Further research: ancient and modern mathematical thoughts.
The following is a list of famous mathematical works, taken from a blog. They are all classified, and everyone can choose according to their own interests. I hope it helps you.
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List of important mathematical works
Reprinted label: miscellaneous talk
geometry
The elements of geometry (Greek ∑ροι χ ε? α) is a mathematical work written by Euclid, an ancient Greek mathematician, with volume *** 13. This book is the foundation of modern mathematics and the most widely circulated book in the west after the Bible.
Volume 1-6: Plane Geometry
Volume 7-9: Number Theory
Volume 10: irrational number
Volume 1 1- 13: solid geometry.
Publication period: about 300 BC
Online version: interactive Java version
Summary: This is probably not only the most important work in geometry, but also the most important work in mathematics. It contains many important results of geometry and number theory and the first algorithm. It is still a valuable resource and a good guide to algorithms. More important than any particular result in this book, it seems that the greatest achievement of this book is to popularize logic and mathematical proof as a method to solve problems.
Importance: The creation, breakthrough, impact and summary of the project are the most modern and excellent (although it is the first, some achievements are still the most modern).
geometry
Description: The Elements of Geometry was published by Descartes on 1637. This book has a great influence on the development of rectangular coordinate system, especially on the point on the plane represented by real numbers; In addition, there is a discussion about expressing curves through equations.
Importance: Project Pioneer, Breakthrough and Impact
logic
Conceptual text (Begriffsschrift)
Author: Gottlob Frege
Introduction: Published in 1879, the title Begriffsschrift is usually translated into conceptual writing or conceptual symbols; The full title of the overview equates it with "pure thinking formula language, modeling in arithmetic language". Frege's motivation for developing his formal logic system is similar to Leibniz's desire to find a calculator. Frege defined a logical calculation method on the basis of mathematics to support his research. Begriffsschrift is both the title of the book and the name of the calculation method defined in the book.
Importance: It can be said to be the most important logical work since Aristotle.
mathematical formula
By Pialott
Introduction: Formulario mathematico, first published in 1895, is the first complete mathematics book written in a formal language. It contains expressions of mathematical logic and many important theorems of other branches of mathematics. Many concepts introduced in this book have become everyday concepts today.
Importance: influence
Mathematical principles (mathematical principles)
Russell and Whitehead
Introduction: Mathematical Principles is a trilogy based on mathematics by Russell and Whitehead, published in1910-1913. It is an attempt to deduce all mathematical truths by using well-defined axioms and reasoning rules in symbolic logic. Whether the axiomatic principle set can lead to contradictions and whether there are mathematical propositions that cannot be proved or falsified in this system still exist. These problems were solved by Godel's incomplete theorem in 193 1 in a somewhat disappointing way.
number theory
Arithmetic research (research and study of integer translation theory)
Gauss
Introduction: Arithmetic Research is a textbook of number theory written by German mathematician C.F.Gauss. It was first published in 180 1 year. Gauss was 24 years old. In this book, Gauss accepted the number theory achievements of mathematicians such as Fermat, Euler, Lagrange and Legendre, and added his own important new achievements.
On prime numbers less than a given number.
Author riemann
Introduction: About prime numbers less than a given value (? How long is the term of office of the first prime minister? Sse) is a groundbreaking paper by Riemann, published in the monthly report of Berlin Academy of Sciences 1859 1 1. Although this is his only published paper on number theory, it contains the thoughts of dozens of researchers who have influenced 19 century until today. This paper mainly includes the definition, heuristic demonstration, proof outline and the application of powerful analysis methods; These have become the basic concepts and tools of modern analytic number theory.
Lecture notes on number theory
By Dirichlet and Dai Dejin.
Introduction: Lecture Notes on Number Theory is a textbook on number theory compiled by German mathematicians Dirichlet and Dai Dejin, and published in 1863. The handout can be regarded as a watershed between the classical number theory of Fermat, Jacobi and Gauss and the modern number theory of Dai Dejin and riemann sum Hilbert. Dirichlet did not clearly define the central concept group of modern algebra, but many of his proofs showed that he had an implicit understanding of group theory.
Early manuscript
Rand mathematical papyrus (Rhind Mathematical Papyrus)
Introduction: This is one of the oldest mathematical documents, belonging to the second middle period of ancient Egypt. This was copied by the scribe Ames from the older China papyrus. In addition to describing how to get the approximate method of π with the accuracy of 1%, it also describes one of the earliest attempts to turn a circle into a square, and in this process, it shows convincing evidence that the theory that the Egyptians deliberately built pyramids to deify π with its proportion is wrong. Although it is exaggerated to say that papyrus represents the first attempt to analyze geometry, Ames did use a concept similar to cotangent.
Nine chapters of mathematical art
Introduction: China's mathematics book may have been written in 1 century, or it may have been written in 200 BC. Its contents include: solving linear problems with the principle of western pseudo-position law. The solution of multivariate unknowns (involving the method of "seeking a solution by extensive extension" invented by Qin, a mathematician in the Southern Song Dynasty, and the remainder theorem of Sun Tzu's Art of War) adopts the principle similar to Gauss elimination. It involves the principle called Pythagorean Theorem in the West (also called Pythagorean Theorem in China).
Archimedes rewriting
Introduction: Although the author's only mathematical tool is middle school geometry as it seems today, he used these methods with rare wisdom, and explicitly adopted infinitesimal to solve the problems that are now handled by integral calculus. These problems include finding the center of gravity of centripetal hemisphere, finding the center of gravity of circular parabolic table, and the area surrounded by parabola and its secant. Contrary to the ignorance of history in some calculus textbooks in the 20th century, he didn't use anything like riemann sum, including his work in this rewriting and his other work. See how Archimedes used infinitesimal to illustrate the details of his method.
textbook
Pure mathematics course
Author: Hardy
Introduction: A classic textbook of introduction to mathematical analysis, written by Hardy. First published in 1908, with many versions. It aims to help Britain innovate mathematics education, especially the University of Cambridge and the schools preparing to train mathematics students in Cambridge. Therefore, it is directly aimed at students with "scholarship level"-10% to 20% ability. This book contains many difficult problems. The content includes introduction to calculus and infinite series theory.
Importance: Start
The art of solving problems.
Richard Ruschick and Sandel Lehotchki
Introduction: The art of problem solving began with two books co-authored by Richard Ruschick and Sandel Lehotchki. These books, with a total of about 750 pages, are for students who are interested in mathematics and/or want to take part in mathematics competitions.
Primitive logic: an introduction to the theory of standard first-order logic elements
Jeffrey Hunt
Introduction: An excellent introductory book on mathematical theory of formal logic system, involving integrity proof, consistency proof and so on, even including set theory.
arithmetic
Arithmetic K: In other words, the foundation of art.
Author: Robert Recorde
Introduction: Written in 1542, it is the first popular arithmetic book written in English.
Assistant to the principal, summary of practice and theoretical arithmetic
Thomas Dillworth
Introduction: Early popular English textbooks were published in18th century America. This book extends from an introductory topic to five advanced topics.
theory of game
On Numbers and Games
John conway
Introduction: This book is divided into two parts, {0, 1|}. The zero part is about numbers, and the first part is about games-including the value of games and some really playable games, such as Nim, Hackenbush, Col and Snort.
The way to win the math game.
Elwyn Berlekamp, john conway and Richard Guy.
Introduction: Information summary of mathematical games. First published in 1982, it is divided into two parts. One part mainly talks about combinatorial games and super real numbers, and the other part mainly talks about some specific games.
algebraic geometry
Algebraic geometry and analytic geometry
Jean-Pierre Serre
Introduction: Mathematically, algebraic geometry and analytic geometry are closely related topics, in which analytic geometry is the theory of complex manifold, and the more general analytic space is locally defined by the 0-point set of analytic functions of multiple complex variables. The (mathematical) theory of the relationship between the two appeared in the early 1950s as part of the work of laying the foundation for algebraic geometry, such as the technology of Hodge theory. (Note that although using analytic geometry as rectangular coordinates also belongs to the category of algebraic geometry in a sense, this is not the theme of this paper. The main thesis to consolidate this theory is Serre's Gé ometrie Algé brique et Gé omé trie Analyticique, which is now generally expressed by GAGA. The result of GAGA formula now represents the comparison theorem, which makes the object in algebraic geometry and its morphism category establish a channel with the object with strict definition and its holomorphic mapping in a subcategory of analytic geometry.
Importance: project creation, breakthrough and influence.
Fundamentals of Algebraic Geometry (? Algeria General Materials Co.
Gro Dandika (Alexander Grothendieck)
With the help of Jean Dieudonne, this is Gro Dandica's explanation of his reconstruction of algebraic geometry. It has become the most important basic work of modern algebraic geometry. The work explained in EGA, like the famous reasons of these books, changed the field and led to landmark progress.
Importance: Pioneering work in the field of innovation.
analysis situs
analysis situs
James munkres
Introduction: This wonderful introductory textbook is the standard textbook of university point set topology and algebraic topology. Munkres can use the rigor of mathematics to teach many topics and give the source of concepts intuitively.
Topology from the differential point of view.
John Milnor
Introduction: This little book introduces the main concepts of differential topology in Milnor's clear and capable style. Although this book is not very extensive, it explains its theme and clarifies all the details in a beautiful way.
Importance: influence
algebraic topology
Allen Hatch
Publication information: Cambridge University Press, 2002.
Online Edition: http://www.math.cornell.edu/~hatcher/AT/ATpage.html
Introduction: This is the first of three series of textbooks, which is suitable for beginners to read. They hope to cover all the basic contents when they see this topic for the first time. This first book contains basic core themes and some relatively basic optional themes.
Importance: Start