As early as the early stage of cultural development, human beings used scores for measuring and averaging scores. In the earliest ancient documents of various nationalities, there are records about scores; There are also different scoring systems among ethnic groups.

Egyptian: Only the molecule is 1. They made up a table to convert the molecule not 1 into the sum of the molecule 1. For example:

22 1 = 1 14 + 142 2 15 = 1 10 + 130 2 13 = 18 + 152 + 1 104

In Babylon: Because of the creation of sexagesimal's counting system, they used fractions with denominators of 60, 602, 603, etc. The Babylonians also compiled a table, using the numerator 1 to express fractions, for example,154 =160+6602+40603.

Greeks: I have studied Egyptian fractional arithmetic and Babylonian hexadecimal arithmetic, and it is difficult to add, subtract, multiply and divide, and the numerical calculation is not well developed.

In ancient China, the divisor was placed under the dividend, and the quotient of division was placed above the dividend, such as:

23÷7-slice algorithm Remember:, divide the integer 3 into 2 and change it to:, in.

The middle 2 is the numerator, and the back 7 is the denominator. The score reads "three and two-sevenths".

According to the previous data, China's ancient mathematical work "Nine Chapters Arithmetic" (about the first century AD) has a complete algorithm of four fractions, which is also the earliest in the world.

"Nine Chapters Arithmetic" calls fractional addition "division", and the rule is "the mother multiplication is true, the mother multiplication is the method, and the fact is like the first method", that is, Ba+DC = BC+ADAC. The "real" here is the divisor, that is, the numerator, and the "method" is the divisor, that is, the denominator; "The same method is one" is to divide the dividend into several parts according to the divisor and take one. If you add up the scores of the same denominator, there is a law of "the same mother and the same clan", that is, ba+ca = b+ca.

"Nine Chapters of Arithmetic" calls fractional subtraction "subtraction", and the rule is "mother multiplication, with more subtraction, the remainder is real, the mother multiplication is the method, and the real one is the method". Namely: Ba -DC = BC-adak.

"Nine Chapters Arithmetic" refers to fractional multiplication as "multiplication", and the rule is "mother multiplication is the method, child multiplication is the reality, and it is the method and one". That is: ba× DC = bdac.

"Nine Chapters Arithmetic" calls fractional division "division by division", and the rule is that "the denominator of the method is multiplied by real (for real), and the denominator of the real is multiplied by real (for law), and one is the law". That is, Batong DC = BC.

These rules are almost exactly the same as those we use now.

The subtraction rule in "Nine Chapters Arithmetic" is "you can have half, you can add the denominator and the sub-number, so you can reduce less and more, so you can get the same number", which means that when the numerator and denominator are even numbers, you should divide by 2; If it is not an even number, then subtract a smaller number from a larger number by tossing and turning, and finally get a remainder equal to the subtraction, which is the greatest common divisor. This method of tossing and turning to find the greatest common divisor is consistent with Euclid's tossing and turning division in theory.

The mathematical calculations in India are written in contrast. In the middle of the seventh century, Indian mathematician Ramogupu

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In Ta's works, two-seventh of the scores are recorded as 7 (only less than the current scores), and the scores are 3.

three

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Erqi notation: 7. Like the notation system in China, the rules of addition, subtraction, multiplication and division of fractions are the same as those in China.

The Arabs accepted India's fractional notation, but added a horizontal line between the numerator and denominator, and wrote an integer part with a fraction in front of the fraction, for example, three and two sevenths as 3 27.

Arabic fraction algorithm/kloc-was introduced to Italy in the early 3rd century, and15th century became popular in European countries. Now it has been used all over the world.