Jiaxian triangle is a mathematical concept. It is a triangular numerical table, which is usually used for numerical solution of higher-order equations. The number combination law of this triangle is very interesting and can be used to generate various interesting mathematical formulas.
The canonical formula of Jiaxian Triangle can be expressed as follows: the first line has only one number 1, the second line has two numbers 1 and 2, and the third line has three numbers 1, 3 and 6, and so on. The number in each row is the sum of two adjacent numbers in the previous row.
Jiaxian triangle is actually a binomial coefficient expansion in combinatorial mathematics. Each number is a combination of binomial coefficients. For example, the three numbers 1, 3 and 6 in the third row actually correspond to c (3,0), c (3,0/) and c (3,2) respectively, that is, the combination numbers of 0, 1 2 are taken from three different elements. Therefore, Jiaxian Triangle can be regarded as the expression of combinatorial mathematics in the triangle.
The canonical formula of Jiaxian Triangle can also be explained by recursion. For line n, the first number is 1, the second number is the sum of the first and second numbers in the previous line, the third number is the sum of the second and third numbers in the previous line, and so on. This recursive relationship can be expressed by mathematical formula, so that every number of Jia Xian triangle can be found.
The application of jiaxian triangle in mathematics;
1. Numerical solution of higher-order equation: The numerical solution of higher-order equation can use Jiaxian triangle. For a given polynomial equation, we can solve the root of the equation by choosing the appropriate numerical combination. Jiaxian triangle provides a convenient way of numerical combination, which enables us to solve the equation more accurately.
2. Application of Combinatorial Mathematics: Jiaxian Triangle is actually a binomial coefficient expansion in Combinatorial Mathematics. It provides an intuitive method to express combinatorial numbers, which can be used to solve some problems in combinatorial mathematics. For example, the Jiaxian Triangle can be used to calculate the value of the combination number C(n, k), where n and k are positive integers.
3. Application of recursive algorithm: The regular formula of Jia Xian triangle can also be used to realize recursive algorithm. By using recursive algorithm, we can generate a triangle table similar to Jiaxian Triangle. This recursive algorithm can be used to solve some numerical calculation problems, such as calculating factorial and Fibonacci sequence.
4. Application of data structure: Jiaxian Triangle can also be used to realize data structure. For example, we can use Jiaxian Triangle to realize a heap data structure. Heap is a special tree data structure, in which each node has a value, and the value of each node is not greater than that of its children.