses the ratio of the circumference to the length of its diameter. Denoted by the Greek letter π.
First symbol of this number by the Greek letter π was taken by British mathematician Jones in 1706, and it became common after the works of Leonhard Euler in 1737.
This designation comes from the initial letters of the Greek words περιφέρεια - circle, peripherals and περίμετρος - perimeter.
History
History of the number π ran parallel with the development of mathematics. Some authors divide the process into 3 periods: the ancient period during which π was studied from the standpoint of geometry, classical era, which followed the development of mathematical analysis in Europe in the XVII century and the era of digital computers.
Prior to Millennium II was known to 10 digits π. Further major advances in the study of π associated with the development of mathematical analysis, especially with the opening of the series, allowing to calculate π with any accuracy by summing appropriate amount of the series. In the 1400s Madhava of Sangamagrama (born Madhava of Sangamagrama) found the first of these series:
Madhava was able to calculate π as 3.14159265359, correctly identified 11 digits in the number. This record was broken in 1424 by the Persian mathematician Jamshid al-Kashi, who, in his work entitled "A Treatise on the circle" led 17 digits of π, 16 of which are true.
Period of digital technology in the XX century led to an increase in the rate of appearance of computer records. John von Neumann and others used in 1949 ENIAC to compute 2037 digits of π, which took 70 hours. Another one thousand digits was obtained in the following decades, and the mark of one million was passed in 1973. This progress was not only due to faster hardware, but also thanks to algorithms. One of the most significant results was the discovery in 1960, the fast Fourier transform, allowing you to quickly arithmetic operations on very large numbers.
Irrationality and Transcendence
π is an irrational number, that is, its value cannot be expressed exactly as a fraction m / n, where m and n - integers. Consequently, its decimal representation never ends and is not periodic. Irrationality of π was first proved by Johann Lambert in 1761, through an expansion of the number in a continued fraction. In 1794 Legendre led more rigorous proof of the irrationality of π and π^2.
π is a transcendental number, which means it cannot be the root of any polynomial with integer coefficients. Transcendence of π was proved in 1882 by Professor Koenigsberg, and later the University of Munich Lindemann. Proof simplified Felix Klein in 1894.
As in Euclidean geometry and area of a circle circumference are functions of the number π, then the proof of the transcendence π put an end to the dispute of squaring the circle, which lasted more than 2500 years.
October 19, 2011 Alexander Yee and Shigeru Kondo timed sequence with an accuracy of 10 trillion digits after the decimal point.
Expressions
There are many formulas of π:
Viete’s formula:
Wallis’s formula:
Leibnitz’s series:
Other series:
Limits:
And so on and so forth.
Unsolved Problems
- It is unknown whether the number pi and e are algebraically independent.
- It is unknown the exact numbers for the measure of irrationality and (but we know that for does not exceed 7.6063)
- Unknown irrationality measure for any of the following numbers: Neither one of them is not even known whether it is a rational number, irrational algebraic or transcendental number.
- So far nothing is known about the normality of the number; do not even know which of the numbers 0-9 are found in decimal number infinite number of times.
Works Cited
Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2 ed.). Wiley. ISBN 978-0-471-54397-8.
Bronshteĭn, Ilia; Semendiaev, K. A. (1971). A Guide Book to Mathematics. H. Deutsch. ISBN 978-3-871-44095-3.
Blatner, David (1999). The Joy of Pi. Walker & Company. ISBN 978-0-8027-7562-7.
