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Pi

Updated: 12/11/2025, 8:43:44 AM Wikipedia source

The number π ( ; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve. The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22 7 {\displaystyle {\tfrac {22}{7}}} 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 algebraic 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. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706. The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors. Because it relates to a circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. It also appears in areas having little to do with geometry, such as number theory and statistics, and in modern mathematical analysis can be defined without any reference to geometry. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.

Tables

· History › Infinite series › Rate of convergence
π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 4 11 + 4 13 + ⋯ {\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }
π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 4 11 + 4 13 + ⋯ {\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }
Infinite series for π
π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 4 11 + 4 13 + ⋯ {\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }
After 1st term
4.0000
After 2nd term
2.6666 ...
After 3rd term
3.4666 ...
After 4th term
2.8952 ...
After 5th term
3.3396 ...
Converges to:
π = 3.1415 ...
π = 3 + 4 2 × 3 × 4 − 4 4 × 5 × 6 + 4 6 × 7 × 8 − ⋯ {\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }
π = 3 + 4 2 × 3 × 4 − 4 4 × 5 × 6 + 4 6 × 7 × 8 − ⋯ {\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }
Infinite series for π
π = 3 + 4 2 × 3 × 4 − 4 4 × 5 × 6 + 4 6 × 7 × 8 − ⋯ {\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }
After 1st term
3.0000
After 2nd term
3.1666 ...
After 3rd term
3.1333 ...
After 4th term
3.1452 ...
After 5th term
3.1396 ...
Infinite series for π
After 1st term
After 2nd term
After 3rd term
After 4th term
After 5th term
Converges to:
π = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 4 11 + 4 13 + ⋯ {\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }
4.0000
2.6666 ...
3.4666 ...
2.8952 ...
3.3396 ...
π = 3.1415 ...
π = 3 + 4 2 × 3 × 4 − 4 4 × 5 × 6 + 4 6 × 7 × 8 − ⋯ {\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }
3.0000
3.1666 ...
3.1333 ...
3.1452 ...
3.1396 ...

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