The author of Nine Chapters Arithmetic is Liu Hui. He is the first mathematical monograph written by Han scholars in ancient China, and the most important one in the Ten Calculations Classic written around the first century A.D. ... which is very rich in content and systematically summarizes the mathematical achievements in the Warring States, Qin and Han Dynasties. At the same time, Nine Chapters Arithmetic has its unique achievements in mathematics. It not only mentioned the problem of score at the earliest, but also recorded the problem of surplus and deficiency at the earliest. The chapter "Equation" also expounds the negative number and its addition and subtraction algorithm for the first time in the history of world mathematics. It should be noted that "Nine Chapters Arithmetic" has no author. It is a comprehensive historical work and the most concise and effective applied mathematics in the world at that time. Its appearance marks the formation of a complete system of ancient mathematics in China.

The mathematical achievements of "Nine Chapters Arithmetic" are various:

(1), the main achievements in arithmetic are fractional operation, proportional problem and "surplus and deficiency" algorithm. Nine Chapters Arithmetic is the earliest book in the world that systematically describes fractional operation. The second, third and sixth chapters have many problems of proportion, which are also relatively early in the world. The algorithm of "insufficient profit" needs to give two assumptions, which is a creation. It was called "two-way" in medieval Europe, and some people thought it came from China through medieval Arab countries.

There are relatively complete methods for calculating scores in "Nine Chapters Arithmetic", including four operations, total score, score reduction, band change and false score (called total score wife in ancient China, and "internal" is pronounced as acceptance) and so on. Its steps and methods are generally similar to those of modern times.

Fraction addition and subtraction, "Nine Chapters Arithmetic" has clearly proposed to divide the fractions first, so that the denominators of the two fractions are the same, and then add and subtract. The step of addition is "mother multiplication is true to each other, mother multiplication is law, and reality is law." Here, "truth" is a molecule. "Law" is the denominator, and "reality is the same as law", that is, the usage removes reality and carries out division operation. "Nine Chapters Arithmetic" also notes two points: First, if the result of the operation appears "those who refuse to obey the law will be punished by law". That is, when the numerator is less than the denominator, it is retained as a fraction. The second type is "mother-in-law, straight-in-law", that is, the fractions with the same denominator are added and subtracted, and the numerator is added and subtracted directly without division.

There are also methods to find the greatest common divisor and approximate fraction in Nine Chapters Arithmetic. The method of finding the greatest common divisor is called "polyphase loss" method, and its specific steps are "half-and-half, half-and-half, division of mother and son, reduction of more by less, and seeking equality by polyphase loss." The number is equal. "The' equal share' mentioned here is our greatest common denominator now. Can be half means that the denominator of the numerator is even, and can be half of them. First, you can remove 2. If they are not all even numbers, the numerator and denominator are calculated separately (that is, set in pairs), the decimal number is subtracted from the large number, and the remainder and subtraction are equal, thus obtaining equal numbers.

In the second, third and sixth chapters of Nine Chapters Arithmetic, various proportional solutions are widely used to solve application problems. At the beginning of the chapter on millet, the exchange ratio of various grains is listed as follows: "Millet method: millet rate is 50, hulled rice is 30, hulled rice is 27, hulled rice is 24, ………………………………………………………………" (Figure 1-23), which means that five-hulled millet can get three-hulled brown rice, with a 10% discount. For example, the first question in the corn chapter: "There was a corn fight today. If you want to cut rice, you have to be geometric. " Its solution is: "take all the numbers multiplied by the rate as the truth, take all the rates as the method, and it is actually one as the method."

The seventh chapter of Nine Chapters Arithmetic is devoted to profit and loss problem and his solution. The first question is: "Today, there are (people) who buy things, and (everyone) pays eight (money) and the rest (profit) is San Qian; People pay seven (money), less than four (money), ask the number of people and ask the price. " "Answer: Seven people, the price is 53 (money)." "Insufficient profit: fixed rate, insufficient profit. The law is not enough to make the rate of dimensional multiplication (that is, staggered multiplication) true, nor is it enough to be law, but it is as good as the law ... set the rate, reduce more with less, and make up for less. In fact, it is the price and the law is the number of people. " The residual method is a unique creation to solve application problems in the history of Chinese mathematics, and it occupies a very important position in the ancient algorithm of China. The technique of surplus and deficiency also spread to the Arab countries in Central Asia through the Silk Road, and received special attention. It was called "Khitan algorithm" and was later introduced to Europe. In the Middle Ages, "Double Method" once ruled their mathematics kingdom for a long time.

(2) Nine Chapters Arithmetic summarizes a lot of geometric knowledge in production and life practice, and puts forward many formulas for calculating area and volume, as well as the application of Pythagorean theorem in square field, quotient work and Pythagorean chapter.

In "Nine Chapters Arithmetic", Fang mainly discusses the calculation method of the area of straight lines and circles in plane graphics. The first question in "Nine Chapters of Arithmetic" is "There are fifteen steps in Tian Guang today, and sixteen steps away from Zongzong". Find the field geometry. " Answer: one acre. Here, "wide" means wide, and "from" means vertical and refers to its length. Uncle said: multiply the number of steps by the number of steps, and then divide it by 240 steps per mu (in essence, it should be the number of steps), which is the number of acres. One hundred acres is one hectare. At that time, rectangles were called square fields or straight fields. The triangle is called Guitian, and the area formula is "Shu Yue: half width multiplied by heel". Here, width refers to the bottom of a triangle, which means the bottom.

Gao and Liu Hui proved this formula in essence in their notes: "Those who are semi-extensive, make up for the shortage with surplus, and make the field straight." "It can also be multiplied by the width" (Figure 1-30). Surplus is redundant, and deficiency is insufficient. "To make up for the deficiency with surplus" means to make up for the deficiency with surplus, which is the traditional method of "in-and-out supplement" used by China to derive the formula of plane figure area in ancient times. From the above picture, it can be seen that "making up the deficit with surplus" has turned Guitian into a straight field with equal area, so the formula for calculating the area of Guitian has been obtained. In questions 27 and 28, the right-angled trapezoid is called the "evil field" (that is, the oblique field), and its area formula is: "In art, two evils (that is, two inclinations should be understood as the two bottoms of the trapezoid) are combined with one half, so that one half can be multiplied from ..." Liu Hui explained in his notes that his method of proof is still "complementary". In questions 29 and 30, the general trapezoid is called Yoshida, and the upper and lower bottoms are called tongue and heel respectively. The area formula is: "Shu Yue: tongue and half-closed, so as to take advantage of it."

As for the area of a circle, in the thirty-first and thirty-second questions in the Nine Chapters of Arithmetic, the formula for calculating the area is: "multiply the radius of half a circle by a product step." Here "Zhou" is the circumference of a circle, and "diameter" refers to the diameter. This formula for calculating the area of a circle is correct. Only at that time, the diameter was taken as one week and three weeks (that is, π≈3). Therefore, the calculated circular area is not accurate enough.

The business chapters of Nine Chapters Arithmetic are all about volume calculation. However, the business chapter does not discuss the volume algorithm of cuboids or cubes. It seems that "Nine Chapters Arithmetic" is based on the volume calculation formula of cuboid or cube: V=abc to calculate the volume of other three-dimensional graphics.

Cities, walls, dikes, ditches, graben and canals mentioned in the commercial chapter of Nine Chapters Arithmetic have different names because of their different functions. Their essence is a straight prism with isosceles trapezoid cross section. Their volume calculation method is: "The upper and lower parts are half wide, the height is multiplied by the depth, and the width is multiplied by the width, that is, the product ruler". The upper and lower sides here refer to the height or depth (h) of the upper and lower bottom surfaces (a, b) of the cross section and the length (l) of the city wall. Therefore, the formula for calculating the volume of cities and city walls is V = 1/2 (a+b) h 。

In his notes, Liu Hui extended and applied the principle of complementary entry and exit of plane graphics to spatial graphics, which became a "loss of width and compensation of narrowness" to prove geometric shapes.

Cutting and blocking

Product formula.

Liu Hui also derived a more complicated formula for calculating the geometric volume by chess measurement. The so-called chess measurement method, "chess" refers to some geometric models verified by geometric models. For example, the cuboid itself is a "chess" [Figure 1-32 (1)]. By obliquely solving a cuboid, two right-angled triangular prisms are obtained, which were called "sectioning blocks" in ancient China (as shown in the figure), so the volume of sectioning blocks is half that of cuboid.

There are also formulas for calculating the volume of frustum and frustum (called "round pavilion" in ancient times) in the business chapter of Nine Chapters Arithmetic. Even if three sides are isosceles trapezoid and the other two sides are twisted pentahedron [Figure 1-33 (1)], the upper and lower bottom surfaces are rectangular.

The volume of a cylinder (called Chutong in ancient times) and a prism (called Chutong in ancient times) with a line segment at the top and a rectangular bottom can be calculated.

③ Algebra content in Nine Chapters Arithmetic is also very rich, which has the advanced level in the world at that time.

1. square root and publisher

Chapter 9 discusses the methods of square root and square root in arithmetic, and the calculation steps are basically the same. The difference is that in ancient times, calculation was carried out through calculation. Now, taking Zhang Shaoguang's topic 12 as an example, the steps of ancient square root calculation are explained. "Today, there are 55225 steps. Find the square geometry. " Answer: 235 steps. The step mentioned here is the ancient unit of length in China.

"Square (refers to a square, the square area is the length of one side. ) operation: put the product into the real (that is, put the square root in the second line of the calculation, which is called the real), and borrow a calculation (that is, borrow a calculation fund to put it in the last line, as shown in figure 1-25 (1)). Step-by-step (refers to the step-by-step movement of borrowed calculation funds) excellent (refers to the transfer of borrowed calculation funds from the unit to hundreds or from hundreds to thousands). , which is equivalent to the division of the square root of modern pen writing as shown in figure 1-25 (2)). Negotiation income (refers to the initial negotiation, because the real digits are 5 and 22.

The above figures1-25 (1) ~ (10) are all calculated, which seems tedious, but in fact the steps are very clear and easy to operate. Its square root principle is the same as modern square root principle. Among them, the right shift and left shift of "borrowing" can be understood as a transformation and substitution from a modern perspective. Transformation and substitution were not understood in the era of nine chapters of arithmetic, but they had a far-reaching impact on the solution of higher-order equations in the late Song and Yuan Dynasties.

The "equation" in the equation chapter of "Nine Chapters Arithmetic" refers to a system of linear equations with multiple variables, and its meaning is different from that of "equation". The solution of multivariate linear equations in Nine Chapters Arithmetic is to put their coefficients and constant terms into a "square matrix" (hence the name "equation"). The process of elimination is equivalent to the linear transformation in advanced algebra of modern university courses.

Because the problem of positive and negative numbers will inevitably appear in the process of solving equations by direct division in the ninth chapter, the skills of positive and negative numbers are clearly put forward in the third question of the equation chapter. In the annotation of this technique, Liu Hui essentially gave the definitions of positive number and negative number: "The gains and losses of the two calculations are opposite, and' positive' and' negative' should be named". And in the calculation tool, that is, the calculation of funds, it is distinguished that "positive is red and negative is black, otherwise evil is different." This is the rule that positive numbers are raised in red and negative numbers are raised in black. If there are only the same colors, put the money in the right direction when encountering positive numbers, and put it in the wrong direction when encountering negative numbers. After the Song Dynasty, written arithmetic also used red and black numeric characters to distinguish positive numbers from negative numbers, or diagonal strokes on single digits to indicate negative numbers, such as-1824. Later, this kind of China numeral characters, including negative numbers, also spread to Japan.

Regarding the addition and subtraction algorithm of positive and negative numbers, "the technique of positive and negative numbers is: the same name is beneficial, and there are points in different names. Positive does not add negative, and negative does not add positive." There are some synonyms, the same name is beneficial, and there is no positive, negative or negative distinction. "Same name" and "different name" here are equivalent to the same number and different number respectively. Mutual benefit and division refer to the addition and subtraction of two numbers. The first four sentences of the article are subtraction:

(1) If the absolute value of the minuend is greater than the absolute value of the minuend, that is, A >: b≥0,

Then the same name is beneficial: (a)-(b) = (a-b),

Division of synonyms: (a)-(b) = (a+b).

(2) If the absolute value of the minuend is less than the absolute value of the minuend, that is, B >；; a≥0 .

If both numbers are positive.

Then a-b=a-[a+(b-a)]=-(b-a).

In the middle formula, A and A cancel each other, but (b-a) can't cancel each other, so "positive" is changed to "negative", that is, "positive has nothing to lose". "No entry" means no right, that is, no cancellation (or insufficient or zero relief for the other party).

(2) If both numbers are negative.

Then (-a)-(-b) =-a-[(-a)-(b-a)] =+(b-a). In the middle formula, (-a) and (-a) cancel each other, but (b-a) can't cancel each other, so "negative is not positive".

(3) If two numbers are one positive and one negative. It is also a synonym for (1).

The last four sentences of the article are the addition algorithm for correcting negative numbers.

(1) Two numbers with the same sign are added, that is, the same name is favorable, and the absolute value of the sum is equal to the sum of the absolute values of the two numbers.

If a>0, b>0,

Then a+b=a+b, (-a)+(-b)=-(a+b)

(2) The addition of two numbers with different symbols is actually subtraction, that is, division of different names. If the absolute value of a positive number is large, then its sum is positive, that is, "nothing is positive." If the absolute value of a negative number is large, then its sum is negative, that is, "nothing is negative." Symbolize as

(1) if a & gtb≥0,

Then a+(-b)=[b+(a-b)]+(-b)=a-b,

Or (-a)+b = [(-b)-(a-b)]+b =-(a-b).

2 if b & gta≥0,

Then a+(-b)=a+[(-a)-(b-a)]=-(b-a),

Or (-a)+b = (-a)+[a+(b-a)] = b-a.

Regarding the multiplication and division rules of positive and negative numbers, you may encounter multiplication and division operations of positive and negative numbers in the era of "Nine Chapters of Arithmetic". Unfortunately, it was not mentioned in the book, until Zhu Shijie of Yuan Dynasty clearly recorded that "multiplication with the same name is positive and multiplication with different names is negative" and "division with the same name is positive and division with different names is negative" in "Arithmetic Enlightenment" (1299), so at the latest, China made a comprehensive summary of the four algorithms of rational numbers. As for the introduction of the concept of positive and negative numbers and the historical record formed by the addition and subtraction algorithm of positive and negative numbers, China is far ahead. The first foreign country to admit negative numbers was the Indian mathematician Brahman Guido in the seventh century (about 598-? ) Europe did not recognize negative numbers until16th century.