Chapter 11 Section 2 Fractions and Decimals

Liu Hui said: "The quantity of things cannot be fully understood, but must be divided into words." Fractions have been widely used in pre-Qin classics and "Zhoubi", and the complete theory of fractions appears in "Nine Chapters" Fang Tian Zhang.The first is the rule of reduction: If the numerator and denominator can be divisible by 2 at the same time, they will be divided by 2 first.If it cannot be divisible by 2, the numerator and denominator are subtracted from the less to the more (that is, the less is subtracted from the more), and the subtraction is repeated to find the greatest common divisor (called an equal divisor), and this is used to approximate it.This method is consistent with Euclid's method of finding the greatest common divisor.For example, the simplified score of Chapter 6 of Fang Tian is 49/91:

7 is the greatest common divisor, using 7 to approximate the numerator and denominator: 49/91=7/13. The addition of fractions is called combined fractions, and the subtraction is called subtracted fractions. The rule is: the numerators multiply the denominators, the addition (subtraction) is regarded as the real, and the multiplication of the denominators is regarded as the law.That is, a/b±c/d=ad/bd±cb/db=(ad±bc)/db.The common fraction is used here, but the least common multiple is not used as the denominator. The multiplication of fractions is called multiplication, and the rule is: the denominators are multiplied to form the denominator, and the numerators are multiplied to form the numerator.That is, a/b×c/d=ac/bd, which is no different from today.Fractional division is called Jingfen. "Nine Chapters" separates reality and law, and divides the molecules: a/b÷c/d=ad/bd÷cb/db=ad/bc.Liu Hui proposed the inverted multiplication method: a/b÷c/d=a/b×d/c=ad/bc, which is the same as today.

This is the earliest fraction algorithm in the world.Fractional arithmetic only became popular in Europe around the 15th century and is thought to have originated in India.In fact, the Brahman Gupta in India only had the fraction calculation rules in the 7th century, and they are all the same as those in China.There are several fraction application problems in ancient Chinese arithmetic.For example, "Sun Tzu's Suan Jing" and "Zhang Qiu Jian's Suan Jing" both have "swinging cups on the river" question: A woman was washing cups and dishes by the river, someone asked her why there were so many cups and dishes?She replied: There are guests at home, I don't know how many.I only know that 2 people share a dish, 3 people share a soup cup, 4 people share a rice bowl, and share 65 cups and plates.Ask how many guests are there?The algorithm is: 65÷(1/2+1/3+1/4)=65÷(13/12)=60 (people).

In the history of mathematics, the generation of decimals is much later than fractions.Liu Hui used decimal fractions (micronumbers) to approach the approximate value of irrational roots when the root of the formula was endless, and he was the first to open decimal fractions.In ancient times, minutes, centimeters, millimeters, silk, seconds, and suddenly were used to represent the odd zero part below the minute.The counterfeit "Xia Hou Yang Suan Jing" is often expressed in a certain whole unit, and no longer lists the micro-units. For example, if 1525 pieces of silk are converted into 1525 pieces of 9375 pieces (1 piece = 4 feet), in fact is a decimal number.Both Qin Jiushao and Li Ye expressed 1863.2 inches as 18632, which is basically the same as today's notation.Yang Hui and Zhu Shijie successively summed up the folk formula for converting catty two into decimal decimals.China is the first country in the world to use decimals.Al Qasi in Central Asia did not master decimal fractions until the 13th century.The concept of decimal decimals was only available in the West in 1585, and the notation is far inferior to that in China during the Tang and Song Dynasties. For example, the above-mentioned decimals are recorded as ⓪①②③④ or 1525⓪9①3②7③5④15259375.