14 March 2014

Pi

Today is Pi day. And it is a sort of special one being that it is 3.14.14
π


A diagram of a circle, with the width labeled as diameter, and the perimeter labeled as circumference

The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letter π, sometimes spelled out as pi, particularly when foreign fonts are not available.

In English, π is pronounced as "pie"

In mathematical use, the lower-case letter π is distinguished from the capital letter Π, which denotes a product of a sequence.


π is commonly defined as the ratio of a circle's circumference C to its diameter d
 \pi = \frac{C}{d}


The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.

It has been represented by the Greek letter "π" since the mid-18th century. According to Wikipedia's entry for the word, the earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.

The Greek letter first appears there in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one, so Jones may have chosen π because it was the first letter in the Greek spelling of the word periphery.

Jones also wrote that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones We know that it had been used earlier for geometric concepts.

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler started using it, beginning with his 1736 work Mechanica and then in 1748, he used π in his widely read work Introductio in analysin infinitorum saying that "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1".  The usage was universally adopted thereafter in the Western world.

Being an irrational number, π cannot be expressed exactly as a common fraction. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed although no proof of this has yet been discovered.

Animation of the act of unrolling circumference of a circle having diameter 2, illustrating the ratio π.

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