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从记数法到复数域：数系理论的历史发展 |
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时间：2009-7-24 13:44:47 来源：不详
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摘 要：数系理论的历史发展表明，数的概念的每一次扩张都标志着数学的进步，但是这种进步并不是按照数学教科书的逻辑步骤展开的。希腊人关于无理数的发现暴露出有理数系的缺陷，而实数系的完备性一直要到19世纪才得以完成。负数早在《九章算术》中就已被中国数学家所认识，然而，15世纪的欧洲人仍然不愿意承认负数的意义。“四元数”的发明，打开了通向抽象代数的大门，同时也宣告在保持传统运算定律的意义下，复数是数系扩张的终点。人类发明的记数法并没有束缚自己的想象力，中国古代“数穷则变”的思想对于当代数学哲学仍具有积极的意义。
关键词：数系；记数法；实数理论；复数扩张
Abstract: The development of the number system through generation of the number concept is one of the instructive studies in the history of mathematics. The progresses of the number concepts did not match the logical steps that appeared in the textbooks. The irrational numbers, which originated in Greek geometry, exposed the fact that there are many “gaps” in the rational number system, but the perfect of the real number had not been proved until 19th century. In 15th century, the negative numbers were took as the kind absurd number, although its concepts and operation rules were completed by ancient Chinese mathematicians in the Nine Chapters of Arithmetic. From the integers to the complex numbers, the generalization of the number operations, the associative, commutative, and distributive laws of addition and multiplication remained unchanged. Further development of the number concept was brought about through changes in the fundamental postulates of algebra. Weierstrass proved the it is impossible to construct a class of numbers more general than the complex number if all the postulates are retained without change.
Key words: numeral system；numeration；theory of real number；the expansion of complex number
引 言
数，是数学中的基本概念，也是人类文明的重要组成部分。数的概念的每一次扩充都标志着数学的巨大飞跃。一个时代人们对于数的认识与应用，以及数系理论的完善程度，反映了当时数学发展的水平。今天，我们所应用的数系，已经构造的如此完备和缜密，以致于在科学技术和[1] [2] [3] [4] [5] [6] [7] 下一页 |
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