History And Educational Application Of In-and-out Complemen Tary Principle

In-and-out complementary principle not only is one of most basic principles,but also con-tains thoughts and methods of the combination of number and shape. The studying of in-and-out complementary principle has a certain theoretical significance and practical value.So,based on existing literature,the paper trace back to the historical evolution process and application of In-and-out complementary principle.The main work is as follows.Firstly, the historical background of in-and-out complementary principle,that is origin,the earliest application times,scope of application of in-and-out complementary principle,what’s more, Bolyai and Gerwien theorem as the theory of planar polygon’s stone and the fact that in-and-out complementary principle can not be used as the theoretical basis of three-dimensional volume,that is Hilbert’s third problem.It is concluded that the earliest application times of In-and-out complementary principle is the Spring and Autumn and it is in the core position of Chinese ancient mathematics.Secondly, the application of in-and-out complementary principle in Chinese ancient math-ematics is discussed and it is disclosed that Liu hui and Qin Jiushao have abundant deduc-tive thinking,that is they use a lot of graphics proof to prove in the process of reasoning,both strengthening theoretical intuition and showing the reliability of reasoning.Thirdly,through the analysis of specific cases,the paper illustrate that in-and-out comple-mentary principle contains ideas and methods of transformation of number and shape,reflecting intuitionistic character of algebra, geometry and calculus,making abstract questions be specific in graphs and providing important information for the middle school mathematics education.