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He also gave two other approximations of π: π ≈ 22 ⁄ 7 and π ≈ 355 ⁄ 113, which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of π using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic ...
355 / 113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places. It is within 0.000 009 % of the value of π, or in terms of common fractions overestimates π by less than 1 / 3 748 629 .
The number π (/ p aɪ / ⓘ; 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 table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant pi (π).For more detailed explanations for some of these calculations, see Approximations of π.
mathematical constant π; 3.14159 26535 89793 23846 26433... Uses; Area of a circle ... (by combining Stirling's approximation with Wallis product) = (where is ...
The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π. Sequences of constants
Comparison of the convergence of the Leibniz formula ( ) and several historical infinite series for π. S n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail) Leibniz's formula converges extremely slowly: it exhibits sublinear convergence.
Systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < 22 / 7 , which is approximately 3.142857. But it takes much less work to show that π < 22 / 7 by the method used in this proof than to show that π is approximately 3.14159.