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Pi

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The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.

Archimedes developed the polygonal approach to approximating π.

Isaac Newton used infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".

The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706

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

Mathematical constant

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A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, statistics, and calculus.

The area between the two curves (red) tends to a limit, namely the Euler-Mascheroni constant.

This Babylonian clay tablet gives an approximation of the square root of 2 in four sexagesimal figures: 1; 24, 51, 10, which is accurate to about six decimal figures.

Image: Different constants of integration