People's Education Edition seventh grade mathematics I. Rational number Rational number: ① integer → positive integer /0/ negative integer ② score → positive score/negative score.
Number axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis. ② Any rational number can be represented by a point on the number axis. (3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other. On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal. The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.
Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (2) The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
Operation of rational numbers: addition: ① Add the same sign, take the same sign, and add the absolute values. ② When the absolute values are equal, the sum of different symbols is 0; When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) A number and 0 add up unchanged.
Subtraction: Subtracting a number equals adding the reciprocal of this number.
Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value. ② Multiply any number by 0 to get 0. ③ Two rational numbers whose product is 1 are reciprocal.
Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible.
Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree.
Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first.
2. Real numbers
Irrational number: Infinitely circulating decimals are called irrational numbers.
Square root: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. If the square of a number X is equal to A, then this number X is called the square root of A. (3) A positive number has two square roots /0 square root is 0/ negative number without square root. (4) Find the square root of a number, which is called the square root, where a is called the square root.
Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ② The cube root of a positive number is positive, the cube root of 0 is 0, and the cube root of a negative number is negative. The operation of finding the cube root of a number is called square root, where a is called square root.
Real numbers: ① Real numbers are divided into rational numbers and irrational numbers. ② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range. ③ Every real number can be represented by a point on the number axis.
3. Algebraic expressions
Algebraic expression: A single number or letter is also an algebraic expression.
Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items. (3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
Algebraic expressions and fractional algebraic expressions of seventh grade mathematics published by People's Education Press: ① The algebraic expression of the product of numbers and letters is called monomial, and the sum of several monomials is called polynomial. Monomial and polynomial are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.
Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items.
Power operation: am+an = a (m+n) (am) n = amn (a/b) n = an/bn division.
Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.
There are two formulas: square difference formula/complete square formula.
Algebraic division: ① monomial division, which divides the coefficient and the power of the same base as the factor of quotient respectively; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients. Factorization: transforming a polynomial into the product of several algebraic expressions. This change is called factorization of this polynomial.
Methods: Common factor method, formula method, grouping decomposition method and cross multiplication were used.
Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged. Fractional operation: multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product.
Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction.
Addition and subtraction: ① Addition and subtraction with denominator fraction, denominator unchanged, numerator addition and subtraction. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted. Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation. B. Equations and inequalities 1, equations and equations
One-dimensional linear equation of seventh grade mathematics in People's Education Press: ① In an equation, there is only one unknown, and the index of the unknown is 1. This equation is called one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation. Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1.
Binary linear equation: An equation that contains two unknowns and whose terms are 1 is called binary linear equation.
Binary linear equations: The equations composed of two binary linear equations are called binary linear equations.
A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation. The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.
Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method.
One-dimensional quadratic equation: an equation with only one unknown term and the highest coefficient of the unknown term is 2. 1) The relationship between quadratic functions of one-dimensional quadratic equation has been studied by everyone, and we have a deep understanding of it, such as the solution and the representation in the image. In fact, the quadratic equation of one variable can also be expressed by quadratic function. In fact, the quadratic equation of one variable is also a special case of quadratic function. Then, if expressed in a plane rectangular coordinate system, the quadratic equation of one variable is the intersection of the X axis in the image and the quadratic function. Which is the solution of the equation. 2) The solution of a quadratic equation. As we all know, the quadratic function has a vertex (-b/2a, 4ac-b2/4a), which is very important for everyone to remember, because as mentioned above, the quadratic equation with one variable is also a part of the quadratic function, so he also has his own solution, with which he can find all the solutions of the quadratic equation with one variable (65434). Make the equation into a complete square formula and then solve it by direct Kaiping method. (2) The common factor is extracted by factorization method, and the formula method is applied to cross multiplication. The same is true for solving quadratic equations with one variable. Using this, the equation is transformed into several products to solve (3) formula method. This method can also be used as a general method to solve quadratic equations with one variable. The roots of the equation are x 1 = {-b+√ [B2-4ac]}/2a, and x2 = {-b-√ [B2-4ac].
3) Step of solving the unary quadratic equation: (1) Step of matching method: First, move the constant term to the right side of the equation, then change the coefficient of the quadratic term into 1, and add the square of half the coefficient of the quadratic term 1 at the same time, and finally match it into a complete square formula (2) Step of factorization method: change the right side of the equation into. Formula method (here refers to the formula method in factorization) or cross multiplication, if possible, can be converted into the form of product. (3) Substitute the formula method into the coefficient of a quadratic equation, in which the coefficient of the quadratic term is A, the coefficient of the linear term is B, and the coefficient of the constant term is c 4. Vieta Theorem Understanding Vieta Theorem with Vieta Theorem In a quadratic equation with one variable, the sum of two roots =-b/. Vieta theorem can be used to find out all the coefficients in the quadratic equation of one variable commonly used in the topic. 5) The root of the quadratic equation of one variable is understood by the discriminant of the root, which can be written as "δ" and read as "Tiao ta", and δ= B2-4ac, which can be divided into three cases: i When δ> 0, the quadratic equation of one variable has two unequal real roots; II When △=0, the quadratic equation of one variable has two identical real roots; Three dang △
2. Inequalities and inequality groups Inequalities: ① Use symbols > =,