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<p>In the early fifth century AD, ancient Chinese calendar-makers developed an algorithm for calculating astronomical constants, which functioned efficiently for deriving precise fractional approximations and is mathematically equivalent to the continued fraction algorithm. A series of&#xa0;highly accurate fractions could be easily obtained by repeatedly applying this algorithm, which suggested that these highly accurate fractions did not appear randomly or coincidentally in ancient Chinese mathematics or calendar systems; however, was established using a systematic method. Based on historical and mathematical analysis, it is highly probable that Zu Chongzhi 祖冲之 (429–500) derived his famous approximation of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>(<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\pi =355/113\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>=</mo> <mn>355</mn> <mo stretchy="false">/</mo> <mn>113</mn> </mrow> </math></EquationSource> </InlineEquation>, as known as the Zu ratio 祖率) using this algorithm.</p>

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How did Zu Chongzhi calculate \(\pi =355/113\)?: a reconstruction of an ancient Chinese algorithm for fractional approximation

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Wei Chen,
  • Anjing Qu

摘要

In the early fifth century AD, ancient Chinese calendar-makers developed an algorithm for calculating astronomical constants, which functioned efficiently for deriving precise fractional approximations and is mathematically equivalent to the continued fraction algorithm. A series of highly accurate fractions could be easily obtained by repeatedly applying this algorithm, which suggested that these highly accurate fractions did not appear randomly or coincidentally in ancient Chinese mathematics or calendar systems; however, was established using a systematic method. Based on historical and mathematical analysis, it is highly probable that Zu Chongzhi 祖冲之 (429–500) derived his famous approximation of \(\pi \) π ( \(\pi =355/113\) π = 355 / 113 , as known as the Zu ratio 祖率) using this algorithm.