The Field Q and the Equality 0.999. . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics

IF 0.3 Q4 MATHEMATICS
Benoît Rittaud, L. Vivier
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引用次数: 0

Abstract

We reconsider the classical equality 0.999. .. = 1 with the tool of circular words, that is: finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the xviii th century, who appear to be the firsts to assert the equality 0.999. .. = 1. Their level of understanding show links with Dubinsky et al.'s apos theory in mathematics education. Eventually, we rebuilt the field Q from circular words, and provide an original proof of the fact that an algebraic integer is either an integer or an irrational number.
域Q与等式0.999…= 1,出自《循环词组合学》和《实用算术史》
我们重新考虑经典等式0.999…= 1使用循环词的工具,即:有限的单词,其最后一个字母被认为是第一个字母。这样的循环词自然嵌入了代数结构,揭示了这个有问题的等式,允许它在Q中而不是在r中被考虑。我们评论了这种结构的早期历史,涉及18世纪上半叶的英语教师和会计,他们似乎是第一个断言等式0.999…= 1。他们的理解水平与杜宾斯基等人在数学教育中的apos理论有关。最后,我们从循环词中重建了域Q,并提供了代数整数要么是整数要么是无理数的原始证明。
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来源期刊
Mathematics Enthusiast
Mathematics Enthusiast MATHEMATICS-
CiteScore
1.40
自引率
0.00%
发文量
43
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.
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