河图、洛书数学内涵考

The Historical Research on the Connotation of Hetu and Luoshu

《汉书·律历志》将一至万的发展分为“由数起”和“大备”两阶段。结合各方面证据可确定“由数起”阶段只产生了个位、十位数字。河图是《易·系辞》中的“天地之数”,内涵是一至十的顺序排列,作用是作为标准向公众推广一至十这些数的概念及数数的计算方法。“大备”阶段,“百”字的出现形成了位值制的雏形。洛书是《易·系辞》中的“大衍之数”,伴随着“千”字出现,其内涵为加法的进位规则集。洛书的出现也标志着加法的形成。

In History of Han Dynasty,the Chapter of Music and Calendar,the development of one to ten thousand is divided into two phases named“You Shu Qi”(number as the origin)and“Da Bei”(greatly developed).With the Eight Diagrams,Zhu Xi and Cai Yuanding made the conclusion that Hetu was“the number of the world”.It is sure that only units digits and tens digits were created at the phase of“You Shu Qi”.Then through in-depth analysis,it can be made sure that the connotation of Hetu is just a seriation of ten numbers of single digits without any combination relationship.The function of Hetu is to generalize the concept of numbers from number one to number ten as a standard and the counting methods,which laid a foundation for the development of decimal number system and characters.At the phase of“Da Bei”in the development of numbers,the appearance of number Bai(one hundred)became the rudiment of place value system.Luoshu appeared along with the number Qian(one thousand),meaning a collection of value placing rules of decimal number system.The appearance of the book symbolizes a complete development of decimal number system in the field of positive integer and the formation of addition.