The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has continued to make small progress. But algebra and geometry at that time were still in an independent state for a long time.
Algebra can be said to be the most widely accepted "mathematics". It can be said that the first mathematics he came into contact with was algebra since everyone began to learn to count when he was a child. Mathematics is a subject that studies numbers, and algebra is also one of the most important parts of mathematics. Geometry is the earliest branch of mathematics studied by people.
Until the Renaissance in16th century, Descartes founded analytic geometry, which linked algebra and geometry which were completely separated at that time. From then on, we can finally prove the theorem of geometry through calculation; At the same time, abstract algebraic equations and trigonometric functions can also be graphically represented. Then more subtle calculus was developed.
At present, mathematics has included many branches. French Bourbaki School, founded in 1930s, thinks that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. They think that mathematics has three basic parent structures: algebraic structure (group, ring, field, lattice, …), ordered structure (partial order, total order, …) and topological structure (neighborhood, limit, connectivity, dimension, …).
Mathematics is applied in many different fields, including science, engineering, medicine and economics. The application of mathematics in these fields is generally called applied mathematics, which sometimes arouses new mathematical discoveries and promotes the development of new mathematical disciplines. Mathematicians also study pure mathematics, that is, mathematics itself, without any practical application. Although a lot of work started from the study of pure mathematics, it may find a suitable application later.
Specifically, there are some sub-fields used to explore the relationship between the core of mathematics and other fields: from logic and set theory (mathematical basis) to empirical mathematics in different sciences (applied mathematics), and more recently, uncertainty research (chaos and fuzzy mathematics). As far as verticality is concerned, the exploration in each field of mathematics is getting deeper and deeper.