Among the four civilizations in the ancient world, China's mathematics has the longest lasting prosperity. In14th century BC, China classical mathematics experienced three development climaxes: Han Dynasty, Wei, Jin, Southern and Northern Dynasties and Song and Yuan Dynasties, and reached its peak in Song and Yuan Dynasties.

Unlike Greek classical mathematics, which centered on proving theorems, China ancient mathematics focused on creating algorithms, especially various algorithms for solving equations. From linear equations to high-order polynomial equations, even indefinite equations, ancient mathematicians in China created a series of advanced algorithms (called "technology" by mathematicians in China), and they used these algorithms to solve corresponding types of algebraic equations, thus solving various scientific and practical problems that led to these equations. In particular, geometric problems are also reduced to algebraic equations, and then solved by programmed algorithms. Therefore, China ancient mathematics has obvious algorithmic and mechanized characteristics. Here are some examples to illustrate this feature of the development of ancient mathematics in China.

1. 1 linear equations and "equation skills"

The "Equation Technique" in the eighth volume of Nine Chapters Arithmetic, the most important mathematical classic in ancient China, is an algorithm for solving linear equations. Taking the title of this volume 1 as an example, expressed by modern symbols, this problem is equivalent to solving a system of linear equations with three variables:

3x+2y+z=39

2x+3y+z=34

x+2y+3z=26

"Chapter 9" has no symbols for unknowns, but arranges the coefficients and constant terms of xyz into a (long) square:

1 2 3

2 3 2

3 1 1

26 34 39

The key algorithm of "equation technology" is called "multiplication and direct division". In this example, the calculation process is as follows: multiply the numbers of this row and the left row by the coefficient of the right row (x), and then "divide" the right row separately, that is, subtract the corresponding numbers of the right row continuously, so the coefficients of this row and the left row will become 0. This equation can be solved by repeatedly executing this "multiplication and division" algorithm. Obviously, the "multiplication and direct division" algorithm of the equation technology in "Nine Chapters Arithmetic" is essentially the elimination method for solving linear equations that we use today. It used to be called "gauss elimination" in western literature, but in recent years it began to change its title. For example, Professor P.Gabriel, an academician of the French Academy of Sciences and former head of the Department of Mathematics at the University of Zurich, called the elimination method for solving linear equations "Zhang Cang method" in his textbook.

1.2 polynomial equation of higher degree and "positive and negative square root"

There are "Fang" and "Fang" in Volume 4 of Nine Chapters Arithmetic. These algorithms in "Nine Chapters Arithmetic" were gradually extended to high-order cases, and developed into numerical solutions of general high-order polynomial equations in the Song and Yuan Dynasties. Qin is a master in this field. In his book Nine Chapters of Mathematics (1247), he gave a complete algorithm for numerical solution of polynomial equation of higher order, which he called "plus and minus square extraction".

Represented by modern symbols, Qin's idea of "positive and negative square roots" is as follows: for any given equation,

f(x)=a[0]x^n+a[ 1]x^(n- 1)+……+a[n-2]x^2+a[n- 1]x+a[n]=0( 1)

Where a[0]≠0, a [n]

f(c+h)=a[0](c+h)^n+a[ 1](c+h)^(n- 1)+……+a[n- 1](c+h)+a[n]=0

The equation about h can be obtained by combining similar terms according to the power of h:

f(h)=a[0]h^n+a[ 1]h^(n- 1)+……+a[n- 1]h+a[n]=0(2)

(Note: a[i] in formula (2) and (1) here is generally different. )

Then the highest number of roots satisfying the new equation (2) can be estimated again. In this way, if the constant term of a new equation is 0, then the root is a rational number; Otherwise, the above process can be continued and the approximate value of the root can be obtained according to the required accuracy.

If we calculate the coefficients A [0], a [1], …, a[n] and estimate c of the original equation (1), we need to reuse the algorithm of the new equation (2). In Shu Shu Jiu Zhang, he used this algorithm to solve various practical problems that can be attributed to algebraic equations, and the number of equations involved was as high as 10. Qin's algorithm to solve these problems is unified and clear, and it is a model of algorithmic and mechanization of ancient mathematics in China.

1.3 multivariate higher order equations and "quaternary technology"

Not all problems can be reduced to a linear equation or a polynomial equation with unknown quantities. In fact, it can be said that if a larger number of practical problems can be solved by algebraic equations, there will be higher-order equations with multiple unknowns.

Even today, it is not easy to solve higher-order multivariate equations. Zhu Shijie, a mathematician of Yuan Dynasty in China, was the first person to systematically deal with multivariate higher-order equations in history. The higher-order equation involved in Zhu Shijie's Four Lessons of Yu Juan (1303) has reached four unknowns. Zhu Shijie used "four elements" to solve these equations. The "four-element method" firstly represents different unknowns with "heaven", "earth", "people" and "things", and at the same time establishes equations, and then solves the equations with the general sequential elimination method. Zhu Shijie created many kinds in Meet with Siyuan.