Overview of three core areas of Mathematics: algebra

Mathematics has developed to the present , It has become owned in the scientific world 100 A large number of major sub disciplines “ republic ”. Generally speaking, mathematics has three core areas ：

1. The part of mathematics that studies numbers belongs to the category of algebra ;

2. Study the part of form , It belongs to the fan Chou of geometry ;

3. The part that communicates form and number and involves limit operation , It belongs to the scope of analysis .

These three kinds of mathematics constitute the ontology and core of the whole mathematics . Around this core , Because mathematics passes the two concepts of number and shape , Permeate with other sciences , And there are many marginal and interdisciplinary disciplines . This paper briefly introduces the historical development of more than ten main branches of mathematics in the three core fields .

1. The arithmetic

Arithmetic has two meanings , One is handed down from China , Equivalent to what is generally said “ mathematics ”, Such as 《 Nine chapters on arithmetic 》 etc. . The other is translated from European Mathematics , From Greek , Yes “ Computing technology ” The meaning of . Now generally speaking “ The arithmetic ”, It often refers to the four operations of natural numbers ; If it is in Higher Mathematics , Then there are “ number theory ” The meaning of . Arithmetic as the content of modern primary school curriculum , It mainly talks about natural numbers 、 Positive fractions and their four operations , And it is consolidated by some of the simplest application problems caused by counting and measurement .

Arithmetic is the oldest branch of Mathematics , Some of its conclusions are that for thousands of years , Slowly and gradually established . They reflect the accumulation over many centuries , And constantly solidify the experience in people's consciousness .

Natural numbers are in the process of calculating a finite set of objects , The resulting abstract concepts . In daily life, people are required not only to calculate a single object , Also calculate various quantities , For example, length 、 Weight and time . To meet these simple measurement needs , Scores are needed .

The development of modern elementary arithmetic operation methods , It originated in India , The time may be 10 Century or 11 century . It was later adopted by Arabs , Then it spread to Western Europe .15 century , It was transformed into its present form . After Indian arithmetic , Obviously, there is the influence of ancient China .

19 In the middle of the century , Glassman successfully selected a basic axiom system for the first time , To define addition and multiplication ; And other propositions of arithmetic , It can be the result of logic , Derived from this system . later , Piano further improved Glassman's system .

Basic concepts of arithmetic and rules of logical inference , Based on human practical activities , It profoundly reflects the objective regularity of the world . Although it is highly abstract , But because the raw materials it summarizes are so extensive , So we can hardly leave it . meanwhile , It also constitutes the most solid foundation for other branches of Mathematics .

2. Elementary algebra

Elementary algebra, as the main content of middle school mathematics curriculum , Its central content is equation theory . The Latin original meaning of algebra is “ place ”. The theory of algebraic equations in elementary algebra is extended from univariate linear equations to two aspects ： One is to increase the number of unknowns , Consider a system of binary or ternary equations consisting of several equations with several unknowns ( It is mainly a system of first-order equations ); The second is to increase the number of unknowns , Investigate the univariate quadratic equation or quasi quadratic equation . The main content of elementary algebra is 16 The century has been basically developed .

Babylon, Cuba ( B.c. 19 century ～ front 17 century ) Solve the problem of first and second order equations , Euclidean 《 Original 》( B.c. 4 century ) There is a method of solving quadratic equations in geometric form . Our country 《 Nine chapters on arithmetic 》( A.D. 1 century ) There are solutions of cubic equations and first-order simultaneous equations , And uses negative numbers .3 Diophantine of the th century is calculated once with rational numbers 、 Solutions of quadratic indefinite equations .13 Tianyuan technique appeared in China in the th century ( Li Ye 《 Round sea mirror 》) It is about the numerical solution of higher-order equations in one variable .16 In the th century, Italian mathematicians discovered the solution of cubic and quartic equations .

The history of the development of algebraic symbols , It can be divided into three stages . The first stage was three centuries ago , There is no abbreviation or symbol for the solution of the problem , Instead, write a paper , It is called literal narrative algebra . The second stage is from the third century to 16 century , Abbreviations are used for some common quantities and operations , It is called simplified algebra . One of the outstanding contributions of tifantu in the third century , Is to simplify Greek algebra , Pioneered simplified algebra . However, since then, text narrative algebra , In other parts of the world except India , It has been very common for hundreds of years , Especially in Western Europe until 15 century . The third stage is 16 After the century , The solution to the problem is mostly expressed in mathematical shorthand composed of symbols , These symbols have no obvious connection with the content they represent , It is called symbolic algebra .16 Weida's masterpiece in the century 《 Introduction to analytical methods 》, He has made many contributions to the development of symbolic algebra .16 The end of the century , Vyte pioneered symbolic algebra , It has become a modern form after Descartes' improvement .

“＋”、“－” No. 1 first appeared in a math book , yes 1489 Weidman's work in . But officially recognized by everyone , As a plus 、 The sign of subtraction , That's from 1514 Started by Hoek in .1540 year , Recod started using it now “＝”. To 1591 year , After Veda's extensive use in his works , It is gradually accepted by people .1600 Harryott created the greater than sign “＞” And the less than sign “＜”.1631 year , Autre gives “×”、“÷” As a multiplication and division operator .1637 year , Descartes used the root sign for the first time , And introduce the letters in front of the alphabet to represent known numbers 、 The following letters indicate the custom of unknown numbers . as for “≮”、“≯”、“≠” The appearance of these three symbols , That's a modern thing .

Extension of the concept of number , In history, it is not all caused by solving algebraic equations , But it is still used to put it in elementary algebra , In order to be consistent with the arrangement of this course . B.c. 4 century , The ancient Greeks discovered irrational numbers . B.c. 2 century ( In the Western Han Dynasty ), China began to use negative numbers .1545 year , Cardano of Italy began to use imaginary numbers .1614 year , Britain's nipple invented logarithm .17 The end of the century , The general concept of real number index is gradually formed .

3. Advanced algebra

In Higher Algebra , First order equations （ Linear equations ） Developed into linear algebra theory ; and —、 Quadratic equations developed into polynomial theory . The former is vector space 、 linear transformation 、 Type theory 、 Invariant theory and tensor algebra, a branch of modern algebra , The latter is a branch of modern algebra that studies equations of any degree with only one unknown quantity . Advanced algebra as a college course , Only study their basis .

1683 Nian guanxiaohe ( The Japanese ) The concept of determinant was first introduced . The most systematic discussion about Determinant Theory , Jacobi 1841 Year of 《 On the formation and nature of determinant 》 A Book . Logically , The concept of matrix precedes the concept of determinant ; And in history , The order is just the opposite . Carlyle is 1855 In, the concept of matrix was introduced , stay 1858 In, he published the first important article on this subject 《 Research Report of matrix theory 》.

19 century , Determinants and matrices have received great attention , There have been more than a thousand articles on these two topics . however , They are not big reforms in Mathematics , It's an expression of shorthand . However, they have proved to be highly useful tools .

The study of polynomial algebra began with 3、4 Exploration of the formula for finding the root of sub equation .1515 year , Filo solved the problem of being simplified to lack 2 Secondary 3 The problem of solving sub equation .1540 year , Ferrari succeeded in discovering the general 4 Algebraic solution of sub equation . People continue to seek 5 Time 、6 The formula for finding the root of an equation of degree or higher , But these efforts are 200 It has been lost for many years .

1746 year , D'Alembert first gave “ The fundamental theorem of algebra ” The proof of ( There are imperfections ). This theorem asserts ： For each real coefficient or complex coefficient n Subalgebral equation , At least one solid root or compound root . therefore , Generally speaking ,n Subalgebral equations should have n A root .1799 year ,22 Gauss is writing his doctoral thesis , The first strict proof of this theorem is given .1824 year ,22 Year old Abel proved ： higher than 4 The radical formula composed of all the coefficients of the general equation of degree , It can't be its root .1828 year , Years old 17 Galois, aged, founded “ Galois theory ”, It includes the necessary and sufficient conditions for the equation to be solved with the root sign .

4. number theory

Number theory with positive integers as its research object , It can be seen as part of arithmetic , But it is not from an operational point of view , But from the point of view of the structure of numbers , That is, a number can be expressed by other numbers with simpler properties . So we can say , Number theory is the science of studying the number system composed of integers in a certain form .

As early as BC 3 century , Euclidean 《 Original 》 Some properties of integers are discussed . He proved that the number of primes is infinite , He also gave a toss and turn division method for finding the common divisor of two numbers . This has nothing to do with our country 《 Nine chapters on arithmetic 》 Medium “ More derogation ” It's the same . Eratoseni gave the idea of finding a natural number no greater than a given number N All primes of “ Sieve method ”： In writing from 1 To N On the papyrus of all integers , Dig out in turn 2、3、5、7…… Multiple ( Respective 2 times ,3 times ,……) as well as 1, All that is left on this sieve of papyrus is prime .

When the difference between two integers can be positive integers m Except when , Call these two numbers for “ model ”m congruence . In our country 《 Sun Tzu's Sutra 》( A.D. 4 century ) Calculate the group of primary congruences in “ Seek a skill ”, Yes “ Chinese remainder theorem ” Known as the .13 century , Qin Jiushao has established a relatively complete congruence theory ——“ Da Yan seeks a technique ”, This is one of the contents of number theory .

Throwing pictures 《 The arithmetic 》 The solution is given in x?＋y?＝z? All integer solutions . Ferma points out x^n＋y^n＝z^n stay n＞3 There is no integer solution , The research on this problem has produced 19 Number theory of the century . Then Gauss 《 Number theory research 》(1801 year ) Formed a systematic number theory .

The classical content of number theory basically does not rely on the methods of other branches of Mathematics , It is called elementary number theory .17 After the middle of the century , Algebra developed under the influence of number theory 、 The geometric 、 analysis 、 Branches of mathematics such as probability , In turn, it promoted the development of number theory , Algebraic number theory appeared ( Study the roots of polynomials with integral coefficients —“ Algebraic number ”)、 Geometric number theory ( Study all coordinates that are integers in the linear coordinate system “ On the hour ”—“ Space grid ”).19 Analytic number theory appeared in the second half of the century , Study the distribution of prime numbers by analytical methods . In the 20th century, complete number theory appeared .

5. Abstract algebra

1843 year , Hamilton invented an algebra in which the commutative law of multiplication does not hold —— Quaternion algebra . In the second year , Glassman deduces several more general types of algebra .1857 year , Carlyle designed another kind of non commutative algebra —— Matrix algebra . Their research opened up abstract algebra ( Also called modern algebra ) Gate . actually , Weaken or delete some assumptions of ordinary algebra , Or replace some assumptions with other assumptions ( Compatible with other assumptions ), You can study many algebraic systems .

1870 year , Clonic gives the abstract definition of finite Abelian group ; Didkin began to use “ body ” That's what I'm saying , And we study algebras ;1893 year , Weber defined abstract body ;1910 year , Stannitz developed the general abstract theory of body ; Didkin and clonic founded ring theory ;1910 year , Stannitz summed up the inclusion Group 、 Algebra. 、 Study of algebraic systems including fields , Created abstract algebra .

1926 year , Nott achieved his ideal ( Count ) theory ;1930 year , Bilhoff established lattice theory , It comes from 1847 Boolean algebra of ; After the Second World War , There have been theories of various algebraic systems and the bulbaki School ;1955 year , Jiadang 、 Glucindick and ellenburger established homology algebra theory .

Up to now , Mathematicians have studied 200 There are many such algebraic structures , Among them, the most important derodan algebras and Lie algebras are examples of algebras that do not obey the law of Association . Most of these jobs belong to 20 century , They make the thoughts of generalization and abstraction fully reflected in Modern Mathematics .

Abstract algebra is a mathematical discipline that studies various abstract axiomatic algebraic systems . Typical algebraic systems have groups 、 Ring 、 Domain etc , They mainly originated from 19 Group theory of the century , Including group theory 、 Ring theory 、 Galois theory 、 Case theory 、 Linear algebra and many other branches , Combined with other branches of mathematics, algebraic geometry 、 Algebraic number theory 、 Algebraic topology 、 Topological groups and other new mathematical disciplines . Abstract algebra has become the universal language of most contemporary mathematics .

Now? , Algebra can be generally interpreted as the theory of letter calculation , But the meaning of letters is constantly expanding . In elementary algebra , Letters indicate numbers ; In Higher Algebra and abstract algebra , Letters represent vectors ( or n Meta ordered array )、 matrix 、 tensor 、 Spinor 、 Hypercomplex and other forms of quantity . so to speak , Algebra has developed into a general theory of formal operations .