代數教學輔助系統

A Supporting System for Algebraic Education

吳忠穎(Chung-Ying Wu)

指導教授 ： 廖賀田

淡江大學/管理學院/資訊管理學系碩士班/碩士(2006年)

https://doi.org/10.6846/TKU.2006.00114

摘要

代數是中學數學中ㄧ門基礎且重要的課程，將電腦科技運用在代數教育，有助於數學教師引導學生的解題思路；但市面上之數學套裝軟體過於昂貴，功能也不完整。本論文依據教育部之數學課程標準，用Java語言實作出一套代數教學輔助函式庫，其中包含無限制位數的整數、有理數、根數、多項式及矩陣，並透過TwFC為輸出入介面。
本論文特別著重於根數的運算，尤其強調倒數運算時分母有理化的部份；我們提出的演算法，能成功地對複雜的根數完成運算。經過實驗證實：本系統的根數運算能力，在效率上大幅度優於Maple及一些著名的數學系統。

關鍵字

代數教學輔助函式庫 ； 根數 ； 分母有理化

並列摘要

Algebra is a fundamental topic in junior mathematic course. In computer aided education on mathematics, it is helpful for a teacher to training student‘s thinking. However, most of the mathematic application programs are very expansive and with incomplete functions. In this paper, a supporting system for algebra is proposed and implemented in Java language. In accordance with syllabus from the Department of Education, R.O.C., topics including unbounded integers, rational numbers, radical numbers, polynomials and matrices are implemented. The package TwFC is introduced to the I/O interface of our system.
This paper emphasizes the arithmetic ability of radical numbers, especially focusing on the operation of denominator rationalization while inverting a radical number. The proposed algorithm can successfully execute the inverting operation with simplification of radical numbers. According to the result of our experiments, the ability of our system is more efficient than Maple and other well-known applications.

並列關鍵字

supporting system for algebraic education ； radical number ； denominator rationalization

參考文獻

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[Bu 74] David M. Burton, “Abstract and Linear Algebra”, Wm. C. Brown Publishers, 1974, p.287

[BD 04] Paul T Bateman, Harold G Diamond, “Analytic Number Theory”, World Scientific, 2004, p.93

[Kn 81] Donald Ervin Knuth, “The Art of Computer Programming: Seminumerical Algorithms, volume 2”, Addison-Wesley, Second edition, 1981.

[Po 50] Harry Pollard, “The Theory of Algebraic Numbers”, The Mathematical Association of America, 1950, p.38