雙語科技百科(數學) 第33期 分數和小數的最早運用
The Use of FractionAl Number and Decimal Number
Fractional number
分數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 r，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.
最初分數的出現，並非由除法而來。分數被看作一個整體的一部分。“分”在漢語中有“分開”“分割”之意。後來運算過程中也出現了分數，它表示兩整數比。在中國古代，東漢時期的《九章算術》中有系統的分數運算方法，這比歐洲大約早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.
從後來劉徽所作的《九章算術注》可以知道，在《九章算術》中，講到約分、合分(分數加法)、減分(分數減法)、乘分(分數乘法)、除分(分數除法)的法則，與我們現在的分數運算法則完全相同。另外，《九章算術》還記載了課分(比較分數大小)、平分(求分數的平均值)等關於分數的知識，是世界上最早的系統敘述分數的著作。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 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.
分數運算，大約在巧世紀纔在歐洲流行。歐洲人普遍認爲，這種算法起源於印度。實際上，印度在7世紀的著作中才開始有分數運算法則，這些法則都與《九章算術》中介紹的法則相同。而劉徽的《九章算術注》成書於魏景元四年(263 )，所以，即使與劉徽的時代相比，印度也要晚400年左右纔有分數運算法則。Decimal number
小數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 notion of decimal number appeared in the West in 1585 and the expression is rather instance，the same number mentioned above would be expressed as (0) ① ② ③ ④or106368 06①3②l③2④
劉徽在(九章算術注》中介紹，開方不盡時用十進分數(徽數，即小數)去逼近，首先提出了關於十進小數的概念。到公元1300年前後，元代劉瑾(公元1271一1368年)所著《律呂成書》中，已將106368.6312寫成:一 □ ⊥ Ⅲ ⊥ Ⅲ M ＿Ⅱ。而西方直到1585年纔有十進小數的概念，且表示方法很複雜，如上述小數，寫法爲:(0) ① ② ③ ④或106368 06①3②l③2④。
