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Numerical approximation of π: as points are randomly scattered inside the unit square, some fall within the unit circle. The fraction of points inside the circle approaches π/4 as points are added. Pi can be obtained from a circle if its radius and area are known using the relationship: =.
Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed π to be between 3.1415926 and 3.1415927 [a] and gave two rational approximations of π, 22 / 7 and 355 / 113 , naming them respectively Yuelü (Chinese: 约率; pinyin: yuēlǜ; "approximate ratio") and Milü. [1]
Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22 / 7 , 333 / 106 , and 355 / 113 . These numbers are among the best-known and most widely used historical approximations of the constant.
The digits of pi extend into infinity, and pi is itself an irrational number, meaning it can’t be truly represented by an integer fraction (the one we often learn in school, 22/7, is not very ...
Proofs of the mathematical result that the rational number 22 / 7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the theory of Diophantine approximations.
Since we only care about the fractional part of the sum, we look at our two terms and realise that only the first sum contains terms with an integer part; conversely, the second sum doesn't contain terms with an integer part, since the numerator can never be larger than the denominator for k > n. Therefore, we need a trick to remove the integer ...
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π.