The origin of mathematics:
Ancient Babylonians had accumulated some mathematical knowledge and could apply it to practical problems. As far as mathematics itself is concerned, their mathematical knowledge is only obtained through observation and experience, and there is no comprehensive conclusion and proof, but their contribution to mathematics should be fully affirmed.
The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be seen in ancient mathematical classics of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress. But algebra and geometry at that time were still independent for a long time.
The development of mathematics:
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 can also be graphically represented. Then more subtle calculus was developed.
Space and structure of mathematics;
Space:
The study of space originated from Euclidean geometry. Trigonometry combines space and numbers, including the famous Pythagorean theorem, trigonometric function and so on. Now the research on space is extended to high-dimensional geometry, non-Euclidean geometry and topology. Numbers and spaces play an important role in analytic geometry, differential geometry and algebraic geometry.
In differential geometry, there are concepts such as fiber bundle and calculation on manifold. Algebraic geometry has the description of geometric objects such as polynomial equation solution set, which combines the concepts of number and space; There is also the study of topological groups, which combines structure and space. Lie groups are used to study space, structure and change.
Structure:
Many mathematical objects, such as numbers, functions and geometry, reflect the internal structure of continuous operations or the relationships defined in them. Mathematics studies the properties of these structures, for example, number theory studies how integers are represented under arithmetic operations.
In addition, things with similar properties often occur in different structures, which makes it possible for a class of structures to describe their state through further abstraction and then axioms. What needs to be studied is to find out the structures that satisfy these axioms among all structures. Therefore, we can learn abstract systems such as groups, rings and domains.
These studies (structures defined by algebraic operations) can form the field of abstract algebra. Because abstract algebra has great universality, it can often be applied to some seemingly unrelated problems.