Chapter 35 Section 5 Positive and Negative Alchemy

Zu Chongzhi's data on solving negative coefficient equations has been lost.Among the existing historical materials, Liu Yi, a mathematician in the 12th century of the Northern Song Dynasty, was the first to break through the limit of the positive coefficient of the equation.According to Yang Hui's "Field Mu Comparison, Multiplication and Division Method", Liu Yi proposed an equation in the form of x-12x=864, -5x+228x=2592, -5x+52x+128x=4096 in "Yigu Yuanyuan", which can be seen He also breaks through the limitation that the coefficient of the first term is 1.In order to solve these negative coefficient equations, Liu Yi proposed the method of formulating the formula of benefit product and formula of subtraction.Yang Hui said that Liu Yi's method "really surpasses all previous ones".These two methods are not multiplication methods, and the latter is closer to multiplication and opening methods.

Qin Jiushao proposed the plus-minus formula, which developed to a very complete level the numerical solution of higher-order equations based on the augmentation-multiplication-opening method.Some of his equations are as high as 10, and the coefficients of the equations are not limited within the range of rational numbers.He stipulates that it is always negative, which is actually the positive root of solving the following equation: f(x)=ax+ax+ax+...+ax+a=0(a≠0,a<0) The complete expression of Qin’s positive and negative alchemy can be found in the "Shushu Nine Chapters" in the field category "Jian Tianqiuqian": It is known that a section of field composed of two sharp fields has 39 steps in the big oblique, 25 steps in the small oblique, and Zhongguang Step 30, find its area (as shown in Figure 31).This problem boils down to the three powers of Kai Linglong: -x+763200x-40642560000=0

Its prescribing process is: (1) List the method, open Linglong three times. (2) Shanglianchao has one place, Yiyuchao has three places, and quotient has entered one place.Shanglian surpasses one more place, Yiyu surpasses another three places, and the quotient enters another place, and Shangshang is determined to be 800. (3) Use commerce to generate corners, enter benefits and lower injustice, use commerce to produce inefficiency and eliminate it from the upper inconsistency, use commerce to generate superior incorruptibility, enter the square, use commerce to produce the square, and get positive accumulation, which is to eliminate with reality.Use the negative reality to eliminate the positive product, and the product will be more than enough, which is positive and real, which is called "changing bones".

(4) One change, the corner of the business is born, and the bottom is cheap.Lian is born with business, enters into Lian, and eliminates each other.Use positive and negative to eliminate each other.Use business to make money, enter the party, and eliminate each other.Eliminate positive and negative squares. (5) Two changes: use business to generate corners, enter the lower level; use business to generate integrity, and enter the upper level. (6) Three changes: use business to make a corner, and enter the lower class. (7) Four changes: Fang retreats once, Shanglian retreats twice, Xialian retreats three times, corners retreat four times; business continues to reset.

(8) With the agreement of the parties, the continuation of the business will be set for forty, and the corner will be entered into Xialian.Lian was born with business, and entered into Lian.Use business to make money, and enter the party.With the method of continuing the business forty lives, eliminate the reality and make the best use of it.The resulting quotient of eight hundred and forty steps is the plot of land. Qin Jiushao pointed out that "the next chapter follows this", indicating that this is a common method.In this formula, when the constant term changes from negative to positive, Qin Jiushao calls it "Bone Change".In the process of rooting, there will also be a situation where the absolute value of the constant term increases, which is called "reincarnation" by Qin.The purpose of Qin's raising these situations is to instruct people not to stand still when encountering abnormal situations, but to continue.

Positive and negative formulas are the consensus of mathematicians in Song and Yuan Dynasties in the 13th century.Yang Hui in the Southern Song Dynasty, Li Ye and Zhu Shijie in the Jin and Yuan Dynasties all contributed to this.Li Ye and Zhu Shijie no longer stipulate that the real value is always negative, but can be positive or negative, and they also put forward suggestions on how to deal with the situation where the constant term changes sign or the absolute value increases.Mathematicians also proposed the method of division.For example, the 40th question of "Yigu Yanduan" requires the positive root of -22.5x-648x-23002=0, and the original formula can not be used to find the exact root, so Li Ye multiplied the constant term 23002 by 22.5 to get 517545, once The coefficient of the term remains unchanged, the coefficient of the quadratic term becomes -1, and then the square root is obtained, that is, the positive root of -y-648y+517545=0 is found.This actually transforms y=22.5x to find y=465, therefore, x=466÷22.5=20(2/3).Zhu Shijie extended this method to higher powers. The opening method of the 13th question of "Hefen Suoyin" in "Siyuan Yujian" is 576x-2640x+1729x+3960x-1695252=0.After the integer part of the positive root is 8, he still has the remaining formula 576x+15792x+159553+704392x-545300=0

Then multiply the constant term by 5763, multiply the coefficient of the first term by 5762, and multiply the coefficient of the second term by 576. The coefficient of the third term remains unchanged, with 1 as the coefficient of the first term, and the open method is converted into y+15792y+91902528y+233700360192y-104208452812800=0 This actually transforms y=576x, and the root is y=384, so x=y/576=384/576=2/3, x=8(2/3).The division method is also known as the art of connecting branches with one body, which is very ingenious.

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