Pi
The number pi (denoted with the lower-case Greek letter π) is a mathematical constant which occurs in many areas of mathematics and physics. It is also known as Archimedes' constant or Ludolph's number and is equal to the ratio of a circle's circumference to its diameter in Euclidean geometry. Alternatively, π can be defined as the area of a circle of radius 1, or as the smallest positive number x for which sin(x) = 0.The value of π to the first sixty-four digits is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 592...
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2 Formulae involving π 3 Computing the value of pi 4 Open questions 5 The nature of π 6 Pi culture 7 Related articles 8 External Links |
Properties
The number π is an irrational number: that is, it cannot be written as the ratio of two integers. This was proved in 1761 by Johann Heinrich Lambert. In fact, the number is transcendental, as was proved by Ferdinand Lindemann in 1882. This means that there is no polynomial with integer (or rational) coefficients of which π is a root. As a consequence, it is impossible to express π using only a finite number of integers, fractions and their roots.
This result establishes the impossibility of squaring the circle: it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The reason is that the coordinates of all points that can be constructed with ruler and compass are special algebraic numbers.