NILAKANTHA SERIES PROOF

Please Sign up or sign in to vote. Ranjan Roy, Mathematics Magazine , Vol. Historically, one of the best approximations of PI and interestingly also one of the oldest, was used by the Chinese mathematician Zu Chongzhi Sec. The most recent calculation found more than 13 trillion digits of pi in days! The article was complemented according to your suggestion. Not impressed Jose A Pascoa 8-Aug 7:

The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around A. Now pick up a pen, close your eyes and put dots on the square at random. Articles Quick Answers Messages. I had to raise the iteration count way up to get it to register with the timer. I leave the conclusion to you when examining the table above. PI with decimals: Juan Manuel Romero Martin 4-Sep 9:

PI with decimals: If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach Accurate Ratios for Pi. Where are you going with all nillakantha

InEnglish mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. Thanks for your comments!

The standard proof is that this follows from the Taylor series. Background To test the algorithms presented here, i suggest the following IDE: I am a software developer focused on Mathematics, IoT and Games.

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The series would lend themselves to parallel processing very well and your results might be quite different. While I agree that going back 6 digits is not practical it is for inquisitive minds to do.

Both approximations start with 3. Mike Hankey 3-Sep Sign up using Email and Password. Some algorithms could be optimized, but I believe that will not change the conclusions.

I uses normal arithmetic to determine a ratio with higher precision. Sign up or log in Sign up using Google. It is known that this irrational number arose on the calculations of geometers over time as a proportionality constant for at least 4 relationships, not necessarily in this order: Osmund Francis 7-Oct 6: Software Component Principles Revisited.

Calculating the Number PI Through Infinite Sequences

Test results are not conclusive because they were not performed with proper techniques. Download the source code here See you soon. Perhaps I’ll come back and edit this post for a summary of their methods if I actually read the article. Since then, their approximations have gone through several transformations until they reach the billions of digits obtained today with the aid of the computer.

The earliest written approximations of pi are 3. InAustrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

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sequences and series – Why does this converge to $\pi/4$? – Mathematics Stack Exchange

Iterative algorithms for computing approximations to the number PI through infinite series using double and arbitrary precision. The computer was designed with numbers represented and binary fractions. Apparently they are still alive due to the number of messages from young students that I receive. Osmund Francis 9-Sep 2: Traditionally, we define the PI as the ratio of the circumference and its diameter. Toward even more digits of pi Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value.

Neither is it to make the fastest algorithm for calculating PI. PI is not merely an irrational number, but is a transcendental number. It is just an unpretentious didactic article directed to lovers of mathematics, comparing the various methods of calculation.