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Happy Pi Day!

The mathematical constant Pi is a real number which is defined as the ratio of a circle's circumference (Greek Piεριφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. The name of the Greek letter Pi is pi (pronounced pie) in English. This spelling can be used in typographical contexts where the Greek letter is not available. Pi is also known as Archimedes's constant (not to be confused with Archimedes's number) and Ludolph's number.

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. All these definitions are equivalent.

The numerical value of Pi, truncated to 50 decimal places (sequence A000796 in OEIS), is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

Although this precision is more than sufficient for use in engineering and science, the exact value of Pi has decimal places that never end. Much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, in addition to supercomputer calculations that have determined over 1 trillion digits of Pi, no pattern in the digits has ever been found. Digits of Pi are available from multiple resources on the Internet, and a regular personal computer can compute billions of digits with available software.

Pi is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert.

Pi is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no 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. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle.

The value of Pi has been known in some form since antiquity. As early as the 20th century BC, Babylonian mathematicians were using Pi=25/8, which is within 0.5% of the exact value.

It is sometimes claimed that the Bible states that Pi=3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular.

The most pressing open question about Pi is whether it is a normal number -- whether any digit block occurs in the expansion of Pi just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of Pi.

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulæ imply that the normality in base 2 of Pi and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.

It is also unknown whether Pi and e are algebraically independent. However it is known that at least one of Pie and Pi + e is transcendental.

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than Pi radians, and the ratio of a circle's circumference to its diameter may also differ from Pi. This does not change the definition of Pi, but it does affect many formulæ in which Pi appears. So, in particular, Pi is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics.

The above stolen liberally from the Wikipedia article on the subject. The text of this entry falls under the GNU Free Documentation License. Now you are edified. Go forth and spread the wisdom of Pi.

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