So, what are mathematicians studying? Or what parts does mathematics consist of? Traditionally, we can divide mathematics into two categories: pure mathematics that studies mathematics itself and applied mathematics that is applied to solving practical problems. However, this classification is not very clear. Many fields developed according to pure mathematics at first, but later they found unexpected applications. Many fields are also closely related, so if we want to classify mathematics accurately, it should be a complex network.

This paper will lead readers to understand the five parts of mathematics: mathematical foundation, algebra, analysis, geometry and applied mathematics.

1. Mathematical Basis

Mathematics foundation studies the problems in logic or set theory, which is the language of mathematics. The field of logic and set theory thinks about the execution framework of mathematics itself. To some extent, it studies the essence of proof and mathematical reality, which is close to philosophy.

Mathematical logic and foundation.

Mathematical logic is the core of this part, but a good understanding of logical laws is after they are first used. In addition to the formal application of basic propositional logic in computer science, philosophy and mathematics, this field also covers general logic and proof theory, and finally forms model theory. Here, some famous results include Godel's incompleteness theorem and the church topic of recursive theory.

2. Algebra

Algebra is developed by refining some key concepts such as counting, arithmetic, algebraic operation and symmetry. Generally speaking, these fields can define their own research objects only through a few axioms, and then consider their examples, structures and applications. Other fields of extraordinary part algebra include algebraic topology, information and communication, and numerical analysis.

number theory

Number theory is one of the oldest and largest branches of pure mathematics. Obviously, it is concerned with problems related to numbers, which are usually integers or rational numbers (fractions). In addition to basic topics such as congruence, divisibility and prime numbers, number theory now includes the study of extraordinary part algebra of rings and number fields; There are also asymptotic estimation and analysis methods and geometric problems of special functions; In addition, it is also closely related to cryptography, mathematical logic and even experimental science.

group theory

Group theory studies the "product" operation set that defines reversible combinations. This includes a group of symmetrical other mathematical objects, which makes group theory occupy a place in all other mathematics. Finite groups may be the easiest to understand, but the symmetry of matrix groups and geometric figures is also the central example of groups.