The Use of Fractional Number and Decimal Number
分数和小数的最早运用
Fractional number
分数
The first appearance of fractional numbers(fen shu in Chinese ) was a result of dividing two numbers. A fraction number was considered a part of a whole number, and in Chinese，the word fen means to split or divide.Later，in the process of calculation，fractional numbers also appeared to denote the ratio of two whole numbers. Systematic fractional number algorithm appeared in the ancient math classic Jiuzhang Suanshu(Nine Chapters on Mathematical Art)in the Eastern Han Dynasty around the 1 st century，which was about 1,400 years ear-tier than in Europe.
最初分数的出现，并非由除法而来。分数被看作一个整体的一部分。“分”在汉语中有“分开”“分割”之意。后来运算过程中也出现了分数，它表示两整数比。在中国古代，东汉时期的《九章算术》中有系统的分数运算方法，这比欧洲大约早1400年。
In the annotated version of Jiuzhang Suanshu by Liu Hui，it clearly shows the algorithm of the subtraction，addition，multiplication，and division of fractional numbers，which is identical to todays fractional calculations. Besides, other knowledge about fractions, including how to compare two fractions，and how to calculate the average of fractions，were also included in the book, making Jiuzhang Suanshu the earliest monograph that systematically discussed fractional numbers.
从后来刘徽所作的《九章算术注》可以知道，在《九章算术》中，讲到约分、合分(分数加法)、减分(分数减法)、乘分(分数乘法)、除分(分数除法)的法则，与我们现在的分数运算法则完全相同。另外，《九章算术》还记载了课分(比较分数大小)、平分(求分数的平均值)等关于分数的知识，是世界上最早的系统叙述分数的著作。
It was not until the 15th century that calculations with fractional numbers be-came popular in Europe. It is generally believed in Europe that the algorithm of frac-tional numbers originated in India，while the fact is the earliest Indian book on the algorithm of fractions appeared in the seventh century，in which were introduced the same rules as the those in Jiuzhang Suanshu .And even the annotated version of Jiuzhang Suanshu by Liu Hui was completed in 263 in the Wei Dynasty(220-265)，which means that the algorithm of fractions is at least 400 years earlier than its counterpart in India.
分数运算，大约在巧世纪才在欧洲流行。欧洲人普遍认为，这种算法起源于印度。实际上，印度在7世纪的著作中才开始有分数运算法则，这些法则都与《九章算术》中介绍的法则相同。而刘徽的《九章算术注》成书于魏景元四年(263 )，所以，即使与刘徽的时代相比，印度也要晚400年左右才有分数运算法则。
Decimal number
小数
Also in the annotated version of Jiuzhang Suanshu by Liu Hui，the notion of the decimal number was first raised up. Liu invented the decimal number to denote the approximate number value of root extradions. Around the year 1300，Liu Jin in the Yuan Dynasty wrote the number 106368. 6312 as 一 □ ⊥　Ⅲ ⊥ Ⅲ M ＿ⅡIn his book Lulu Chengshu.The notion of decimal number appeared in the West in 1585 and the expression is rather difficult.For instance，the same number mentioned above would be expressed as (0) ①　②　③　④or106368 06①3②l③2④
刘徽在(九章算术注》中介绍，开方不尽时用十进分数(徽数，即小数)去逼近，首先提出了关于十进小数的概念。到公元1300年前后，元代刘瑾(公元1271一1368年)所著《律吕成书》中，已将106368.6312写成:一 □ ⊥　Ⅲ ⊥ Ⅲ M ＿Ⅱ。而西方直到1585年才有十进小数的概念，且表示方法很复杂，如上述小数，写法为:(0) ①　②　③　④或106368 06①3②l③2④。