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Since the middle of the 20th century, the approximation of π has been the task of electronic digital computers (for a comprehensive account, see Chronology of computation of π). On June 28, 2024, the current record was established by the StorageReview Lab team with Alexander Yee's y-cruncher with 202 trillion (2.02× 10 14) digits. [2]
Rewriting the approximation for and using the approximation for gives + + = + Thus, rearranging terms gives e π − π ≈ 20. {\displaystyle e^{\pi }-\pi \approx 20.} Ironically, the crude approximation for 7 π {\displaystyle 7\pi } yields an additional order of magnitude of precision.
The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52 163 / 16 604 , though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as 86 953 / 27 678 . For eight, 102 928 / 32 763 is needed. [2]
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
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 ...
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b 2N m, when struck by the other object. [1] (This gives the digits of π in base b up to N digits past the radix point.)
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.
In practical implementations such as y-cruncher, there is a relatively large constant overhead per term plus a time proportional to / , and a point of diminishing returns appears beyond three or four arctangent terms in the sum; this is why the supercomputer calculation above used only a four-term version.