TOPICS IN ELEMENTARY MAHEMATICS

课程名称：

Topics in Elementary Math

课程概述：

内容简介：

People have been using real numbers everywhere, from daily life to scientific research, even in mathematics --- that’s right: mathematics is NOT part of science! However, people seldom questioned the existence and consistency of the theory of real numbers, which is an issue that exclusively bothered mathematicians, who are “like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.” (Johann Wolfgang von Goethe)

In this course, we will explore the system of real numbers from its very definition, an axiomatic one, to obtain a comprehensive understanding of these all-too-familiar figures. Topics that will be covered in the semester include but not limited to:

1. The notion of limit and epsilon-N statements;
2. Dedekind’s cut;
3. Existence of the exact bounds;
4. Nested interval lemma;
5. Heine-Borel theorem;
6. Bolzano-Weierstrass principle;
7. Cauchy’s convergence criterion;
8. Monotonic sequence theorem;
9. Completeness and continuity.

Remark: Bilingual lectures.

推荐阅读：

1. Zorich, Mathematical Analysis I, Ch 2~3【教材】
2. Dedekind, Essays on the Theory of Numbers
3. 克莱因，《古今数学思想》，第40-41章
4. 波耶，微积分概念史——对导数与积分的历史性的评论，第七章
5. 亚历山大罗夫，集与函数的泛论初阶
6. Dieudonné, Foundations of Modern Analysis

课程评价：

1. 课堂参与的积极程度与表现情况占40%。
2. 课程大作业占30%。
3. 期末提交小论文一篇，主题为实数公理体系的发展史，中英文不限。成绩占30%。