What is mathematics? Some people would say: "Isn't mathematics only the knowledge of number?" Such kind of saying isn't correct, because mathematics studies not only the number, but also the shape. The well-known triangle and square are the object which mathematics studies, too. Historically, the definitions about what is mathematics are more omnifarious. Someone says that mathematics is relationship. And someone says that mathematics is logic. "logic is the young stage of mathematics, and mathematics is the mature stage of logic." What on earth is mathematics? Great revolutionary mentor Engels gives a series of scientific judgment at the theory of dialectic materialism through the profound analysis of the origin and the essence of mathematics. Ehgels points out that mathematics is the science of quantity and the object of pure mathematics is the spatial format and numerical relationship of the real world. On the basis of the viewpoint of Engels, more exact defination about mathematics is that mathematics is the science of studying the numerical relationship and spatial format of the real world. Mathematics can be divided to two categories, one is pure mathematics, and the orher is applied mathematics. Pure mathematics is also called base mathematics that specially studies the internal rule of mathematics itself. Algebra, geometry, calculus and probability introduced in books for primary and high school students are all part of pure mathematics. One distinct feature of pure mathematics is to study the numerical relationship and spatial format of the object in its pure format without the practical content. For example, whatever the trapezoid calculation is for paddy field or for mechanical accessory, what people concern is the numerical relationship implied in this geometrical graph. Applied mathematics is a huge system. One says that applied mathematics is the part of our whole knowledge that can be expressed by the mathematical language. Applied mathematics focuses on explaining natural phenomena and solving practical problem, and it is the bridge between pure mathematics and scientific technology. Today is information society. The information thesis, which studies information specially, is an important embranchment in applied mathematics. Mathematics has three distinct features. High abstract is one of the distinct features of mathematics. All mathematical theories have most abstract format. This kind of abstract is formed during a series of phases, and it is very different from the common abstract in natural science. Further more, not only the concept is abstract, but the mathematical method itself is also abstract. For instance, physical scientists can prove their theories through experiments, but mathematical scientists cannot prove the theorem through experiments. They have to use logistic reasoning and calculation. Geometry, which was thought to be intuitive in mathematics in the past, has developed toward the direction of abstract. According to the theories of axiom, geometrical graph is not necessary to be acknowledged; whatever it is round, or square, we can even use desk, chair and beer cup to replace dot, line and area, as long as they satisfy the relationship of combination, consequence and contract, have consistency, independence and completion. That is geometry. System preciseness is another distinct feature of mathematics. The correction of mathematical thought expresses in a way of logistic preciseness. As early as 2000years ago, mathematical scientist had sorted out massive geometric knowledge into a rigorously systematic theory using the logistic method from several basic conclusions. It looks like an elegant logistic chain that every link connects perfectly. As a result, mathematics is reputed as the model of precise science. Broad applicability is also a distinct feature of mathematics. In the world of large cosmos, tiny atom, speed rocket, skillful chemistry, changeable earth, mystery living beings and complicated daily lives, mathematics exists anywhere, anytime. In the 20th century, mathematics had infiltrated into almost every scientific branch with the massive springing up of branches of applied mathematics. Not only physics and chemistry enjoyed the achievements of mathematics, but biology, linguistics and history, which seldom used mathematics before, have integrated with mathematics and form many borderline sciences such as biology-mathematics, mathematical-economy, mathematics-psychology, mathematical-linguistics and mathematics-history. Mathematicalizaion of every science is the trend for the development of modern science.