The number π 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 "π" since the mid-18th century, though
it is also sometimes spelled out as "pi".
Being an irrational number, π cannot be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate π. 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, π is a transcendental number – a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π 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 π 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 π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava-Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computerr scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to, as of late 2013, over 13.3 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π 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, π 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 π 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 π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 67,000 digits.
Being an irrational number, π cannot be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate π. 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, π is a transcendental number – a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π 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 π 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 π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava-Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computerr scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to, as of late 2013, over 13.3 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π 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, π 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 π 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 π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 67,000 digits.
The digits of π have no apparent pattern and have passed
tests for statistical randomness, including tests for normality; a number of
infinite length is called normal when all possible sequences of digits (of any
given length) appear equally often. The conjecture that π is normal has not been proven or
disproven. Since the advent of
computers, a large number of digits of π have been available on which to perform
statistical analysis. Yasumasa Kanada
has performed detailed statistical analyses on the decimal digits of π and found them consistent with normality;
for example, the frequency of the ten digits 0 to 9 were subjected to statistical
significance tests, and no evidence of a pattern was found. Despite the fact that π's digits pass statistical tests for
randomness, π contains some sequences of digits that may appear non-random to
non-mathematicians, such as the Feynman point, which is a sequence of six
consecutive 9s that begins at the 762nd decimal place of the decimal
representation of π.
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