Pi's transcendence forbids squaring the circle by compass alone
Pi (π) is the constant ratio of a circle's circumference to its diameter—about 3.14159… Irational and transcendental, it never repeats as a decimal and is not a root of any nonzero rational polynomial. That transcendence proves classical squaring of the circle impossible. Archimedes chased it about 250 BC.
The usual definition divides circumference by diameter, a ratio that stays fixed whatever the circle's size: double the diameter and the circumference doubles too. Because arc length needs limits to define rigorously, mathematicians have also defined π in other ways. Karl Weierstrass did so in 1841 with an integral, and π can equally be pinned down through the complex exponential, as the unique positive real number for which exp z equals 1 precisely at the integer multiples of 2πi.
The constant turns up in many formulae across mathematics and physics, which is why some of them double as alternative definitions. It is irrational, so no ratio of two integers equals it; 22/7 and 355/113 come close without being exact. It is also transcendental, a consequence of the Lindemann–Weierstrass theorem, which settles the same question for e. The symbol is lowercase Greek pi, said like pie in English, and is kept distinct from capital Π, which marks a product the way Σ marks a sum.
Egyptians and Babylonians used working approximations, Archimedes devised an algorithm around 250 BC that could reach any desired accuracy, and Chinese mathematicians had seven digits by the fifth century AD. Computers have since supplied vast numbers of digits, which show no apparent pattern and pass tests for randomness. Yasumasa Kanada's statistical analyses found the frequencies of 0 through 9 consistent with normality, meaning every string of digits appears equally often, but whether π is actually normal remains unproven.
Source: Pi
