Absolute value (1) concept: the distance between a number and the origin on the number axis is called the absolute value of this number.
(1) The absolute values of two opposite numbers are equal;
② There are two numbers whose absolute values are equal to positive numbers, one number whose absolute values are equal to 0, and no number whose absolute values are equal to negative numbers.
③ The absolute values of rational numbers are all non-negative.
(2) If the letter A is used to represent rational numbers, the absolute value of the number A should be determined by the value of the letter A itself:
(1) When a is a positive rational number, the absolute value of a is itself a;
(2) When A is a negative rational number, the absolute value of A is its inverse-A;
③ When a is zero, the absolute value of a is zero.
That is | a | = {a (a > 0) 0 (a = 0) | a (a < 0)
Fraction (1) Fraction Operation
Keywords fractional four operations, sequential multiplication, division, addition and subtraction,
Multiplication and division are at the same level, and the sign of division must be changed (multiplication).
Keywords multiplication simplification, factorization priority,
Molecules and denominators meet and then operate.
The addition and subtraction denominator should be the same, and the product of denominator is the key.
It is not difficult to find that simpl common ground,
The sign must be changed in two places, and the result is the simplest.
(2) the algorithm of score
(1) approximate score
(1) If the numerator and denominator of a fraction are both monomials or products of several factors, their common factors are removed.
(2) The numerator and denominator of a fraction are polynomials, which are decomposed into factors respectively, and then the common factor is removed.
(2) Common factor extraction method
The coefficient takes the greatest common divisor of the coefficient of the numerator denominator, the letter takes the letter of the numerator denominator * * *, and the index takes the smallest exponent of the letter of the public denominator * * *, which is their common factor.
(3) division
Divide two fractions, invert the numerator and denominator of the divisor, and then multiply by the divisor.
(4) Power
Multiply the numerator by the numerator and the denominator by the denominator to simplify the complex.
Plane Cartesian coordinate system 1. Definition: Draw two mutually perpendicular number axes on a plane, and their origins coincide to form a plane Cartesian coordinate system. The horizontal axis is called the X axis or the horizontal axis, and it is customary to take the right as the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation direction is positive. The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
2. Any point on the plane can be represented by an ordered number pair, which is denoted as (a, b), where a is the abscissa and b is the ordinate.
3. The coordinate of the origin is (0,0);
The connection line with the ordinate point is parallel to the X axis;
The connecting line of the points with the same abscissa is parallel to the Y axis;
The ordinate of this point on the X axis is 0, which is expressed as (x, 0);
The abscissa of the point on the Y axis is 0, which is expressed as (0, y).
4. After the plane rectangular coordinate system is established, the coordinate plane is divided into four parts, I, II, III and IV, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant respectively. The points on the coordinate axis do not belong to any quadrant.
5. The characteristics of points in several quadrants:
The first quadrant (+,+); The second quadrant (-,+);
The third quadrant (-,-); The fourth quadrant (+,-).
6.(x, y) The point symmetrical about the origin is (-x,-y);
(x, y) The point of symmetry about x is (x,-y);
The point where (x, y) is symmetrical about y is (-x, y).
7. Distance from point to two axes: the distance from point P(x, y) to X axis is ︱ y ︳;
The distance from the point P(x, y) to the y axis is ︱x︳.
8. The coordinates of the points on the bisector of the first and third quadrants are (m, m);
The coordinates of the points on the bisectors of the second and fourth quadrants are (m, -m).
After congruent triangles (1) flips and translates, two triangles that can completely overlap are called congruent triangles, and the three sides and three angles of the two triangles are equal.
(2) The nature of congruent triangles
1. The angles corresponding to congruent triangles are equal.
2. The corresponding sides of congruent triangles are equal.
3. Vertices that can completely coincide are called corresponding vertices.
4. The heights of the corresponding sides of congruent triangles are equal.
5. The bisectors of the corresponding angles of congruent triangles are equal.
6. The median lines of the corresponding sides of congruent triangles are equal.
7. congruent triangles is equal in area and circumference.
8. The trigonometric functions of congruent triangles corresponding angles are equal.
(3) congruent triangles's judgment
(1)SSS (side by side)
A triangle with three equal sides is congruent triangles.
(2)SAS (edge)
A triangle with two equal corners is congruent triangles.
(3)ASA (corner)
Two angles and their sides correspond to the congruence of a triangle.
(4)AAS (corner)
The opposite sides of two angles and one angle correspond to congruences of equal triangles.
(5)RHS (right angle, hypotenuse, edge)
In a pair of right-angled triangles, the hypotenuse is equal to the other right-angled side.
One-dimensional linear inequality (group) 1. Inequality: a formula connecting two algebraic expressions with inequality symbol >
2. The basic properties of inequality:
Add (or subtract) the same number or the same algebraic expression on both sides of inequality A, and the direction of inequality remains unchanged;
Both sides of inequality B multiply (or divide) the same positive number, and the direction of inequality remains unchanged;
Both sides of C inequality are multiplied by (or divided by) the same negative number, and the direction of inequality should be changed.
3. Solution set of inequality: the value of the unknown quantity that can make the inequality hold is called the solution of this inequality; The set of all solutions of an inequality is called the solution set of this inequality.
4. One-dimensional linear inequality: an inequality with only one unknown number, degree 1 and coefficient not equal to zero is called one-dimensional linear inequality; Its standard form is ax+b > 0 or ax+b < 0, (a≠0).
5. Represent by inequality, and solve the inequality group by number axis or formula (formula (simple inequality): the same big takes the big, the same small takes the small, the big (middle) small (middle) big takes the middle, the big (middle) size (middle) is small, and the solution is gone.
Intersecting lines and parallel lines 1. Properties of parallel lines
Property 1: Two straight lines are parallel and equal to the complementary angle. Property 2: Two straight lines are parallel and the internal dislocation angles are equal. Property 3: Two straight lines are parallel and complementary. Determination of parallel lines:
Judgment 1: congruent angles are equal and two straight lines are parallel. Decision 2: The internal dislocation angles are equal and the two straight lines are parallel. Judgment 3: The internal angles on the same side are equal and the two straight lines are parallel.
2. Adjacent complementary angles: among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are adjacent complementary angles.
Diagonal: Two sides of one angle are relative extension lines of another angle, and two angles like this are diagonal to each other.
Perpendicular: When two straight lines intersect at right angles, they are said to be perpendicular to each other, and one of them is said to be perpendicular to the other.
Parallel lines: In the same plane, two disjoint lines are called parallel lines. Conformal angle, internal dislocation angle and ipsilateral internal angle:
3. Isomorphism angles: ∠1and ∠ 5. Diagonal lines with the same positional relationship like this are called isomorphic angles.
Internal angles: ∠2 and ∠6 A pair of angles like this is called an internal angle.
The diagonal lines such as ∠ 2 and ∠ 5 are called ipsilateral internal angles. Proposition: A statement that judges a thing is called a proposition.
4. Translation: In a plane, a figure moves a certain distance in a certain direction. This movement of graphics is called translation transformation, or translation for short.
Corresponding point: every point in the new graphic after translation is obtained by moving a point in the original graphic. Such two points are called corresponding points.
Evaluation of algebraic expression 1. Algebraic expression: replace the letters in algebraic expression with numerical values, and the calculated result is called algebraic value.
2. Evaluation of algebraic expression: the value of algebraic expression can be directly substituted into calculation. If a given algebraic expression can be simplified, it should be simplified before evaluation.
The required questions briefly summarize the following three types:
① The known conditions are not simplified, but the given algebraic expression is simplified;
② given conditions are simplified, given algebraic expressions are not simplified;
③ The known conditions and given algebraic expressions should be simplified.
chapter one
A dollar at a time is different from a dollar at a time.
1, i