Algebra is a branch of mathematics that studies the theory and method of algebraic operation of numbers and words, more precisely, it is a branch of mathematics that studies the theory and method of algebraic operation of real numbers and complex numbers and polynomials and their coefficients. Elementary algebra is the extension and development of old arithmetic. 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. Then, this kind of "algebra" was developed in the sixteenth century.
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 Diao Fan, an ancient Greek mathematician in the third century BC, as the originator of 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 of the Qing Dynasty, and Leali, an Englishman, translated a book written by Di Yaogan, an Englishman, and the translated name was 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".
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. However, some equations still have no solution within the scope of rational numbers. Thus, the concept of number was once extended to real numbers, and then further extended to complex numbers.
Then, in the range of complex numbers, is there still an equation that has no solution and must be extended again? Mathematicians say, no need. 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.
Combined with the above analysis, the basic content of elementary algebra is:
Three Numbers-Rational Number, Irrational Number and Imaginary 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.
These ten rules are:
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; The power of the index is equal to the constant base index; 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.
Algebraic simplification:
Algebra simplified evaluation is an important and difficult content in junior high school mathematics teaching. Students can't find the starting point when solving problems, and choosing the wrong method often gets twice the result with half the effort. How to improve learning efficiency and tide over difficulties smoothly, the author classifies and summarizes this problem and discusses its solutions for students' reference.
1. The given algebraic expression is simplified if the known conditions are not simplified.
2. Given the simplification of conditions, given algebra is not simplified.
3. Known conditions and given algebraic expressions should be simplified.
Lesson 3 Algebraic Expressions
Learning point
Algebraic formula, algebraic formula's value, algebraic expression, similar terms, merging similar terms, bracket removal and bracket removal rules, power operation rules, algebraic expression's addition, subtraction, multiplication, division and power multiplication formulas, positive integer exponential power, zero exponential power and negative integer exponential power.
Outline requirements
1, to understand the concept of algebra, simple algebra will be listed. If we understand the concept of algebraic value, we can get the algebraic value correctly.
2. Understanding the concepts of algebra, monomial and polynomial will arrange polynomials in descending order (or ascending order), and understanding the concepts of similar items will merge similar items;
3. Master same base powers's multiplication, division, power multiplication algorithm and product multiplication algorithm, and be proficient in the operation of digital exponential power;
4. Be able to skillfully use multiplication formulas (square difference formula, complete square formula and (x+a)(x+b)=x2+(a+b)x+ab) for operation;
5. Master the operations of addition, subtraction, multiplication, division and power of algebraic expressions, and can perform simple mixed operations of addition, subtraction, multiplication, division and power of algebraic expressions.
Key points of inspection
Related concepts of 1. algebraic expression.
(1) Algebraic formula: Algebraic formula is a formula that connects numbers or letters representing numbers with operation symbols (addition, subtraction, multiplication, division, multiplication and root). A single number or letter is also an algebraic formula. It has "
(2) the value of algebraic expression; Replace the letters in the algebraic expression with numerical values, and the calculated result p is called algebraic value.
The algebraic value can be directly substituted into the calculation. If a given algebraic expression can be simplified, it should be simplified before evaluation.
(3) Classification of algebraic expressions
2. Related concepts of algebraic expressions
(1) monomial: An algebraic expression containing only the product of numbers and letters is called a monomial.
For a given monomial, we should pay attention to analyze its coefficient, which letters it contains and the index of each letter.
(2) Polynomial: The sum of several monomials is called polynomial.
For a given polynomial, we should pay attention to analyze how many terms it has and what each term is, and then analyze each term as a monomial.
(3) The descending order and ascending order of polynomials
The exponent of a polynomial in descending alphabetical order is called a polynomial in descending alphabetical order.
Arranging a polynomial according to the exponent of letters from small to large is called arranging this polynomial in ascending order.
Give a polynomial and arrange it in descending or ascending order as needed.
(4) Similar projects
Items with the same letters and the same index are called similar terms.
To judge whether a given item belongs to the same category, we should know that items in the same category can be merged, that is, X can represent the letter part in the formula of single item.
3. Algebraic expression operation
Addition and subtraction of (1) algebraic expressions: addition and subtraction of several algebraic expressions. Usually, each algebraic expression is enclosed in parentheses and then connected by an addition and subtraction sign. The general steps of algebraic expression addition and subtraction are as follows:
(1) If you encounter brackets, remove the brackets according to the rule of removing brackets: there is a "ten" in front of the brackets, and remove the brackets and the "+"in front. Everything in brackets has the same symbol, and the "one" comes before the brackets. Remove the brackets and the preceding "one". Everything in brackets has the same symbol.
(2) Merging similar items: adding the coefficients of similar items, and taking the result as the coefficient. Letters and the index of letters remain unchanged.
(2) Multiplication and division of algebraic expressions: the multiplication (division) of monomials refers to the multiplication (division) of their coefficients and the same letters respectively. For the letters contained in the monomial (division), the multiplication (division) of the same letter and its exponent as a factor of the product (quotient) need the same operational properties of the base:
Multiply (divide) a polynomial by a monomial, first multiply (divide) each term of the polynomial by the monomial, and then add the products (quotients).
Multiply polynomials by multiplying each term of one polynomial by each term of another polynomial, and then add the products.
When encountering a special form of polynomial multiplication, you can also directly calculate:
(3) The efficacy of algebraic expressions
The single power takes the power of the coefficient as the coefficient of the result, and then takes the power obtained by multiplying the number of powers and the index of letters as the factor of the result.