Pi

The number pi (symbol: π is a mathematical constant that is the ratio of a circle's circumference to its diameter, and is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes written as pi. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers. 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. 

In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that – when combined with increasing computational power – extended the decimal representation of π to, as of late 2011, over 10 trillion (10¹³) digits

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals,thermodynamics, mechanics, and electromagnetism.

Fundamentals

Definition

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

π is commonly defined as the ratio of a circle's circumference C to its diameter d:  

The ratio C/d is constant, regardless of the circle's size.

Name

Leonhard Euler popularized the use of the Greek letter π in works he published in 1736 and 1748.

Properties

π is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.^[9] Since π is irrational, it has an infinite number of digits in its decimal representation, and it does not end with an infinitely repeating pattern of digits.

 

 

Antiquity

The Great Pyramid at Giza, constructed c. 2589–2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologists concluded that the pyramid builders had knowledge of π and deliberately designed the pyramid to incorporate the proportions of a circle.^[24] Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.

The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250. In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)² ≈ 3.1605.

In India around 600 BC, the Shulba Sutras (Sanskrit texts that are rich in mathematical contents) treat π as (9785/5568)² ≈ 3.088. In 150 BC, or perhaps earlier, Indian sources treat π as  ≈ 3.1622.

Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomonwith a diameter of ten cubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular. Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh. 

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD).

 Here is π with the first 100 decimal places:

3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679...