Chapter 27 Section Three Positive and Negative Techniques

Equation arithmetic can result in negative numbers.There are two ways here. One is that in the process of direct division and elimination, the situation of subtracting large numbers from decimals often occurs, such as in the third question of the equation chapter; the second is that the equation itself often has negative coefficients, such as in the fourth question. According to the meaning of the question, list 5x- 11=7y, the transposition will be 5x-7y=11.Therefore, "Nine Chapters" introduced the concept of negative numbers and positive and negative techniques.Liu Hui said: "Nowadays, the gains and losses of the two calculations are opposite, and the positive and negative numbers must be named." ("Nine Chapters of Arithmetic · Equation Chapter Note") This is a clear definition of positive and negative numbers. "Nine Chapters" gives:

Positive and negative art says: the same name divides each other, different names benefit each other, the positive does not enter the negative, and the negative does not enter the positive.Its different names get rid of each other, and the same name benefits each other. If there is no positive, it will be positive, and if there is negative, it will be negative. This is the famous law of addition and subtraction of positive and negative numbers.The same name and different names are the same name and different name today.The first four sentences talk about subtraction, and its meaning is: if the two numbers have the same sign, then the absolute value of the difference is the difference of their absolute values: (±a)-(±b)=±(ab), a≥b>0; (±a)-(±b)=∓(ba), 0<a≤b.If the two numbers have different signs, the absolute value of their difference is the sum of their absolute values: (±a)-(∓b)=±(b+a); if there is no subtraction number for a positive number, it is a negative number: 0-(+a)=-a, a>0; a negative number without a subtraction number is a positive number: 0-(-a)=a, a>0.The last four sentences are addition rules: (±a)+(∓b)=±(ab), a≥b>0; (±a)+(∓b)=∓(ba), 0<a≤b;( ±a)+(±b)=±(a+b), a, b>0; 0+(+a)=a; 0+(-a)=-a, a>0.

Positive and negative surgery was originally used only in equations, and it was used in other branches of mathematics after the Song and Yuan Dynasties.Taking the 8th question of the equation chapter of "Nine Chapters" as an example, the equation is listed as