Pi
π (pi) is a fundamental mathematical constant defined as the ratio of the circumference of a circle to its diameter in Euclidean geometry, approximately equal to 3.141592653589793.[1] This value remains constant regardless of the circle's size, making it a universal property of circles.[2] π is an irrational number, proven incapable of being expressed as a fraction of two integers by Johann Heinrich Lambert in 1761, and it is also transcendental, meaning it is not the root of any non-zero polynomial equation with rational coefficients, as established by Ferdinand von Lindemann in 1882.[3][1] The recognition of π dates back to ancient civilizations, with the Babylonians around 1800 BCE approximating it as 3.125 using geometric methods, and the Egyptians employing a value of about 3.16 in practical calculations.[4] In the 3rd century BCE, Archimedes provided more precise bounds for π (between 3 10/71 and 3 1/7) by inscribing and circumscribing regular polygons around a circle.[3] The modern symbol π was introduced by Welsh mathematician William Jones in 1706 and popularized by Leonhard Euler in the 18th century, becoming the standard notation thereafter.[1] As one of the most important irrational constants in mathematics, π appears in diverse areas including trigonometry, calculus, complex analysis, and physics, underpinning formulas for circle areas (πr²), sphere volumes ((4/3)πr³), sine and cosine functions, and infinite series like the Leibniz formula (π/4 = 1 - 1/3 + 1/5 - ...).[5][6] Its computation has advanced significantly, with 300 trillion digits calculated as of 2025 using sophisticated algorithms, reflecting ongoing interest in its properties and approximations.[7]Fundamentals
Definition
In geometry, the constant \pi is defined as the ratio of the circumference C of a circle to its diameter d.[1] This ratio is independent of the circle's size, making \pi a universal constant for all circles in Euclidean space.[8] Ancient mathematicians, such as Archimedes in his work Measurement of a Circle (c. 250 BCE), approximated \pi by calculating the perimeters of regular polygons inscribed in and circumscribed about a circle; as the number of sides increases, these perimeters converge to the circumference, bounding \pi between \frac{223}{71} and \frac{22}{7}.[9] Analytically, \pi can be expressed as \pi = 4 \arctan(1), where \arctan is the inverse tangent function, or as the smallest positive real number x such that \sin(x) = 0, with \sin denoting the sine function.[1] These definitions arise from the properties of trigonometric functions and their inverses, linking \pi to the solutions of transcendental equations.[10] In the context of the unit circle (radius r=1), \pi relates arc length to angular measure in radians: the full circumference is $2\pi, so an angle of \theta radians subtends an arc of length \theta./06%3A_Radians/6.00%3A_Arclength_and_Radians) Basic inequalities provide simple bounds, such as $3.14 < \pi < \frac{22}{7} \approx 3.142857, reflecting early approximations like those from Archimedes and confirming \pi's irrationality without delving into proofs.[1]Numerical approximations
The decimal expansion of π is infinite and non-terminating, beginning with 3.14159265358979323846264338327950288419716939937510 (the first 50 digits after the decimal point).[11] This non-repeating nature follows from the irrationality of π, first proved by Johann Heinrich Lambert in 1761 and later simplified by Ivan Niven in 1947.[12] Early numerical approximations relied on geometric methods, such as Archimedes' use of inscribed and circumscribed regular polygons around a circle in the 3rd century BCE. By considering 96-sided polygons, he derived the bounds \frac{223}{71} < \pi < \frac{22}{7}, where \frac{223}{71} \approx 3.140845 and \frac{22}{7} \approx 3.142857.[13] These inequalities place π between approximately 3.1408 and 3.1429, accurate to three decimal places and marking the first rigorous bounds obtained without infinite processes. Simple rational approximations like \frac{22}{7} and \frac{355}{113} arise as convergents from the continued fraction expansion of π, providing successively better estimates.[14] The fraction \frac{22}{7} approximates π to about two decimal places, while \frac{355}{113} \approx 3.141593 matches π to six decimal places. For such convergents p/q, the approximation error satisfies |\pi - p/q| < 1/q^2; thus, for \frac{22}{7}, the bound is |\pi - 22/7| < 1/49 \approx 0.0204, though the actual error is roughly 0.001264.[15] Similarly, for \frac{355}{113}, the bound |\pi - 355/113| < 1/113^2 \approx 7.8 \times 10^{-5} reflects its high accuracy relative to the denominator size.[15]| Fraction | Decimal Approximation | Digits Accurate | Error Bound ($1/q^2) |
|---|---|---|---|
| 22/7 | 3.142857... | 2 | ≈0.0204 |
| 355/113 | 3.14159292... | 6 | ≈7.8×10⁻⁵ |
Irrationality
An irrational number is a real number that cannot be expressed as a ratio of two integers, where the denominator is nonzero.[16] The irrationality of \pi was first established by Johann Heinrich Lambert in 1761, who employed continued fractions to show that \pi cannot be rational.[17] Lambert's approach demonstrated that assuming \pi is rational leads to a contradiction in the continued fraction expansion of the tangent function evaluated at rational multiples of \pi.[17] A more accessible proof was provided by Ivan Niven in 1947, which assumes \pi = a/b for positive integers a and b.[12] Niven defines a polynomial f(x) = x^n (\pi - x)^n / n! for large integer n, and considers the integral I_n = \int_0^\pi f(x) \sin x \, dx.[18] By integrating by parts repeatedly, this integral equals an integer, as it relates to values of derivatives of f at 0 and \pi, which are integers under the assumption.[18] However, $0 < I_n < \pi^{2n+1} / n! for $0 < x < \pi, and for sufficiently large n, this upper bound is less than 1, contradicting the integral being a nonzero integer.[18] Thus, the assumption that \pi is rational must be false.[12] The irrationality of \pi implies that its decimal expansion is infinite and non-repeating, and it has no exact representation as a finite fraction.[16] This property underscores why approximations like 22/7 or 355/113, while useful, are inherently inexact.[16] Niven extended his method in his 1956 book Irrational Numbers to prove that \pi^2 is also irrational.[19] Assuming \pi^2 = a/b, a similar integral involving \sin(\pi x) and polynomials yields a positive quantity bounded above by less than 1 for large n, yet equal to an integer, leading to a contradiction.[19] This result highlights the depth of \pi's algebraic properties beyond mere irrationality.[19]Transcendence
A transcendental number is a complex number that is not algebraic, meaning it is not a root of any non-zero polynomial equation with rational coefficients. The number π is transcendental, a result first established by Ferdinand Lindemann in 1882.[20] His proof utilized a special case of the Lindemann–Weierstrass theorem, which asserts that if α is a non-zero algebraic number, thene^\alpha
is transcendental.[21] Lindemann applied this to Euler's identity e^{i\pi} + 1 = 0, which rearranges to e^{i\pi} = -1. If π were algebraic, then i\pi would also be algebraic (as i is algebraic), implying e^{i\pi} is transcendental and contradicting its equality to the algebraic number -1.[22] The transcendence of π has significant consequences in geometry and algebra. Since π is not algebraic, straightedge-and-compass constructions cannot produce a length equal to \sqrt{\pi}, rendering the classical problem of squaring the circle—constructing a square with area equal to that of a given unit circle—impossible.[23] This property underscores π's fundamental incompatibility with the algebraic numbers generatable by such Euclidean tools. Extensions of these results include the transcendence of e, proved by Charles Hermite in 1873 using integral approximations and contradiction arguments involving assumed algebraic relations.[24] However, the transcendence of sums and products like π + e remains unresolved; it is unknown whether π + e is transcendental.[25] Likewise, the algebraic independence of π and e over the rationals—meaning no non-trivial polynomial relation with rational coefficients connects them—is an open problem, despite partial results on related constants like π and e^\pi.[25]