Mediaboss Marketing

The number is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "" since the mid-18th century, though it is also sometimes spelled out as "pi" (//).

Being an irrational number, cannot be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate . Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed; however, to date, no proof of this has been discovered. Also, is a transcendental number a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Although ancient civilizations needed the value of to be computed accurately for practical reasons, it was not calculated to more than seven digits, using geometrical techniques, in Chinese mathematics and to about five in Indian mathematics in the 5th century CE. The historically first exact formula for , based on infinite series, was not available until a millennium later, when in the 14th century the MadhavaLeibniz series was discovered in Indian mathematics.[1][2] 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 to, as of late 2013, over 13.3 trillion (1013) digits.[3] Scientific applications generally require no more than 40 digits of so the primary motivation for these computations is the human desire to break records. However, the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

Because its definition relates to the circle, is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses or spheres. It is also found in formulae used in other branches of science such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics and electromagnetism. The ubiquity of makes it one of the most widely known mathematical constants both inside and outside the scientific community: Several books devoted to it have been published, the number is celebrated on Pi Day and record-setting calculations of the digits of often result in news headlines. Attempts to memorize the value of with increasing precision have led to records of over 67,000 digits.

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter , sometimes spelled out as pi. In English, is pronounced as "pie" ( //, pa).[4] In mathematical use, the lowercase letter (or in sans-serif font) is distinguished from its capital counterpart , which denotes a product of a sequence.

The choice of the symbol is discussed in the section Adoption of the symbol .

is commonly defined as the ratio of a circle's circumference C to its diameter d:[5]

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula = C/d.[5] There are also other definitions of that do not immediately involve circles at all. For example, is twice the smallest positive x for which cos(x) equals 0.[5][6]

is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as 22/7 are commonly used to approximate ; no common fraction (ratio of whole numbers) can be its exact value).[7] Since is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits. There are several proofs that is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2) but smaller than the measure of Liouville numbers.[8]

More strongly, is a transcendental number, which means that it is not the solution of any non-constant polynomial with rational coefficients, such as x5/120 x3/6 + x = 0.[9][10] The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or n-th roots such as 331 or 10. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[11] Squaring a circle was one of the important geometry problems of the classical antiquity.[12] Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is impossible.[13]

Link:
Pi - Wikipedia, the free encyclopedia