A formal justification of the Ancient Chinese Method of Computing Square Roots

IF 0.3 Q4 MATHEMATICS

Mathematics Enthusiast Pub Date : 2021-01-01 DOI:10.54870/1551-3440.1513

Edilberto Nájera, Leslie Cristina Najera-Benitez

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引用次数: 0

Abstract

: In this paper a formal justiﬁcation of the ancient Chinese method for computing square roots is given. As a result, some already known properties of the square root which is computed with this method are deduced. If any other number base is used, the justiﬁcation given shows that the method is applied in the same way and that the deduced properties are still being fulﬁlled, facts that highlight the importance of positional number systems. It also shows how to generalize the method to compute high orders roots. Although with this elementary method you can compute the square root of any real number, with the exact number of decimal places that you want, the mathematicians of ancient China were not able to generalize it for the purpose of computing irrational roots, because they did not know a positional number system. Finally, in order for high school students gain a better understanding of number systems, the examples given in this paper show how they can use the square root calculus with this method to practice elementary operations with positional number systems with diﬀerent bases, and also to explore some relationships between them.

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中国古代计算平方根方法的形式论证

本文对中国古代计算平方根的方法进行了形式化论证。最后，推导出了用该方法计算的平方根的一些已知性质。如果使用任何其他数字基数，给出的理由表明该方法以相同的方式应用，并且推导出的性质仍然满足，这些事实突出了位置数系统的重要性。并给出了如何推广计算高次根的方法。虽然用这个初等的方法，你可以计算任何实数的平方根，精确到你想要的小数位数，但古代中国的数学家无法推广它来计算无理根，因为他们不知道一个位置数系统。最后，为了使高中生更好地理解数字系统，本文给出的例子说明了他们如何用这种方法使用平方根微积分来练习不同基数的位置数系统的初等运算，并探讨了它们之间的一些关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics Enthusiast MATHEMATICS-

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Mathematics Enthusiast (TME) is an eclectic internationally circulated peer reviewed journal which focuses on mathematics content, mathematics education research, innovation, interdisciplinary issues and pedagogy. The journal exists as an independent entity. The electronic version is hosted by the Department of Mathematical Sciences- University of Montana. The journal is NOT affiliated to nor subsidized by any professional organizations but supports PMENA [Psychology of Mathematics Education- North America] through special issues on various research topics. TME strives to promote equity internationally by adopting an open access policy, as well as allowing authors to retain full copyright of their scholarship contingent on the journals’ publication ethics guidelines. Authors do not need to be affiliated with the University of Montana in order to publish in this journal. Journal articles cover a wide spectrum of topics such as mathematics content (including advanced mathematics), educational studies related to mathematics, and reports of innovative pedagogical practices with the hope of stimulating dialogue between pre-service and practicing teachers, university educators and mathematicians. The journal is interested in research based articles as well as historical, philosophical, political, cross-cultural and systems perspectives on mathematics content, its teaching and learning. The journal also includes a monograph series on special topics of interest to the community of readers.