Pi

+Pi (π) is a fundamental mathematical [constant](/wiki/constant) representing the ratio of a [circle's](/wiki/circle) [circumference](/wiki/circumference) to its [diameter](/wiki/diameter). Its value is approximately 3.14159. For many practical purposes, a truncated value such as 3.14 or the [fraction](/wiki/fraction) 22/7 is used as an approximation. This [irrational](/wiki/irrational) [number](/wiki/number) has a decimal expansion that continues infinitely without repetition. It is central to [geometry](/wiki/geometry) and [trigonometry](/wiki/trigonometry), appearing in formulas for the area of a [circle](/wiki/circle) and the volume of a [sphere](/wiki/sphere), for example.

+The [history](/wiki/history) of Pi dates back to ancient civilizations, who approximated its value for practical purposes in [construction](/wiki/construction) and [measurement](/wiki/measurement). Ancient [Babylonians](/wiki/Babylonians) and [Egyptians](/wiki/Egyptians) found early approximations, with the [Rhind Papyrus](/wiki/Rhind_Papyrus) from ancient [Egypt](/wiki/Egypt) providing an early method. Early estimations involved [polygons](/wiki/polygon) inscribed within and circumscribed around a [circle](/wiki/circle), a method notably refined by [Archimedes](/wiki/Archimedes) (c. 287–212 BC), a [Mathematician](/wiki/Mathematician) from [Ancient Greece](/wiki/Ancient_Greece). He developed a rigorous method by using a 96-sided regular polygon, establishing bounds for its value. The symbol 'π' was introduced in the 18th century by [William Jones](/wiki/William_Jones) and popularized by [Euler](/wiki/Euler).

+The computation of Pi to an ever-increasing number of [decimal place](/wiki/decimal_place)s has been a significant challenge for mathematicians and [computer science](/wiki/computer_science) throughout [history](/wiki/history). Modern efforts leverage powerful [supercomputers](/wiki/supercomputer) and sophisticated [algorithms](/wiki/algorithm), such as the [Chudnovsky algorithm](/wiki/Chudnovsky_algorithm) and the [BBP formula](/wiki/BBP_formula), to calculate trillions of digits. While 39 digits are sufficient for most cosmological calculations, the pursuit continues as a benchmark for [computation](/wiki/computation) and algorithmic efficiency, serving also as tests for new [computer hardware](/wiki/computer_hardware) and tools for studying the [randomness](/wiki/randomness) of Pi's digits.

+Pi appears in countless [formulas](/wiki/formula) across [physics](/wiki/physics), [engineering](/wiki/engineering), and other [sciences](/wiki/science). It emerges in contexts seemingly unrelated to circles, such as [probability](/wiki/probability), [number theory](/wiki/number_theory), and [harmonic motion](/wiki/harmonic_motion). For instance, Pi is crucial in [Fourier analysis](/wiki/Fourier_analysis) for decomposing complex [waveforms](/wiki/waveform) into simpler [sine](/wiki/sine) and [cosine](/wiki/cosine) components, and it appears in the [Normal distribution](/wiki/Normal_distribution) which is fundamental in [statistics](/wiki/statistics) and describes many natural phenomena. Its presence extends to [quantum mechanics](/wiki/quantum_mechanics), [cosmology](/wiki/cosmology), and [electrical engineering](/wiki/Electrical_engineering). Its omnipresence highlights its fundamental role in describing the universe.

+## Fundamentals

+Pi (π) is a fundamental mathematical [constant](/wiki/constant) representing the ratio of a [circle's](/wiki/circle) [circumference](/wiki/circumference) to its [diameter](/wiki/diameter). This [irrational](/wiki/irrational) [number](/wiki/number) has a decimal expansion that continues infinitely without repetition. It is central to [geometry](/wiki/geometry) and [trigonometry](/wiki/trigonometry), appearing in formulas for the area of a [circle](/wiki/circle) and the volume of a [sphere](/wiki/sphere), for example.

+## Mathematical roles and characterizations

+## Historical development

+## Computation of digits

+## Definition

+## Mathematical Properties

+## History

+## Occurrence and Applications

+## Computation and Cultural Impact

+As an [irrational](/wiki/irrational) [number](/wiki/number), Pi cannot be expressed as a simple [fraction](/wiki/fraction). Beyond being irrational, Pi is also a [transcendental number](/wiki/transcendental_number), meaning it is not the [root](/wiki/root) of any non-zero [polynomial](/wiki/polynomial) with [rational number](/wiki/rational_number) coefficients. This transcendental nature has profound implications in various areas of [mathematics](/wiki/mathematics), including the impossibility of [squaring the circle](/wiki/squaring_the_circle) with only a [compass and straightedge](/wiki/compass_and_straightedge).

+The [history](/wiki/history) of Pi dates back to ancient civilizations, who approximated its value for practical purposes in [construction](/wiki/construction) and [measurement](/wiki/measurement). Early estimations involved [polygons](/wiki/polygon) inscribed within and circumscribed around a [circle](/wiki/circle), a method notably refined by [Archimedes](/wiki/Archimedes). The symbol 'π' was introduced in the 18th century by [William Jones](/wiki/William_Jones) and popularized by [Euler](/wiki/Euler).

+Pi appears in countless [formulas](/wiki/formula) across [mathematics](/wiki/mathematics), [physics](/wiki/physics), [engineering](/wiki/engineering), and other [sciences](/wiki/science). It is not only central to [geometry](/wiki/geometry) and [trigonometry](/wiki/trigonometry) (e.g., area of a circle, volume of a [sphere](/wiki/sphere)), but also emerges in contexts seemingly unrelated to circles, such as [probability](/wiki/probability), [number theory](/wiki/number_theory), and [harmonic motion](/wiki/harmonic_motion). Its omnipresence highlights its fundamental role in describing the universe.

+The computation of Pi to an ever-increasing number of [decimal place](/wiki/decimal_place)s has been a significant challenge for mathematicians and [computer science](/wiki/computer_science) throughout [history](/wiki/history). While 39 digits are sufficient for most cosmological calculations, the pursuit continues as a benchmark for [computation](/wiki/computation) and algorithmic efficiency. Pi also holds cultural significance, inspiring memorization challenges and featuring in popular culture, often celebrated on [Pi Day](/wiki/Pi_Day) (March 14th).

+Pi (π) is a fundamental mathematical [constant](/wiki/constant) representing the ratio of a [circle's](/wiki/circle) [circumference](/wiki/circumference) to its [diameter](/wiki/diameter). This [irrational](/wiki/irrational) [number](/wiki/number) has a decimal expansion that continues infinitely without repetition, a core concept in [geometry](/wiki/geometry).

+## See also

+- [Circle](/wiki/circle)

+- [Constant](/wiki/constant)

+- [Number](/wiki/number)