Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine
{"title":"A rational approximation of the two-term Machin-like formula for $π$","authors":"Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine","doi":"arxiv-2406.08510","DOIUrl":null,"url":null,"abstract":"In this work, we consider the properties of the two-term Machin-like formula\nand develop an algorithm for computing digits of $\\pi$ by using its rational\napproximation. In this approximation, both terms are constructed by using a\nrepresentation of $1/\\pi$ in the binary form. This approach provides the\nsquared convergence in computing digits of $\\pi$ without any trigonometric\nfunctions and surd numbers. The Mathematica codes showing some examples are\npresented.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"42 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - General Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.08510","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we consider the properties of the two-term Machin-like formula
and develop an algorithm for computing digits of $\pi$ by using its rational
approximation. In this approximation, both terms are constructed by using a
representation of $1/\pi$ in the binary form. This approach provides the
squared convergence in computing digits of $\pi$ without any trigonometric
functions and surd numbers. The Mathematica codes showing some examples are
presented.
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