Summarize and sort out the nature of numbers and the experience of four operations between numbers in the application process, and form the oldest mathematics-arithmetic. The set of rational numbers, the four operations of addition, subtraction, multiplication and division form a closed set of numbers, which looks very complete. More than 2500 years ago, many people, even some mathematicians at that time, thought so.
In the 5th century BC, Pythagoras school attached great importance to integers and wanted to explain everything with integers. "Number is the foundation of all things" became their philosophy. The discovery of irrational numbers is a fatal blow to Pythagoras' philosophy based on integers, which is called the "first mathematical crisis" in the history of mathematics. Later, many irrational numbers were discovered, and pi is the most important one. /kloc-Leonardo da Vinci, a famous Italian painter in the 0/5th century, called it an "unreasonable number". Now people combine rational numbers with irrational numbers and call them "real numbers". From this, two solutions are obtained: and. Are they still the solutions of (2)? If not, (2) there is no solution, and solving the equation is like walking into a dead end. In order to solve this problem, mathematicians have to expand the range of numbers again and introduce the symbol "I" to represent "the square root of 1", which is called imaginary number; Then the real numbers A, B and the imaginary array are combined into a formal number, which is called "complex number". For a long time, people can't find the quantities represented by imaginary numbers and complex numbers in real life, which makes people feel a little illusory. With the development of science, imaginary number has been widely used in hydraulic, drawing, aviation and other fields. In this way, the family of numbers is further expanded, including real numbers and complex numbers, and addition, subtraction, multiplication and division are extended to include powers and roots, forming a new branch of mathematics "algebra". Algebra has made further progress in two aspects. One is to study linear equations with many unknowns and introduce symbols and concepts such as matrix, vector and space to form "linear algebra"; The other is to study higher-order equations with high unknowns and form "polynomial algebra" (also known as "polynomial theory"). In this way, the object of algebra research is not only numbers, but also matrices, vectors, vector spaces and their transformations. Can carry out "operation", although it is also called addition or multiplication, but the basic operation law of numbers is sometimes no longer effective. Therefore, the content of algebra can be summarized as a set of algebraic structures with operations, such as groups, rings, fields and so on. , but also contains abstract algebra, Boolean algebra, relational algebra, computer algebra and many other branches. Due to the development of science and technology, the range of numbers is constantly expanding, from positive integers, natural numbers, integers, real numbers to complex numbers, to vectors, tensors, matrices, groups, rings, fields and so on. In order to distinguish, people call real numbers and complex numbers "narrow numbers" and vectors, tensors and matrices "generalized numbers". Although people still have some different views on how to classify numbers, they all admit that the concept of numbers will continue to expand and develop.