The main subjects of mathematics mainly come from the needs of commercial calculation, understanding the relationship between numbers, measuring land and predicting astronomical events. These four requirements are generally related to a wide range of sub-fields in mathematics, such as quantity, structure, space and change (that is, arithmetic, algebra, geometry and analysis). In addition to the above-mentioned main concerns, there are sub-fields used to explore the relationship between the core of mathematics and other fields: to logic, to set theory (foundation), to empirical mathematics in different sciences (applied mathematics), and to the rigorous study of uncertainty in modern times. The study of quantity and quantity begins with numbers, and at the beginning is the familiar arithmetic operations of natural numbers and integers and those described in arithmetic. In number theory, the deeper properties of integers are studied, including famous results, such as Fermat's last theorem. When the number system is further developed, integers are considered as a subset of rational numbers and included in real numbers, and continuous quantities are represented by real numbers. Real numbers can be further extended to complex numbers. Further generalization of numbers can continue to include quaternions and octal numbers. The consideration of natural numbers will also lead to over-limit numbers, which formulizes the concept of counting to infinity. Another research field is its size, which leads to cardinality and another infinite concept: Avery number, which allows meaningful comparison between the sizes of infinite sets. Structure Many mathematical objects, such as sets of numbers and functions, have embedded structures. The structural properties of these objects are discussed in groups, rings, bodies and other abstract systems that are themselves objects. This is the field of abstract algebra. Here is a very important concept, that is, vector, which is extended to vector space and studied in linear algebra. The study of vector combines three basic fields of mathematics: quantity, structure and space. Vector analysis extends it to the fourth basic field, namely change. The study of space comes from geometry, especially Euclidean geometry. Trigonometry combines space and numbers, including the famous Pythagorean theorem. Nowadays, the study of space extends to higher dimensional geometry, non-Euclidean geometry (which plays a central role in general relativity) 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. Foundation and Philosophy In order to understand the mathematical foundation, mathematical logic and set theory have been developed. German mathematician Georg Cantor (1845- 19 18) initiated the set theory, boldly marched into infinity, provided a solid foundation for all branches of mathematics, and its own content was quite rich, put forward the existence of real infinity, and made inestimable contributions to the future development of mathematics. Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Pioncare also compared set theory to an interesting "pathological situation". Kroneck refuted Cantor's "neuroticism" and "walked into the hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said, "My theory is rock solid. Whoever opposes it will lift a rock and lick his own feet. " Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. At the beginning of the 20th century, Hilbert, the greatest mathematician in the world, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Russell praised Cantor's works as "the greatest works that can be boasted in this era". Mathematical logic focuses on putting mathematics on a solid axiomatic framework and studying the results of this framework. It is the birthplace of Godel's second incomplete theorem, which is perhaps the most widely spread achievement in logic-there is always a true theorem that cannot be proved. Modern logic is divided into recursion theory, model theory and proof theory, which are closely related to theoretical computer science.

Branch of mathematics

1 .arithmetic 2. Elementary algebra 3. Advanced algebra iv. Number theory 5. Euclidean geometry 6. Non-Euclidean geometry 7. Analytic geometry 8. Differential geometry 9. Algebraic geometry 10. Projective geometry 1 1. Geometric topology 12. Topology 65438. 3. Fractal geometry 14. Calculus 15. Theory of real variable function 16. Probability statistics 17. Complex variable function theory 18. Functional analysis 19. Partial differential equation 20. Ordinary differential equation 2 1. Mathematical logic.

Mathematical numbers, basic natural numbers, negative numbers, positive numbers, integer fractions, binary fractions, unit fractions, decimals, finite decimals, infinite decimals, cyclic decimals, rational numbers, irrational numbers, quadratic irrational numbers, composite numbers, normal numbers, real numbers, imaginary numbers, complex numbers and Gaussian integers. Eisenstein Integer, Algebraic Number, Algebraic Integer, Regular Number, Transcendental Number, Extension, Double Complex Number, Hypercomplex Number, Quaternion * * * Quaternion, Complex Quaternion, Octal Number, Hexadecimal Number, Tessarin, Supernumber, Large Real Number, Extremely Real Number, Even Logarithm, Nominal Value, Hyperbolic Complex Number, Serial Number, Overlimit Number, Ordinal Number, Cardinality. Mathematical constant, large number, pi = 3.14159265358 ... e = 2.718281828 ... imaginary unit I2 =–1(the square of I) is infinite.

List of Mathematical Theorems (in alphabetical order) A Abel-Ruffini Theorem Atia-Singer Exponential Theorem Abel Theorem Adal Theorem Abel Binomial Theorem Abel Curve Theorem Eisenstein Theorem Orr Theorem Archimedes Midpoint Theorem B Porzano-Virsh Trass Theorem Barna-Taskey Paradox Bertrand Chebyshev Theorem Bayati Theorem Bertrand Periodic Theorem Closed Image Theorem Bernstein Theorem Fixed Point Theorem Bryan Sang Theorem Brown Theorem Bezo Theorem Bosuke-ulam Theorem C Vertical. Diameter Theorem Chen Theorem Sampling Theorem D Dini Theorem Isoperimetric Theorem Algebra Basic Theorem Polynomial Remainder Theorem Law of Large Numbers Dirichlet Theorem Di Meff Theorem-Laplace Theorem Descartes Theorem multinomial theorem Girard Girard Desargues Theorem E Binomial Theorem F Fubini Theorem Vander Waals Deng Theorem Fermat's Great Theorem Normal Graph Lemma Fermat's Square Sum Theorem Fayette Thompson Theorem Frobenius Theorem Fermat's Minor Theorem Van oberth Theorem Fernsler-Harder Weigel Theorem Inverse Function Theorem Fermat Polygon Number Theorem G Green's Formula Pigeon Nest Principle Gauss-Markov Theorem is superior to Gu Shan-intellectual village Theorem Godel Completeness Theorem Inertia Theorem Godel Incompleteness Theorem Generalized Orthogonality Theorem Golding Theorem Gauss Divergence Theorem Gustav. Sen theorem * * * Yoke Complex Root Theorem Gauss-Lucas Theorem Goldbach-euler theorem Pythagoras Theorem Gelfond-Schneider Theorem Burden Theorem Cantor-Bernstein-Schroeder Theorem H Helen Formula Gerbrand Theorem Hellinger-Teplic Theorem Wallace-Boer-Gewiener Theorem hopf-Reynolds Theorem Heine-Bohr Theorem Helmholtz Theorem Butterfly Theorem J tikhonov Theorem Wonderful Theorem The compactness of the first mean value theorem integral. Theorem Integral Second Mean Value Theorem Convolution Theorem Extreme Value Theorem Kirchhoff Theorem Angular bisector Theorem K Cauchy Theorem Klein Inequality Fixed Point Theorem Cantor Dingrich Mean Value Reliability Theorem Cramer Theorem Cauchy-Lipshits Theorem Kelly Hamilton Theorem Knut-Taskey Theorem carmichael Theorem Cauchy Integral Theorem Kroneck Theorem Kroneck Weber Theorem Cano Theorem L Zero Consistency Lucien Theorem Leberg Control Convergence Theorem Loewenheim Skran. Dinglirol Theorem Lagrange Theorem (Group Theory) Lagrange Mean Value Theorem Ramsey Theorem Lax-milgram Theorem Riemann Mapping Theorem Lhuilier Theorem Lebesgue Theorem Lagrange Theorem (Number Theory) Lebesgue Differential Theorem, Levi Convergence Theorem, Liuville Theorem Menelaus Theorem morera Theorem Nash Embedding Theorem n Napoleon Theorem O euler theorem (Number Theory) Euler Rotation Theorem Euclid Theorem euler theorem (Geometry) P Poincare-hopf Theorem Pique Theorem Spectrum Theorem Bramagouta Theorem Pascal Theorem Pappus Theorem Picard Theorem Average Principle Cut-off Theorem Q Homogeneous Kendoff Theorem Curve Basic Theorem S Four-color Theorem fundamental theorem of arithmetic Steiner-Rams Theorem Four Vertices Theorem Siping Square Theorem Stokes Theorem Prime Theorem Stolz-cesaro Theorem Si Tong Boolean Algebra Representation Theorem Sun-Ne Theorem Stewart Theorem Seva Theorem Projective Theorem T Thales Theorem Isomorphism Basic. Theorem Taylor Mean Value Theorem Taylor Formula Thuren Theorem tabor Theorem Toure Theorem Ptolemy Richard W Wollsten Holm Theorem Infinite Monkey Theorem Wilson Theorem Wilstras Approximation Theorem Calculus Fundamental Theorem Vieta Theorem Viviani Theorem Five-color Theorem Weber Theorem X Silodine Sim Theorem Sylvester-Calais Theorem Linear Algebra Fundamental Theorem Linear Congruence Theorem Y Swallowtail Theorem Noise Channel Coding Theorem Finite Single Group Classification Deduction Theorem Circular Power Theorem Friends. Yi Theorem Factorial Theorem Implicit Function Theorem Rational Root Theorem Cosine Theorem Z China Remainder Theorem Proves the Sum of Reciprocals of All Prime Numbers Divergent Rank-Zero Degree Theorem Ancestor Principle Line Length Formula Central Limit Theorem James Theorem Maximum Flow Minimum Cut Theorem Spindle Theorem Tangent Theorem Inverse Sine Theorem.