1. Algebraic expression: the expression of the number of connections, and the letter indicating this number with the operation symbol "+-×××" is called algebraic expression.

Note: There are certain restrictions on the numbers expressed by letters. First, the number obtained by letters should ensure that its formula is meaningful, and second, the number obtained by letters should also make real life or production meaningful; A single number or letter is also algebraic.

2. Some points for attention in column algebra:

(1) When the band fraction is multiplied by letters, the band fraction should be changed to a false fraction, for example, a× should be written as a;

(2) When there is a division operation in the algebraic expression, the division method and the division method are generally connected by a fractional line, such as the form written in 3 A;

3. Several important algebraic expressions: (m and n represent integers)

(1) The square difference between A and B is: A2-B2; The square of the difference between a and b is: (a-b) 2;

(2) If a, b and c are positive integers, the two-digit integer is 10a+b and the three-digit integer is10a+10b+c;

(3) If both m and n are integers, the quotient m is divided by 5, and the remainder n is 5m+n; Even number is 2n, and odd number is 2n+1; Three consecutive integers are: n- 1, n, n+1;

4. Rational number:

(1) Any number that can be written in form is a rational number. ? Not a rational number.

(2) Classification of rational numbers: ① ②

(3) Note: Among rational numbers, 1, 0 and-1 are three special numbers.

(4) Natural numbers include: 0 and positive integers.

5. Absolute value:

(1) The absolute value of a positive number is itself, the absolute value of 0 is 0, and the absolute value of a negative number is its inverse;

(2) The absolute value can be expressed as: or; The problem of absolute value is often discussed in categories;

(3) ; ;

(4) |a| is an important non-negative number, that is | a | ≥ 0; Note: |a|? |b|=|a？ b|，.

(3)a2 is an important non-negative number, that is, A2 ≥ 0; If a2+|b|=0? a=0，b = 0；

(4) According to the law, the decimal point of the cardinal number moves by one place and the decimal point of the square number moves by two places.

6. Scientific notation: Write numbers greater than 10 in the form of a× 10n, where a is a number with only one integer bit. This notation is called scientific notation.

7. Approximation precision: a divisor, rounded to that bit, that is, the divisor is accurate to that bit.

8. Significant digits: All digits from the first non-zero digit on the left to the exact digit are called significant digits of this approximation.

9. Hybrid algorithm: multiply first, then multiply and divide, and finally add and subtract;

10. Properties of the equation:

Properties of the equation 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still an equation;

Property 2 of the equation: both sides of the equation are multiplied (or divided) by the same non-zero number, and the result is still an equation.

1 1. One-dimensional linear equation: An integral equation with only one unknown number and a degree of 1 and a non-zero coefficient is a one-dimensional linear equation.

(1) The standard form of linear equation with one variable: ax+b=0(x is unknown, a and b are known numbers, a≠0).

②. The simplest form of linear equation with one variable: ax=b(x is unknown, a and b are known numbers, a≠0).

(3) The general steps of solving a linear equation with one variable: sorting out the equation, removing the denominator, removing brackets, moving terms, merging similar terms, and converting them into 1 (testing the solution of the equation).

④ Shifting term: after changing the sign, moving the term of the equation from one side to the other is called shifting term. The shift term is based on the equality attribute 1.

12. Common formulas for solving application problems with column equations:

(1) Travel Problem: Distance = Speed? Time;

(2) Engineering problem: workload = work efficiency? Working hours;

(3) Proportion: Part = All? Ratio;

(4) Downstream problem: Downstream velocity = still water velocity+water velocity, and countercurrent velocity = still water velocity-water velocity;

(5) Commodity price problem: selling price = pricing? Fold? Profit = price-cost;

(6) Perimeter, area and volume: C circle =2πR, S circle =πR2, C rectangle =2(a+b), S rectangle =ab, C square =4a,

S square =a2, S ring =π(R2-r2), V cuboid =abc, V cube =a3, V cylinder = πR2h, V cone =πR2h.

Summary of knowledge points in the second volume of grade one

1. Multiplication with the same base: am? An=am+n, constant radix, exponential addition.

2. same base powers's division: am÷an=am-n, constant base, exponential subtraction.

3. The power of the power and the power of the product: (am)n=amn, the base is unchanged, multiplied by the index; (ab)n=anbn, and the power of the product is equal to the product of the power of each factor.

4. Zero exponent and negative exponent formulas:

( 1)A0 = 1(a≠0)； A-n=, (a≠0). Note: 00, 0-2 is meaningless.

(2) For negative exponent, numbers less than 1 can be recorded by scientific notation, such as 0.0000201= 2.01×10-5.

5.( 1) square difference formula: (a+b)(a-b)= a2-b2, and the product of the sum of two numbers and the difference between these two numbers is equal to the square difference between these two numbers;

(2) Complete square formula:

(1) (A+B) 2 = A2+2AB+B2, and the square of the sum of two numbers is equal to the sum of their squares, plus twice their product;

(2) (a-b) 2 = A2-2AB+B2, and the square of the difference between two numbers is equal to the sum of their squares, minus twice their product;

③(a+b-c)2 = a2+B2+C2+2ab-2ac-2bc※

6. Formula:

(1) If the quadratic trinomial x2+px+q is completely flat, there is a relation:

(2) The quadratic trinomial ax2+bx+c can always be transformed into the form of a(x-h)2+k after the formula. ※.

Note: When x=h, the maximum (or minimum) value k of ax2+bx+c can be obtained.

(3) Note. ※.

7. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, referred to as the coefficient of single item for short;

When the coefficient is not zero, the sum of all the letter indexes in the single item is called the number of times of the single item.

8. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term;

In polynomials, the degree of the term with the highest degree is called the degree of polynomials;

Note: (If A, B, C, P and Q are constants) ax2+bx+c and x2+px+q are two common quadratic trinomials.

9. Similar items: monomials with the same letters and the same letter index are similar items.

10. Rules for merging similar items: when the coefficients are added, the letter index remains unchanged.

1 1. Rules for deleting (adding) brackets: When deleting (adding) brackets, if there is a "+"sign before the brackets, all items in the brackets remain unchanged; If there is a "-"before the brackets, all items in the brackets should be changed.

Note: In general, the final results of polynomial calculation should be arranged in ascending power (or descending power).

Plane geometric part

1, the important property of complementary angle: the complementary angle of the same angle or equal angle is equal.

The important property of complementary angle: the complementary angle of the same angle or equal angle is equal.

2. ① axiom of straight line: there is only one straight line after two o'clock.

Axiom of line segment: The line segment between two points is the shortest.

(2) Theorem about vertical line: (1) There is one and only one straight line perpendicular to the known straight line;

(2) Of all the line segments connecting a point outside and a point on the line, the vertical line segment is the shortest.

Scale: In the scale of 1:m, 1 represents the distance on the map, and m represents the actual distance. If it is 1 cm on the map, it represents the actual distance of m cm.

3. The sum of the internal angles of the triangle is equal to 180.

The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

4. Diagonal formula of n polygon:

An equilateral polygon is called a regular polygon.

5. The internal angle and formula of N-polygon:180 (n-2); The sum of the outer angles of a polygon is equal to 360 degrees.

6, judge whether three line segments can form a triangle:

①a+b & gt； C(a b is the shortest two line segments) 2A-B

7. Value range of the third party:

A-b<c< If the two sides of A+B are 5 and 8 respectively, the value range of the third side is 3.

8, the corresponding perimeter range:

If both sides are A and B, the range of perimeter is 2a.

If the two sides are 5 and 7 respectively, the range of perimeter is 14.

9. Related propositions:

A (1) triangle has at most 1 right angles or obtuse angles, at most 3 acute angles and at least 2 acute angles.

(2) The range of the maximum acute angle in the acute triangle is 60 ≤ X.

(3) The included angle of bisectors of two corners of any triangle = 90+half of the third triangle.

(4) The obtuse triangle has two heights on the outside.

(5) congruent figures have the same size (area, perimeter) and shape.

(6) Two triangles with equal areas are not necessarily congruent figures.

(7) Triangle stability.

(8) The distance from the bisector of the angle to both sides of the angle is equal.

(9) An isosceles triangle with an angle of 60 is an equilateral triangle.