48.第一章

After the decline of ancient Greek civilization, the development of mathematics entered a trough in Europe, while the ancient Chinese, Indian, and Arabian civilizations continued to promote the progress of mathematics. Ancient Chinese mathematics focused on practical arithmetic and algorithm thinking. The ancient mathematical classic The Nine Chapters on the Mathematical Art systematically summarized the algorithms of field measurement, grain transportation, engineering calculation, and equation solving in ancient China, forming a complete ancient arithmetic system. Chinese mathematicians successively made world-leading achievements in fraction operation, positive and negative number operation, remainder theorem, and high-order equation solution. Indian mathematicians invented the modern Arabic numeral system (0-9), perfected the decimal counting method, and laid the foundation for global unified numerical calculation. Arabian mathematicians inherited and integrated the mathematical achievements of Greece, India, and China, sorted out algebraic theories, and preserved and spread the mathematical classics of ancient civilizations during the medieval European dark ages, becoming an important bridge for the inheritance and development of mathematics.

The Renaissance in the 16th century ushered in the golden age of modern mathematics. With the liberation of human thought and the development of astronomy, physics, and navigation, mathematics faced new development demands. In the 17th century, Descartes founded analytic geometry, unified geometry and algebra for the first time, and realized the transformation from geometric figure research to algebraic equation research, opening up a new field of mathematical analysis. Subsequently, Newton and Leibniz independently invented calculus, which solved the core mathematical problems of variable motion, instantaneous velocity, and curved area calculation that could not be solved by classical mathematics. The birth of calculus marked the entry of mathematics into the era of variable mathematics from the era of constant mathematics, and greatly expanded the boundary of mathematical research.

The 18th and 19th centuries witnessed the systematization and refinement of modern mathematics. Mathematicians such as Euler, Lagrange, and Laplace perfected the theoretical system of calculus and established disciplines such as differential equations, analytic mechanics mathematics, and probability theory. At the same time, number theory, projective geometry, non-Euclidean geometry, and group theory were successively established, making mathematics branch more diversified and theoretical system more rigorous. In the late 19th century, the axiomatization movement of mathematics rose, and mathematicians sorted out the logical foundation of various mathematical branches, making modern mathematics form a complete and standardized theoretical system.

In the 20th century, mathematics entered the era of modern mathematics with rapid explosion. With the progress of quantum mechanics, relativity, computer technology, and information science, mathematical research expanded to more abstract and cutting-edge fields. Topology, functional analysis, abstract algebra, differential geometry, mathematical logic, and other emerging disciplines developed rapidly. At the same time, the intersection of mathematics and other disciplines became increasingly close, and interdisciplinary mathematics such as mathematical physics, mathematical biology, financial mathematic