Cauchy inequality is an inequality discovered by Cauchy during his research. It is widely used to solve the related problems of inequality proof, so it is very important in the promotion and research of advanced mathematics, and it is one of the research contents of advanced mathematics.
Generally, the pure greater than sign ">" is used. Usually, the number in inequality is a real number, and letters also represent real numbers. The general form of inequality is F(x, y, z)≤G(x, y, …, z) (in which inequality can also be one of them), and the common domain of analytic expressions on both sides is called the domain of inequality. Inequality can express both a proposition and a problem.
Similarly, a b here represents the dot product (inner product) of a and vector b, |a| represents the length (modulus length) of vector a, and |b| represents the length (modulus length) of vector b.
The intuitive meaning of Cauchy inequality is that the absolute value of the dot product of two vectors will not exceed the product of their lengths. When the directions of two vectors are close to the same, their dot products get the maximum; When the directions of two vectors are almost opposite, their dot product is the smallest.
Cauchy inequality is widely used in high school mathematics, involving various mathematical concepts and problems such as vectors, complex numbers and trigonometric functions. It is an important tool for learning linear algebra and solving various mathematical problems.
Introduction to Cauchy:
He is a French mathematician and mechanic. At the age of 27, he became a professor at the Paris Institute of Technology and was elected as an academician of the French Academy of Sciences. He has made many important achievements in his life. Cauchy inequality is one of his very important achievements. In addition, he also conducted in-depth research in many fields of mathematics, including number theory, algebra, mathematical analysis and differential equations.
Cauchy's contributions to higher mathematics include: convergence and divergence of infinite series, theory of real variable function and complex variable function, differential equation, determinant, probability and mathematical equation. At present, we have learned the definitions of limit and continuity, derivative, limit of differential and definite integral, and infinite sum, which are basically given by Cauchy.