Basic knowledge points: 1. Classification of real numbers: 1. Rational number: Any rational number can always be written, in which P and Q are coprime integers, which is an important feature of rational numbers. 2. Irrational numbers: There are three kinds of irrational numbers encountered in junior high school: infinite square roots, for example,; Infinite decimal of infinite ring with specific structure, such as1.10/00100010001...; Numbers with specific meanings, such as π, and so on. 3. To judge the number of a real number, we can't just rely on the superficial feeling, but often we have to sort out and simplify it before we can draw a conclusion. Second, some concepts in real number 1, inverse number: only two numbers with different signs are called inverse numbers. The inverse of (1) real number a is-a; (2)a and B are reciprocal a+b=0 2, reciprocal: (1) The reciprocal of the real number a(a≠0) is; (2)a and B are reciprocal; (3) Note that 0 has no reciprocal 3. Absolute value: (1) The absolute value of a number has the following three situations: (2) The absolute value of a real number is non-negative. From the number axis, the absolute value of a real number is the distance from the point representing this number to the origin. (3) To remove the absolute value sign (simplification), you need to confirm the real number (positive and negative) in the absolute value sign, and then remove the absolute value sign. 4. Square root of square root of degree n (1), arithmetic square root: let a≥0, call it the square root of A, and call it the arithmetic square root of A. (2) Positive numbers have two square roots in opposite directions; The square root of 0 is 0; Negative numbers have no square root. (3) Cubic root: Cubic root called real number A (4) Positive numbers have positive cubes; The cube root of 0 is 0; Negative numbers have negative cubic roots. Third, the real number and the number axis 1, the number axis: the straight line defining the origin, positive direction and unit length is called the number axis. Origin, positive direction and unit length are the three elements of the number axis. 2. Correspondence between points on the number axis and real numbers: each point on the number axis represents a real number, and each real number can be represented by a unique point on the number axis. There is a one-to-one correspondence between real numbers and points on the number axis. Fourth, the comparison of real numbers 1 represents two numbers on the number axis, and the number on the right is always greater than the number on the left. 2. A positive number is greater than 0; Negative number is less than 0; Positive numbers are greater than all negative numbers; The absolute values of two negative numbers are larger but smaller. 5. Operation and addition of real number 1: (1) Add two numbers with the same symbol, take the original symbol and add their absolute values; (2) Add two numbers with different signs, take the sign of the addend with large absolute value, and subtract the one with small absolute value from the one with large absolute value. Additive commutative law and the law of association can be used. 2. subtraction: subtracting a number is equal to adding the reciprocal of this number. 3. Multiplication: (1) Multiplies two numbers, the same sign is positive, the different sign is negative, and the absolute value is multiplied. (2) multiply n real numbers, and if one factor is 0, the product is 0; If n nonzero real numbers are multiplied, the sign of the product is determined by the number of negative factors. When there are even negative factors, the product is positive. When the negative factor is odd, the product is negative. (3) Multiplication can use multiplicative commutative law, multiplicative associative law and multiplicative distributive law. 4. Division: (1) Divide two numbers, with the same sign being positive and the different sign being negative, and divide by the absolute value. (2) Dividing by a number is equal to multiplying the reciprocal of this number. (3)0 divided by any number equals 0, and 0 cannot be a dividend. 5. Power sum roots: Power sum roots are reciprocal operations. 6. Operation sequence of real numbers: power sum roots are three-level operations, multiplication and division are two-level operations, and addition and subtraction are one-level operations. If there are no parentheses, the operations at the same level should be performed from left to right in turn. For different levels of operations, the high-level operations are calculated first and then the low-level operations, and those with brackets are calculated first. No matter what kind of operation, we should pay attention to the preoperative signs. Effective numbers of intransitive verbs and scientific notation 1. Scientific notation: let n > 0, then N= a× (where 1 ≤ a 0, so we can get the solution: Example 2. If, compare the size analysis of A, B and C: ; c > 0； So it is easy to draw: a < b < C. Solution: Example 3. If it is the opposite number, find the value analysis of a+b: from the non-negative characteristics of absolute value, it can only be: a-2=0, b+2=0, that is, a=2, b= -2, so a+b=0 solution: Example 4. As we all know, A and B are opposite to each other. Solution: Original formula = Example 5. Calculation: (1) (2) Solution: (1) Original formula = (2) Algebra part Chapter 2: Basic knowledge of algebra: 1. Algebraic formula 1. Algebraic formula: connecting numbers or letters representing numbers with operational symbols. A single number or letter is also algebraic. 2. Algebraic value: replace the letters in algebra with numerical values, and the result obtained after calculation is called algebraic value. 3. Classification of algebraic expressions: 2. Related concepts and operations of algebraic expressions 1, concept (1) monomial: such as x, 7, and, the product of numbers and letters is called monomial. A single number or letter is also a monomial. The degree of monomial: In the monomial, the exponent of all letters is called the degree of monomial. Single factor: the numerical factor in a single item is called a single factor. (2) Polynomial: The sum of several monomials is called polynomial. Polynomial term: Every monomial in a polynomial is called a polynomial term. A polynomial contains several terms, so it is called a polynomial. Polynomial Degree: In a polynomial, the degree of the term with the highest degree is the degree of the polynomial. Items without letters are called constant items. Ascending (descending) power arrangement: Arranging a polynomial according to the exponent of a letter from small (large) to large (small) is called arranging polynomials according to the ascending (descending) power of the letter. (3) Similar items: items with the same letters and the same letter index are called similar items. 2. Add and subtract (1) algebraic expressions: merge similar items: add the coefficients of similar items, and the obtained results are taken as coefficients, and the letters and their indexes remain unchanged. Rules for removing brackets: brackets are preceded by a "+"sign. Remove the brackets and the "+"sign in front of them, and everything in brackets will remain the same; There is a "-"before the brackets. Remove the brackets and the "-"in front of them, and everything in brackets has changed. Rules of parenthesis: put a "+"sign in front of parentheses, and the things in parentheses remain unchanged; Parentheses are preceded by a "-"symbol, and everything in parentheses will change its symbol. Algebraic addition and subtraction is actually merging similar items. In operation, if you encounter parentheses, remove the parentheses first, and then merge similar items. (2) Algebraic expression multiplication and division: algorithm of power: where m and n are positive integers multiplied by the power of the same base number; The power of the same base number is divided by: The power of power: the power of products:. Multiplying monomial by monomial: take the product of their coefficients as the coefficient of the product, and for the same letter, use the sum of their indices as the index of this letter; For letters contained only in monomials, their exponents are regarded as factors of products. Polynomial Multiplying Single Term: Multiply each term of polynomial by single term, and then add the products. Polynomial Multiplication Polynomial: Multiply each term of one polynomial by each term of another polynomial, and then add the products. One-term division of one term: divide the coefficient and the same base by the factor of quotient, and for the division formula that only contains letters, it is used as the factor of quotient together with its index. Polynomial divided by monomial: divide each term of this polynomial by this monomial, and then add the obtained quotients. Multiplication formula: square difference formula:; Complete square formula: 3. Factorization 1. Concept of factorization: Factorization is the process of transforming a polynomial into the product of several algebraic expressions. 2. Common factorization methods: (1) common factor extraction method: (2) formula method: square difference formula:; Complete square formula: (3) cross multiplication: (4) grouping decomposition method: polynomial terms can be appropriately grouped to extract common factors or decomposed by formulas. (5) Using the root formula method: If both roots of are, there is: 3. General steps of factorization: (1) If each term of a polynomial has a common factor, then the common factor is extracted first; (2) Put forward common factor formula or common factor formula, and then consider whether formula or cross multiplication can be used; (3) For the quadratic trinomial, we should first try to cross-multiply and decompose, if not, then use the radical method. (4) Finally, consider using group decomposition method. Four. Score 1. Definition of fraction: The formula with this shape is called fraction, where A and B are algebraic expressions and B contains letters. (1) score is meaningless: when B=0, the score is meaningless; When B≠0, the score is meaningful. (2) When the score is 0: a = 0:A=0 B≠0, the score is equal to 0. (3) Fraction: The common factor of the numerator and denominator of a fraction is called a fraction. The method is to decompose the numerator and denominator factors and then reduce the common factor. (4) simplest fraction: When the numerator and denominator of a fraction have no common factor, it is called simplest fraction. If the final result of fractional operation is a fraction, it must be simplified to the simplest fraction. (5) Divisibility: The process of changing several fractions with different denominators into fractions with the same denominator equal to the original fraction is called divisibility of fractions. (6) The simplest common denominator: the product of the highest power of the denominator of all factors. (7) Rational formula: Algebraic formula and fractional formula are collectively called rational formula. 2. Basic properties of the score: (1); (2) (3) Sign-changing rule of fractions: If you change any two symbols of numerator, denominator and fraction itself, the value of fraction remains unchanged. 3. Fraction operation: (1) addition and subtraction: addition and subtraction of fractions with the same denominator and the same denominator, and addition and subtraction of molecules; Fractions with different denominators are added and subtracted, and then divided into fractions with the same denominator. (2) Multiplication: factorize the numerator and denominator of each fraction, then multiply the numerator by the numerator and the denominator by the denominator. (3) Division: dividing by a fraction is equal to multiplying its reciprocal. (4) Power: The power of the fraction is the power of the numerator and denominator respectively. V. Quadratic radical 1 and the concept of quadratic radical: the formula is called quadratic radical. (1) The simplest quadratic root: the factor of the square root is an integer and the factor is an algebraic expression. A quadratic root that does not contain a completely openable factor in the square root is called the simplest quadratic root. (2) Similar quadratic roots: The quadratic roots with the same number of roots after being transformed into the simplest quadratic roots are called similar quadratic roots. (3) Rationalization of denominator: the radical sign in denominator is named as denominator rationalization. (4) Physical and chemical factors: two algebras multiplied by quadratic roots. If their product does not contain quadratic roots, we say that these two algebras are physical and chemical factors (commonly used physical and chemical factors are: sum; And) 2. Properties of quadratic radical: (1); (2) ; (3) (a≥0，b≥0)； (4) 3. Operation: (1) addition and subtraction of quadratic roots: after converting each quadratic root into the simplest quadratic root, merge similar quadratic roots. (2) Multiplication of quadratic roots: (a≥0, b≥0). (3) Division of quadratic root: If the final result of quadratic root operation is a root, it should be transformed into the simplest quadratic root. Example: 1. Factorization: 1. Method of extracting common factor: Example 1. Analysis: first extract the common factor, and then use the square difference formula to solve it: omit the [summary law] factorization is based on the first extraction, then the formula, etc. However, the first factor should be decomposed until it can no longer be decomposed, and it is often necessary to make a final review of each factor after decomposition. 2. Cross multiplication: Example 2, (1); (2) Analysis: It can be regarded as a quadratic trinomial of sum (x+y), and it is preliminarily decomposed by cross multiplication. Solution: When applying cross multiplication, pay attention to a single letter, a polynomial or algebraic expression, and sometimes cross multiplication needs to be used continuously. 3. Grouping decomposition method: Example 3. Analysis: the first group, the first and second items are a group, the third and fourth items are a group, and then the formula is extracted. [Summary of Laws] Polynomials are properly grouped into basic method factor groups, and the purpose of grouping is to solve problems by proposing common factors, cross multiplication or formula methods. 4, radical formula method: example 4, solution: omitted 2, clever operation of formula Example 5, calculation: analysis: factorization using square difference formula to simplify decimal operation. Solution: Grasp the characteristics of the three multiplication formulas and use them flexibly. In particular, we should master several variants of the formula, the reverse use of the formula, and master the skills of using the formula to make the operation simple and accurate. 2. Simplification and evaluation: Example 6. Simplify first, then evaluate:, where x =–1y = solution: abbreviation [rule summary] must be simplified first and then substituted for evaluation. Pay attention to the rule of removing brackets. 3. Calculation of scores: Example 7. Simplified analysis:-can be regarded as a solution: omit [law summary] When calculating scores: (1) When division is converted into multiplication, the numerator and denominator should be reversed; (2) pay attention to the negative sign. 4. Calculation example of roots 8. Knowing that the sum of the simplest quadratic roots is the same kind of quadratic roots, find the value of B. Analysis: According to the definition of similar quadratic roots, we can get: 2b+1= 7–B. Solution: Briefly describe the properties and operations of quadratic roots, especially the simplification, evaluation and application of quadratic roots are the main contents of senior high school entrance examination. Chapter 3 of Algebra: Equations and Basic Knowledge Points of Equations: 1. Concepts related to equation 1 Equation: An equation with an unknown number is called an equation. 2. Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation, and the solution of the equation containing the unknown quantity is also called the root of the equation. 3. Solving the equation: The process of finding the solution of the equation or judging that the equation has no solution is called solving the equation. 4. Finding the root of the equation: When the equation is deformed, the root that is not suitable for the original equation is called the root of the original equation.