This paper expounds the general division, simplification, four operations and simplification rules of complex fractions with the same sign and different sign. In the annotation of prescription, he discussed the existence of irrational roots from the infinite meaning of prescription, introduced new numbers, and created a method of infinitely approaching irrational roots with decimals.

In calculus theory, he first gave a clear definition of rate, and then based on three basic operations, such as multiplication, reduction and identity, he established a unified theoretical basis for the operation of numbers and formulas. He also defined the "equation" in China's ancient mathematics by rate, that is, the augmented matrix of linear equations in modern mathematics.

In pythagorean theory, pythagorean theorem and the calculation principle of pythagorean solution are demonstrated one by one, the theory of similar pythagorean shape is established, and pythagorean measurement is developed. Through the analysis of typical figures such as "crossing in the hook" and "straight in the stock", a similar theory with China characteristics was formed.

Circumcision and pi, he wrote in arithmetic in chapter 9? In the annotation of roundness field, the exact formula of circle area is proved by secant technique, and the scientific method of calculating pi is given. He first cuts a circle from the hexagon inscribed in the circle, and every time the number of sides is doubled, he calculates the area of 192 polygon, π= 157/50=3. 14, and then calculates the area of 3072 polygon, π = 3927/1.

Principles in Liu Hui's Nine Chapters of Arithmetic? Yang Equestrian Notes, when he solved the volume of cone by infinite division, he put forward Liu Hui's principle of calculating the volume of polyhedron.

In the annotation of Nine Chapters of Arithmetic Circle Opening, he pointed out the inaccuracy of the formula V=9D3/ 16(D is the diameter of the ball) and introduced the famous geometric model "Mouhe Square Cover". "Mouhe Square Cover" refers to the intersection of inscribed cylinders with two perpendicular axes.

In the annotation of "Nine Chapters of Arithmetic Equations", he put forward a new method to understand linear equations, using the idea of ratio algorithm.

In his "Island Calculation", he put forward the repeated difference technique, and used repeated tables, continuous cables, cumulative moments and other methods to measure the height and distance. He also developed gravity difference technology from two observations to three observations and four observations by analogy. In the 7th century, India and Europe only began to study the problem of two observations in15 ~16th century.