Elementary algebra studies numbers and charactersAlgebraic operationThe theory and method, more specifically, is to study the sum of real numbers andcomplex, and theAlgebraic expressionThe mathematical branch of algebraic operation theory and methodsubject。
Elementary algebra is the extension and development of ancient arithmetic.In ancient times, when a large number of solutions to various quantitative problems were accumulated in arithmetic, in order to find a systematic and more general method to solve various quantitative relations, a solution was createdalgebraic equationThe principle of is the elementary algebra of the central problem.
AlgebraThere is no doubt that (algebra) evolved from arithmetic.As for the generation of algebra, it is not easy to say clearly.For example, if you think "algebra" means solving axtwo+Skills for symbolic algebraic equations such as bx+c=0.Then, this kind of "algebra" was developed in the sixteenth century.
If we do not require algebraic symbols to be as concise as they are today, then algebra can be traced back to an earlier era.Westerners willancient GreekmathematicianDifantuAs the ancestor of algebra.In China, algebraic problems expressed in words appeared earlier.
"Algebra" as a mathematicsProper nounRepresenting a branch of mathematics, it was officially used in China in 1859.In that year, among mathematicians in the Qing DynastyLi ShanlanI translated a book written by the British Demegan with the Englishman Wileyali. The name of the translation is Algebra.Of course, the contents and methods of algebra have long been produced in ancient China, such as《Chapter Nine Arithmetic》InequationQuestion.
From“Chapter Nine Arithmetic”After describing the equation in Volume VIIINumerical algebraChina has always maintained brilliant achievements in the field of.The equation chapter of "Nine Chapters of Arithmetic" first explains that the technique of positive and negative is inexorable, just as people learn elementary algebra from positivenegativeOfFour arithmetic operationsJust like learning, the appearance of negative numbers enriches the content of numbers.In the first century BC, there were multiple equationsUnary quadratic equationandIndefinite equationSeveral.One variable quadratic equation is borrowedGeometryAnd proved.The emergence of the indefinite equation was a subject worthy of attention in China more than 2000 years agoDiophantine equationMore than 300 years earlier.
With xthree+pxtwo+Qx=A formCubic equation, China in the Tang Dynasty in the seventh centuryWang Xiaotong“Seize the Ancient Suanjing”It has been recorded, use "fromOpen cubeIt is not difficult to imagine how happy Wang Xiaotong was when he got this solution. He said that anyone who could change a word in his book would be rewarded with a thousand dollars.Eleventh centuryJia XianIt has invented the same numerical equation solution as Horner's (1786-1837) method, and people can't forget the 13th century Chinese mathematiciansQin JiushaoGreat contribution in this regard.stayHistory of World MathematicsEpigraphequationThe original records ofTianyuan SkillIs simple and clear.QuaternionIt is an inevitable product of the development of Tianyuan Art.Series is an ancient thing, two thousand years ago“Zhou Bi Suanjing”And "Nine Chapters of Arithmetic"arithmatic seriesandGeometric series。Yuan Dynasty of China in the early 14th centuryZhu ShijieThe calculation of his series should be highly valued. Some of his work was recorded in European works in the 18th and 19th centuries.In the 11th century, China had completeBinomial coefficientTable, and also thePreparation method。
Historical documents reveal thatSurplus deficiency techniqueIt was sent from China to Europe.Interpolation methodChina can trace back to the sixth centuryLiu ZhuoAnd at the end of the seventh centurymonka lineInterpolation calculation with unequal spacing.Before the 14th century, many problems in algebra were studied, and China was one of the advanced countries.By the 18th and 19th centuriesLi Rui（1773-1817）、Wang Lai(1768-1813) toLi Shanlan(1811-1882), they also published many famous works on this research.
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Elementary algebraic equation
The central content of elementary algebra is solutionalgebraic equationTherefore, for a long timeAlgebraTo understand the science of algebraic equations, mathematicians also focus on the study of algebraic equations.itsresearch methodIt is highly computational.
To discuss algebraic equations, one of the first problems encountered is how to compose the quantitative relations in practiceAlgebraic expression, and then according toEquimetric relationList the equations.So an important content of elementary algebra is algebraic expression.Due to the different quantitative relations in things, elementary algebra has formedIntegral form、fractionandRadicalThese three kinds of algebraic expressions.Algebraic expressions are the incarnation of numbers, so in algebra, they can beFour arithmetic operations, obeyBasic operationLaw, but also can be carried outPower(This is only limited to the exponential power of rational number) and square root.These six operations are usually calledAlgebraic operation, to distinguish it from theArithmetic operation。
In the process of the generation and development of elementary algebraalgebraic equationThe research on the number also promoted the further development of the concept of numberConceptual extensionreachRational numberThe range of such numbers as positive and negative integers, positive and negative fractions, and zero.This is another important content of elementary algebra, which is the extension of the concept of number.
With rational numbers, the problems that elementary algebra can solve are greatly expanded.But, some one yuanPolynomial equationThere is still no solution in the rational range.Thus, the concept of number is expanded to real number once, and then tocomplex。
Nine Chapters of Arithmetic
Is there still an algebraic equation that has not been solved in the complex number range, and the complex number must be expanded again?Mathematicians said: No, no.This is a famous theorem in algebra——Fundamental theorem of algebra。This theorem is simplynSub equation hasnRoot (s).December 15, 1742SwitzerlandmathematicianEulerHe made a clear statement in a letter, and then another mathematician, Gauss of Germany, gave a strict proof in 1799.
The basic contents of elementary algebra are as follows:
Three kinds of numbers rational numberIrrational number、complex；
The content of elementary algebra is roughly equivalent to the content of algebra courses offered in current middle schools, but it is not completely the same.For example, strictly speaking, the concept, arrangement and combination of numbers should be included in the content of arithmetic;Function is the content of analytical mathematics;InequalityThe solution ofsolve equationsHowever, inequality, as a method of estimating values, essentially belongs toAnalytical mathematicsThe scope of;Coordinate methodIs researchanalytic geometryYes, wait.These are just a kind of arrangement method formed in history.
Elementary algebra is the continuation and promotion of arithmeticAlgebraic expressionAnd solving algebraic equations.Algebraic operationIt is characterized by adding, subtracting, multiplying, dividing, and square root extraction only for a limited number of times.All elementary algebra has ten rules.This is the key point for learning elementary algebra.
Ten Rules
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Elementary algebra is the continuation and promotion of arithmeticAlgebraic expressionAnd solving algebraic equations.Algebraic operationIt is characterized by adding, subtracting, multiplying, dividing, and square root extraction only for a limited number of times.All elementary algebra has ten rules.This is the key point for learning elementary algebra.
TwoequationBasic properties: add (subtract) a number on both sides of the equation at the same time, and the equation remains unchanged;Both sides of the equation are multiplied (divided) by a non-zero number at the same time, and the equation remains unchanged;
Threeindex law：Power of same baseMultiplication,base numberunchangedindexAddition;PowerPower, base constant index multiplication;The power of the product is equal to the product of the power.
primaryAlgebraIt further develops in two aspects. On the one hand, it studies more unknownsEquations；On the other hand, the research on unknowns is more frequent (unitary)Higher order equation。At this time, algebra has moved from elementary algebra toAdvanced AlgebraThe direction oflinear algebraAnd (unary) polynomial algebra.
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Research on Elementary Algebra
Elementary algebra refers to the theory of algebraic equations before the first half of the 19th century. It mainly studies whether an algebraic equation (system) can be solved, how to find all roots (including approximate roots) of algebraic equations, and various properties of the roots of algebraic equations.Algebra can be divided into two parts: elementary algebra and abstract algebra.Elementary algebra is the extension and development of more ancient arithmetic, while abstract algebra is generated and developed on the basis of elementary algebra.
Elementary algebra starts from the simplestUnary linear equationAt the beginning, on the one hand, we will further discuss the linear equations of two and three variables, and on the other hand, we will study the equations of more than two times and can be transformed into quadratic equations.Continuing to develop along these two directions, algebra is discussing any number ofunknown numberThe linear system of equations, also known as linear system of equations, also studies the higher degree monovariate system of equations.At this stage, it is calledAdvanced Algebra。
Advanced algebra is the general name of the advanced stage of algebra, which includes many branches.nowadaysIn collegeAdvanced algebra generally includes two parts:linear algebra, polynomial algebra.Advanced algebra further expands its research objects on the basis of elementary algebra, introducing many new concepts and quantities that are very different from those usually used, such as the most basic ones are set, vector andvector space Etc.These quantities have the operation characteristics similar to numbers, but the research methods and operation methods are more complex and abstract.
aggregateIt is the whole of things with certain attributes;vectorIs a quantity that has both a numerical value and a direction;vector space Also calledlinear space, is a set of many vectors and conforms to the rules of some specific operations.The object of operation in vector space is no longer just a number, but a vector, and its operation properties are also very different.
