brief introduction
In ancient times, when arithmetic accumulated a large number of solutions to various quantitative problems, in order to find a systematic and more general method to solve various quantitative relations, elementary algebra centered on the principle of solving equations was produced.
There is no doubt that algebra is developed from arithmetic. As for when algebra came into being, it's hard to say clearly. For example, you think "algebra" refers to the skills of solving equations such as bx+k=0. This kind of "algebra" was only developed in the sixteenth century.
trace (back) to the source of sth
If algebraic symbols are not required to be as concise as they are now, then the generation of algebra can be traced back to an earlier era.
Westerners regard Diophantu, an ancient Greek mathematician in the 3rd century BC, as the originator of algebra, and it was Mohammed ibn Moussa, a great mathematician of the ancient Arab Empire (China called him "Hua Ci Mi Zi", born and died in 780-850 AD) who really founded algebra. In China, algebraic problems expressed in words appeared earlier.
"Algebra", as a proprietary mathematical term, represents a branch of mathematics. It was officially used in China, and it was first used in 1859. That year, Li, a mathematician in Qing Dynasty, and Leali, an Englishman, translated a book written by Di Yaogan, an Englishman. The name of the translation is algebra. Of course, the contents and methods of algebra have long been produced in ancient China. For example, there are equation problems in "Nine Chapters of Arithmetic".
form
elementary algebra
basic content
Three Numbers-Rational Number, Irrational Number and Complex Number
Three forms-algebra, fraction and root.
The central content is equation-integral equation, fractional equation, radical equation and equation.
The content of elementary algebra is roughly equivalent to the content of algebra courses offered in modern middle schools, but it is not exactly the same. For example, strictly speaking, the concept, arrangement and combination of numbers should be classified as the content of arithmetic; Function is the content of analytical mathematics; The solution of inequality is a bit like the method of solving equations, but inequality, as a method of estimating values, essentially belongs to the category of analytical mathematics; Coordinate method is the study of analytic geometry. These are just a sorting method formed in history.
Elementary algebra is the continuation and expansion of arithmetic, and the research object of elementary algebra is algebraic operation and equation solving. Algebraic operations are characterized by a limited number of operations. All elementary algebra has ten rules. This is the key point to understand and master when learning elementary algebra.
rule
Five basic algorithms: additive commutative law, additive associative law, multiplicative commutative law, multiplicative associative law and distributive law;
The basic properties of the two equations: adding a number on both sides of the equation at the same time, the equation remains unchanged; Both sides of the equation are multiplied by a non-zero number at the same time, and the equation remains unchanged;
Three exponential laws: power with the same base and exponential addition with the same base; Exponential power, constant basis, exponential multiplication; The power of the product is equal to the product of power.
Elementary algebra has been further developed in two aspects: on the one hand, it studies linear equations with many unknowns; On the other hand, it is to study higher-order equations with higher unknowns. At this time, algebra has developed from elementary algebra to advanced algebra.
( 1)a-b=0，a=b
(2)a+b=0，a=-b，b=-a
(3)a*b=0, a=0 or b=0.
(4)(a-b) (a-b)=0，a=b
advanced algebra
research objects
Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes linear algebra and polynomial algebra.
On the basis of elementary algebra, advanced algebra further expands the research object and introduces many new concepts and quantities which are completely different from usual ones, such as set, vector and vector space. These quantities have operational characteristics similar to numbers, but the research methods and operational methods are more complicated. A set is the sum of things with certain attributes; Vector is a quantity with both direction and value; Vector space, also called linear space, is a collection of many vectors, which conforms to the rules of some specific operations. The object of operation in vector space is not just a number, but a vector, and its operation properties are also very different.
Differences and relations between linear algebra and linear algebra
Many people confuse advanced algebra with linear algebra and don't understand the difference.
Advanced algebra is a specialized course for college mathematics majors, while linear algebra is a course for science, engineering and some medical majors besides college mathematics majors.
solve an equation
The central content of elementary algebra is to solve equations, so algebra has long been understood as the science of equations, and mathematicians have also focused on equations. Its research method is highly computational.
When discussing the equation, the first problem is how to form an algebraic expression from the actual quantitative relationship, and then list the equation according to the equivalence relationship. So an important content of elementary algebra is algebra. Because of the different quantitative relations in things, elementary algebra generally forms three algebraic expressions: algebraic expression, fractional expression and radical expression.
Algebraic expressions are the embodiment of numbers, so in algebra, they can all perform four operations, abide by the basic operation rules, and can also perform two new operations: power and root. These six operations are usually called algebraic operations to distinguish them from arithmetic operations that only contain four operations.
In the process of the emergence and development of elementary algebra, the study of solving equations has also promoted the further development of the concept of numbers, extending the concepts of integers and fractions discussed in arithmetic to the scope of rational numbers, so that numbers include positive and negative integers, positive and negative fractions and zero. This is another important content of elementary algebra and an extension of the concept of numbers.
With rational numbers, the problems that elementary algebra can solve are greatly expanded, but some equations still have no solutions within the scope of rational numbers. Thus, the concept of number was once extended to real numbers, and then further extended to complex numbers.
Mathematicians say that there is no need to expand complex numbers. This is a famous theorem in algebra-the basic theorem of algebra. This theorem is simply that an equation of degree n has n roots. Euler, a Swiss mathematician, made it clear in a letter in June 5438+0742+1February 5, and later Gauss, another mathematician in Germany, gave a strict proof in June 5438+0799.
algebra
Algebra, the western name of algebra, comes from the name of the important works of Arab mathematician Hua Lazimi in the 9th century. This book is called "ilm al-jabr wa' 1 muqabalah", which means "the science of reduction and cancellation". This book was translated into algebra after it was spread to Europe. In the early Qing Dynasty, two volumes of works on algebra without authors were introduced into China, which were successively translated into the New Law of Bala in Algar and Algebra (translated by Li, 1853).