1. 1 positive and negative numbers
① Positive Numbers: Numbers greater than 0 are called positive numbers. (When necessary, sometimes "+"is added before the positive number. )
② Negative number: The previously learned numbers other than 0 with a negative sign "-"in front of them are called negative numbers. It has the opposite meaning to a positive number.
③0 is neither positive nor negative. 0 is the boundary between positive and negative numbers and is the only neutral number.
Pay attention to the number of opposite meanings: north and south; Things; Up and down; Left and right; Rise and fall; High and low; Growth decline, etc.
1.2 rational number
1, rational number
(1) Integer: positive integers, 0 and negative integers are collectively called integers; (2) scores; Positive score and negative score are collectively called scores; (3) Rational Numbers: Integers and fractions are collectively called rational numbers.
2. Counting axes
(1) Definition: Numbers are usually represented by points on a straight line, which is called the number axis;
(2) Three elements of the number axis: origin, positive direction and unit length;
(3) Origin: Take any point on the straight line to represent the number 0, and this point is called the origin;
(4) Relationship between points on the number axis and rational numbers: All rational numbers can be represented by points on the number axis, but not all points on the number axis represent rational numbers.
3. Inverse number
Only two numbers with different symbols are reciprocal. (For example, the reciprocal of 2 is -2, and the reciprocal of 0 is 0)
4. Absolute value
The distance between the point representing the number A on the (1) axis and the origin is called the absolute value of the number A, and it is recorded as |a|. Geometrically, the absolute value of a number is the distance between two points.
(2) The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. Two negative numbers, the larger one has the smaller absolute value.
Addition and subtraction of rational number 1.3
Rational number addition rule:
1, two numbers with the same sign are added, the same sign is taken, and the absolute values are added.
2. Add two different symbols with different absolute values, take the symbol of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value. Two opposite numbers add up to 0.
When a number is added to 0, you still get this number.
Commutative law and additive associative law.
Rule of rational number subtraction: subtracting a number is equal to adding the reciprocal of this number.
Multiplication and division of rational number 1.4
Rational number multiplication rule: two numbers are multiplied, the same sign is positive and the different sign is negative, and the multiplication takes the absolute value; Any number multiplied by 0 is 0.
Two numbers whose product is 1 are reciprocal.
Multiplicative commutative law, associative law, distributive law.
(2) Rational number division rule:
Dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number;
Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value;
Divide 0 by any number that is not equal to 0 to get 0.
1.5 power of rational number
1, the operation of finding the product of n identical factors is called power, and the result of power is called power. In the n power of a, a is called the base and n is called the exponent. The odd power of a negative number is negative and the even power of a negative number is positive. Any power of a positive number is a positive number, and any power of 0 is 0.
2. Mixed arithmetic of rational numbers: multiply first, then divide, and finally add and subtract; Operation at the same level, from left to right; If there are brackets, do the operation in brackets first, and then follow the brackets, brackets and braces in turn.
3. Using scientific notation, the number greater than 10 is expressed as the n power of × 10. Note that the range of a is1≤ a.
Chapter II Addition and subtraction of algebraic expressions
2. 1 algebraic expression
1, single
A formula consisting of the product of numbers and letters. Coefficient, degree of monomial. A monomial is an algebraic expression of the product of numbers or letters. A single number or letter is also a monomial. Therefore, the key to judge whether an algebraic expression is a monomial depends on whether the numbers and letters in the algebraic expression are products, that is, the denominator does not contain letters. If the expression contains addition and subtraction, it is not a monomial.
2. Single coefficient
Refers to the numerical factor in a single item.
3. Number of events
Refers to the sum of the indices of all the letters in a monomial.
4.polynomial
The sum of several monomials. The key to judge whether an algebraic expression is a polynomial is whether each term in the algebraic expression is a monomial. The degree of each monomial, constant term and polynomial is the highest degree among polynomials. The degree of polynomial refers to the degree of the highest term in polynomial, here is the highest term, and its degree is 6; The term polynomial refers to every monomial in a polynomial. Pay special attention to the polynomial term, including the attribute symbol before it.
5. Numbers are represented by letters or quantitative relations are represented by columns. Please note that each term of monomial and polynomial is preceded by a symbol.
6. Monomial and polynomial are collectively called algebraic expressions.
2.2 Addition and subtraction of algebraic expressions
1, similar projects
Items with the same letter and the same letter index. It has nothing to do with the coefficient before the letter (not equal to 0).
2. Similar items must meet both conditions.
(1) contains the same letters; (2) The indexes of the same letters are the same. Both are indispensable.
Similar terms have nothing to do with coefficient size and alphabetical order.
3. Merge similar projects
Combine similar terms in polynomials into one term. Reduction of criminal law, combination method and distribution method are all acceptable.
4. Rules for merging similar projects
After merging similar items, the coefficient of the obtained item is the sum of the coefficients of similar items before merging, and the letter part remains unchanged.
5. Rules for removing brackets
Remove the brackets and look at the symbols: it is a plus sign and the same sign; This is a negative sign and a positive sign.
6. The general steps of algebraic addition and subtraction: one to go, two to find, three to connect.
(1) If you encounter brackets, first remove the brackets according to the rules for removing brackets. (2) Combine similar projects. (3) Merge similar items.
Chapter III One-variable Linear Equation
3. 1 linear equation
1, the equation is an unknown equation.
2. The equations all contain only one unknown (element) X, and the exponent of the unknown X is 1 (degree). Such an equation is called a linear equation with one variable.
Note: To judge whether an equation is a linear equation, we should grasp three points:
The formula of (1) unknown is algebraic expression (the equation is an integral equation);
(2) The simplified equation contains only one unknown number;
(3) The number of unknowns in the ranking equation is 1.
3. Solving the equation is to find the value of the unknown quantity that makes the left and right sides of the equation equal, and this value is the solution of the equation.
4. Properties of the equation
Adding (or subtracting) the same number (or formula) on both sides of the (1) equation at the same time, the results are still equal;
(2) Both sides of the equation are multiplied by the same number at the same time, or divided by the same number that is not 0, and the results are still equal.
Note: When using nature, be sure to pay attention to the simultaneous changes on both sides of the equal sign; We must pay attention to the number 0 when using property 2.
3.2, 3.3 Solving linear equations with one variable
In the actual process of solving equations, the following steps may not be fully used, and some steps need to be used repeatedly. Therefore, we should pay attention to the following points when solving the equation:
(1) Denominator removal: multiply both sides of the equation by the least common multiple of each denominator, and do not omit items without denominator; The numerator is a whole, and brackets should be added after removing the denominator; Denominator removal and denominator rounding are two concepts that cannot be confused;
(2) Remove brackets: first remove brackets, then remove brackets, and finally remove braces; Don't omit the items in brackets; Don't get the symbols wrong;
③ Shift the term: move the term containing the unknown to one side of the equation, and all other terms to the other side of the equation (shift the term to change the sign);
(4) Merging similar terms: don't lose terms, the solution equation is homomorphic deformation, and each step is an equation, which can't be written in equivalent form like calculation or simplification;
⑤ The coefficient is 1: the letters and their indices remain unchanged, and the coefficient is 1. Divide the two sides of the equation by the coefficient a of the unknown quantity to get the solution of the equation. Don't confuse numerator and denominator.
3.4 Practical problems and linear equations of one variable
I. Conceptual carding
The general steps to solve practical problems by enumerating one-dimensional linear equations are as follows:
(1) Pay special attention to keywords and meanings in the exam, and find out the relevant quantitative relations;
② Set the unknown number (focus unit);
③ List the equations according to the equality relation;
④ Solve this equation;
⑤ Test and write the answer (including the company name).
Second, the way of thinking (summarize the mathematical thinking methods commonly used in this unit)
(1) modeling idea: through the analysis of the quantitative relationship in practical problems, abstract it into a mathematical model and establish a linear equation.
⑵ Equation thought: The idea of solving practical problems with equations is equation thought.
⑶ Transformation idea: In essence, the process of solving a linear equation with one variable is to transform the unknown coefficient into 1 by removing the denominator, brackets, shifting terms and merging similar terms, constantly replacing the original equation with new and simpler equations, and finally gradually transforming the equation into the form of x=a, which embodies the transformation idea of turning "unknown" into "known".
⑷ The idea of combining numbers with shapes: When solving problems with a series of equations, the quantitative relationship is analyzed with the help of line segments and charts, so that the quantitative relationship in the problem can be displayed intuitively, which embodies the superiority of combining numbers with shapes.
5. Classification idea: In the process of solving the letter coefficient equation and the absolute value symbol equation, classification discussion is often needed, and in the process of solving practical problems related to scheme design, attention should also be paid to the application of classification idea in the process.
Third, the research of mathematical thinking method.
1. When solving a linear equation with one variable, you need to know what deformation each step has made and what problems you need to pay attention to.
2. When looking for the quantitative relationship of practical problems, we should be good at using intuitive analysis methods, such as tabular method, linear analysis method and graphic analysis method.
3. The test of application problems of solving equations includes two aspects:
(1) Test whether the result is the solution of the equation;
⑵ is to judge whether the solution of the equation conforms to the actual meaning in the topic.
Fourth, the application (common equivalence relation)
Trip problem: s=v×t
Engineering problem: total workload = work efficiency × time.
Profit and loss problem: Profit = price-cost.
Interest rate = profit/cost × 100%
Price = list price × discount quantity × 10%
Savings profit: interest = principal× interest rate× time.
Sum of principal and interest = principal+interest