The condition that a fraction is meaningful is that the denominator is not zero, and the condition that the value of a fraction is zero is that the numerator is zero and the denominator is not zero.
2. The basic nature of the fraction: the numerator of the fraction is multiplied by the denominator or divided by the algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.
3. General and approximate fractions of fractions: The key is to find the least common multiple of denominator and factorization factor.
4. Fractional operation: Fractional multiplication rule: the numerator and denominator of two fractions are multiplied respectively, and as the numerator and denominator of the result, the quotation that can be reduced becomes the simplest result.
Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor.
Fractional power law: Fractional power should be numerator and denominator respectively.
Addition and subtraction of fractions: addition and subtraction of fractions with the same denominator and addition and subtraction of molecules with the same denominator. Fractions with different denominators are added and subtracted, first divided by fractions with the same denominator, and then added and subtracted.
Mixed operation: The operation sequence is the same as before. It can be simplified by the operation speed.
5. The zeroth power of any number that is not equal to zero is equal to 1, that is; When n is a positive integer, (
The properties of positive integer exponential power operation (please review it yourself) can also be extended to integer exponential power.
6. Fractional equation: an equation with a fraction and an unknown number in the denominator-fractional equation.
The process of solving the fractional equation is essentially to multiply both sides of the equation by an algebraic expression (the simplest common denominator) and transform the fractional equation into an integral equation.
When solving a fractional equation, when both sides of the equation are multiplied by the simplest common denominator, the simplest common denominator may be 0, which increases the root, so the fractional equation must be tested.
Steps to solve the fractional equation:
(1) Simplification before simplification (2) Multiply both sides of the equation by the simplest common denominator and turn it into an integral equation; (3) solving the integral equation; (4) Root inspection.
There are two conditions to add a root: one is that its value should make the simplest common denominator 0, and the other is that its value should be the root of the whole equation after removing the denominator.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
What are the steps of applying the equation? (1) trial; (2) setting; (3) column; (4) solutions; (5) answer.
There are several types of application problems; What is the basic formula? There are basically five kinds: (1) Travel problems: basic formula: distance = speed × time, and travel problems are divided into meeting problems and catching up problems. (2) numerical problems should master the representation of decimals in numerical problems. (3) Basic formula of engineering problem: workload = working hours × working efficiency. (4) The countercurrent problem is smooth = static.
7. Scientific notation: The notation of expressing a number as a form (where n is an integer) is called scientific notation.
When an n-bit integer whose absolute value is greater than 10 is expressed by scientific notation, the exponent of 10 is
In scientific notation, when the absolute value is less than 1, the exponent of 10 is an inverse proportional function of the number of zeros before the first non-zero number (including a zero before the decimal point).
1. Definition: A function with the shape (k is a constant, k≠0) is called an inverse proportional function.
2. Image: The image of inverse proportional function belongs to hyperbola.
3. Naturally: when k> is 0, the two branches of hyperbola are located in the first and third quadrants respectively, and the y value of each quadrant decreases with the increase of x value;
When k < 0, the two branches of hyperbola are located in the second and fourth quadrants respectively, and the y value of each quadrant increases with the increase of x value. ..
4.| k |: represents the area of a rectangle surrounded by a point on the inverse proportional function image and a vertical line segment formed by two coordinate axes and two coordinate axes. pythagorean theorem
1. Pythagorean Theorem: If the lengths of two right angles of a right triangle are A and B, and the length of the hypotenuse is C, then.
2. The inverse theorem of Pythagorean theorem: If the three sides of a triangle are A, B and C, it is satisfied. Then this triangle is a right triangle.
A proposition that is proved to be correct is called a theorem.
We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Pythagorean Theorem Inverse Theorem) Quadrilateral
Definition of parallelogram: Parallelograms with two groups of parallel opposite sides are called parallelograms.
The nature of parallelogram: the opposite sides of parallelogram are equal; Diagonal angles of parallelogram are equal. Diagonal bisection of parallelogram.
Determination of parallelogram 1. Two sets of quadrilaterals with equal opposite sides are parallelograms. 2. The quadrilateral with bisector is a parallelogram.
3. Two groups of quadrangles with equal diagonal are parallelograms; 4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
The midline of the triangle is parallel to the third side of the triangle and equal to half of the third side.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Definition of rectangle: a parallelogram with a right angle.
The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided.
Rectangular judgement theorem: 1. A parallelogram with a right angle is called a rectangle. 2. Parallelograms with equal diagonals are rectangles.
A quadrilateral with three right angles is a rectangle.
Definition of diamond: parallelogram with equal adjacent sides.
The nature of the diamond: all four sides of the diamond are equal; The two diagonals of a diamond are bisected vertically, and each diagonal bisects a set of diagonals.
Decision theorem of diamond: 1. A set of parallelograms with equal adjacent sides is a diamond. 2. Parallelograms with diagonal lines perpendicular to each other are diamonds.
A quadrilateral with four equilateral sides is a diamond. (A and B are two diagonal lines)
Definition of a square: a diamond with right angles or a rectangle with equal adjacent sides. A parallelogram with a right angle and three equal sides is a square.
The essence of a square: all four sides are equal and all four corners are right angles. A square is both a rectangle and a diamond.
Square judgment theorem: 1. A rectangle with equal adjacent sides is a square. Diamonds with right angles are squares.
Definition of trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid.
Definition of right-angled trapezoid: a trapezoid with a right angle.
Definition of isosceles trapezoid: isosceles trapezoid.
The nature of isosceles trapezoid: the two angles on the same base of isosceles trapezoid are equal; The two diagonals of an isosceles trapezoid are equal.
Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid.
Auxiliary lines commonly used to solve trapezoidal problems are shown in the figure.
The center of gravity of the line segment is the midpoint of the line segment. The center of gravity of a parallelogram is the intersection of its two diagonals. The point of doubt when three center lines of a triangle meet is the center of gravity of the triangle. A rectangle with an aspect ratio (about 0.6 18) is called a golden rectangle. data analysis
1. weighted average: the calculation formula of weighted average. Understanding of weight: It reflects the importance of a certain data in the whole data.
The right to learn does not directly give numbers, but appears in the form of ratio or percentage, and the weighted average is obtained by using the frequency distribution table.
2. Arrange a set of data in order from small to large (or from large to small). If the number of data is odd, the middle number is median); This set of data. If the number of data is even, the average of the middle two data is the median of this set of data.
3. The data with the highest frequency in a set of data is the pattern of this set of data.
4. The difference between the maximum data and the minimum data in a set of data is called the range of this set of data.
5. The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation and the more stable it is.
Data collection and sorting steps: 1. Data collection II. Data arrangement 3. Data description 4. Data analysis. Investigation report writing. Communication.
6. The average value is influenced by the extreme value, and the mode is not influenced by the extreme value, which is an advantage. The calculation of median is rarely affected by extreme value.