论文标题
场Q和平等0.999。 。 。 = 1来自循环单词和实用算术历史的组合学
The Field Q and the Equality 0.999 . . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics
论文作者
论文摘要
我们重新考虑经典平等0.999。 .. = 1带有圆形单词的工具，即：有限单词的最后一个字母被假定为第一个字母。这样的循环单词自然地嵌入了代数结构，这些结构启发了这种有问题的平等，可以在Q而不是在R中进行考虑。我们评论了这种结构的早期历史，涉及英语教师和第十一世纪第一部分的会计师，他们似乎是首先声称等于等于0.999。 .. = 1。他们的理解水平显示了与杜宾斯基等人的数学教育中的Apos理论的联系。最终，我们从圆形单词中重建了字段Q，并提供了代数整数是整数或不合理数字的原始证据。
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.
