People's education edition, seventh grade, the first volume of mathematical knowledge 1
Addition and subtraction of algebraic expressions
I. Algebraic expressions
1, the formula that connects numbers or letters representing numbers with operation symbols is called algebraic expression. A single number or letter is also algebraic.
2. Replace the letters in the algebraic expression with numerical values, and the result calculated according to the operational relationship in the algebraic expression is called the value of the algebraic expression.
Second, algebraic expressions.
1, single item:
(1) An algebraic expression consisting of the product of numbers and letters is called a monomial.
(2) The numerical factor in a single item is called the coefficient of the item.
(3) The sum of the indices of all the letters in the monomial is called the number of times of the monomial.
2.polynomial
The sum of (1) monomials is called a polynomial.
(2) Each monomial is called a polynomial term.
(3) Items without letters are called constant items.
3. Ascending order and descending order
(1) Arranging polynomials according to the exponent of x from large to small is called descending power arrangement.
(2) Arranging polynomials according to the exponent of x from small to large is called ascending power arrangement.
Third, the addition and subtraction of algebraic expressions
The theoretical basis of 1. Algebraic addition and subtraction is: the rule of removing brackets, the rule of merging similar items, and the multiplication distribution rate.
Rules for removing brackets: If there is a "ten" in front of brackets, remove brackets and the "+"in front of them, and all items in brackets will remain unchanged; If there is a "one" in front of the bracket, remove the bracket and the "one" in front, and change the symbols of everything in the bracket.
2. Similar items: items with the same letters and the same letter index are called similar items.
Merge similar projects:
(1) The concept of merging similar terms: merging similar terms in polynomials into one term is called merging similar terms.
(2) Rules for merging similar items: when the coefficients of similar items are added, the result will be taken as the coefficient, and the index of letters will remain unchanged.
(3) Steps to merge similar projects:
A. find similar projects accurately.
B. Reverse the distribution law, and add the coefficients of similar items together (with brackets) to keep the letters and the indexes of letters unchanged.
C. write the results after the merger.
(4) Note:
A. If the coefficients of two similar items are opposite, the result after merging similar items is 0.
B. Don't leave out items that can't be merged.
C. As long as there are no more similar terms, it is the result (which may be a single term or a polynomial).
Note: The key to merging similar items is to correctly judge similar items.
3, several general steps of algebraic expression addition and subtraction:
(1) List algebraic expressions: enclose each algebraic expression in parentheses and then connect it with a plus sign and a minus sign.
(2) Open brackets according to the rules for opening brackets.
(3) Merge similar items.
4, the general steps of algebraic evaluation:
Algebraic simplification of (1)
(2) Substitution calculation
(3) For some special algebraic expressions, "whole substitution" can be used for calculation.
People's Education Press, Grade Seven, Volume One, Mathematical Knowledge II
A preliminary understanding of graphics
A, three-dimensional graphics and plane graphics
1, cuboids, cubes, spheres, cylinders and cones are all three-dimensional figures. In addition, prisms and pyramids are also common three-dimensional figures.
2. Rectangular, square, triangle and circle are all plane figures.
3. Many three-dimensional graphics are surrounded by some plane graphics, which can be expanded into plane graphics by proper cutting.
Second, points and lines
1, there is a straight line after two, and there is only one straight line.
2. The line segment between two points is the shortest.
3. The line segment AB at point C is divided into two equal line segments AM and MB, and point M is called the midpoint of line segment AB. Similarly, line segments have bisectors and quartiles.
4. The figure formed by the infinite extension of line segments in one direction is called ray.
Third, the angle
The 1. angle is a graph composed of two rays with a common endpoint.
2. Rotate around the endpoint until the end edge and the start edge of the angle form a straight line, and the formed angle is called a flat angle.
3. Rotate around the endpoint until the ending edge and the starting edge overlap again, and the angle formed is called fillet.
4. Degrees, minutes and seconds are commonly used units of angle measurement.
Divide a fillet into 360 equal parts, each equal part is an angle of one degree, and record it as1; Divide the angle of 1 degree into 60 equal parts, each part is called an angle of 1 minute, and it is recorded as1'; Divide the angle of 1 into 60 equal parts, and each part is called 1 sec, and it is recorded as 1 ".
Fourth, the comparison of angles.
Starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle. Similarly, there is the so-called bisector.
Verb (abbreviation for verb) complementary angle and complementary angle
1. If the sum of two angles is equal to 90 degrees (right angle), the two angles are said to be complementary.
2. If the sum of two angles is equal to 180 (flat angle), it is said that the two angles are complementary.
3. The complementary angles of equal angles are equal.
4. The complementary angles of equal angles are equal.
Six, the intersection line
1, definition: When two straight lines intersect and one of the four angles formed is a right angle, then the two straight lines are perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot.
2. Note:
(1) The vertical line is a straight line.
⑵ The four angles formed by two straight lines with vertical relationship are all 90.
(3) Verticality is a special case of intersection.
(4) Vertical symbols: a⊥b, AB⊥CD.
3. Draw a known straight line with countless vertical lines.
4. There is one and only one straight line perpendicular to the known straight line.
5. Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest. Simply put: the vertical line segment is the shortest.
6. The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.
7. One vertex has a common * * *, one side has a common * * *, and the other side is an extension line opposite to each other. Such two angles are called adjacent complementary angles.
There are four pairs of adjacent complementary angles when two straight lines intersect.
8. One vertex has a common * * *, and both sides of the corner are opposite extension lines. These two angles are called antipodal angles. Two straight lines intersect and have two opposite angles. The vertex angles are equal.
Seven, parallel lines
1. In the same plane, if two straight lines have no intersection, they are parallel to each other, and it is recorded as: a ∨ b.
2. Parallelism axiom: After passing a point outside a straight line, there is one and only one straight line parallel to this straight line.
3. If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
4, determine the method of two straight lines parallel:
(1) Two straight lines are cut by the third straight line. If congruent angles are equal, two straight lines are parallel. To put it simply: the same angle is equal and two straight lines are parallel.
(2) Two straight lines are cut by a third straight line. If the internal dislocation angles are equal, two straight lines are parallel. To put it simply: the internal dislocation angles are equal and the two straight lines are parallel.
(3) Two straight lines are cut by a third straight line. If they are complementary to each other, the two straight lines are parallel. To put it simply: the internal angles on the same side are complementary and the two straight lines are parallel.
5, the nature of parallel lines
(1) Two parallel lines are cut by a third straight line and have the same angle. To put it simply: two straight lines are parallel and have the same angle.
(2) Two parallel lines are cut by a third line, and the internal dislocation angles are equal. To put it simply: two straight lines are parallel and their internal angles are equal.
(3) The two parallel lines are cut by the third straight line and complement each other. Simply put, two straight lines are parallel and complementary.
People's education printing plate seventh grade first volume mathematics knowledge 3
Definition of formula
1. monomial: in algebraic expressions, if only multiplication (including power) operations are involved. Or algebraic expressions that contain division but do not contain letters in division are called monomials.
2. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, which is simply referred to as the coefficient of single item; 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.
3. Polynomial: The sum of several monomials is called polynomial.
4. 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 polynomial, the degree of the term with the highest degree is called the degree of polynomial.
5. Algebraic expressions: monomials and polynomials are collectively referred to as algebraic expressions.
2.2 Addition and subtraction of algebraic expressions
1. Similar items: monomials with the same letters and the same letter index are similar items.
2. Rules for merging similar items: When the coefficients are added, the letter index remains unchanged.
3. Rules for deleting (adding) brackets: When deleting (adding) brackets, if there is a "+"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.
4. Algebraic addition and subtraction: Algebraic addition and subtraction is actually to combine similar terms of polynomials on the basis of removing brackets.
5. Power-up and power-down permutation of polynomials: arranging the terms of a polynomial according to the exponent of a letter from small to large (or from large to small) is called power-up permutation (or power-down permutation) of this letter.
Note: In general, the final results of polynomial calculation should be arranged in ascending power (or descending power).
People's education printing plate seventh grade first volume mathematics knowledge 4
rational number
1. 1, the concept of rational number:
(1) Positive integers, 0 and negative integers are collectively called integers; Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers.
(2) Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; π is not rational number;
(3) Note: among rational numbers, 1, 0 and-1 are three special numbers with their own characteristics; These three numbers divide the numbers on the number axis into four areas, and the numbers in these four areas also have their own characteristics;
2. Number axis: The number axis is a straight line that defines the origin, positive direction and unit length.
3. The opposite number:
(1) There are only two numbers with different signs, and we say that one of them is opposite to the other; The antonym of 0 is still 0;
(2) Note: The inverse of a-b+c is-A+B-C; The inverse of a-b is b-a; The inverse of a+b is-a -a-b；;
4. 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;
⑵ Note: The absolute value means the distance between the point representing a certain number on the number axis and the origin;
(3) | a | is an important non-negative number, that is | a | ≥ 0; Note: | a || b | = | a b |,
5. Rational number ratio:
(1) The greater the absolute value of a positive number, the greater the number;
(2) Positive numbers are always greater than 0 and negative numbers are always less than 0;
(3) Positive numbers are greater than all negative numbers;
(4) Two negative numbers are larger than the size, and the absolute value is larger but smaller;
5] The two numbers on the axis, the number on the right is always greater than the number on the left;
[6] Large number-decimal number > 0, decimal number-large number < 0.
1.2, rational number algorithm
1. Rational number arithmetic:
(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;
(2) Add two numbers with different symbols, take the symbol with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value;
(3) When a number is added to 0, the number is still obtained.
2. Arithmetic of rational number addition:
The commutative law of (1) addition: a+b = b+a;
(2) The associative law of addition: (a+b)+c=a+(b+c).
3. Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number; That is, a-b=a+(-b).
4. The rational number multiplication rule:
(1) Multiply two numbers, the same sign is positive, the different sign is negative, and the absolute value is multiplied;
(2) Multiply any number by zero to get zero;
(3) When several numbers are multiplied, one factor is zero and the product is zero; Each factor is not zero, and the sign of the product is determined by the number of negative factors.
5. Arithmetic of rational number multiplication:
Commutative law of (1) multiplication: ab = ba.
(2) multiplicative associative law: (ab) c = a (BC);
(3) Distribution law of multiplication: a(b+c)=ab+ac.
6. Rational number division rule: dividing by a number is equal to multiplying the reciprocal of this number; Note: Zero cannot be divisible.
7. The power law of rational numbers: the power of any positive number is positive;
1.3, the definition of power
1. The operation of seeking common ground factor product is called power;
2. In power, the same factor is called base, the number of the same factor is called exponent, and the result of power is called power;
3. Approximation accuracy: a divisor, rounded to that bit, that is, the divisor is accurate to that bit.
4. Significant digits: All digits from the first non-zero digit on the left to the exact digit are called the significant digits of this approximation.
5. Hybrid algorithm: multiply first, then multiply and divide, and finally add and subtract; Note: How to calculate simply and accurately is the most important principle of mathematical calculation.
6. Special value method: it is a method of substituting numbers that meet the requirements of the topic into guesses to verify the establishment of the topic, but it cannot be used for proof.
People's education printing plate seventh grade first volume mathematics knowledge 5
One-dimensional linear equation
3. 1, solve a linear equation.
1. Equality and equivalence: the formula connected by "=" is called equality. Note: "Equal amount can be substituted"!
2. The nature 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.
3. Equation: An equation with an unknown number is called an equation.
4. Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation; Note: "The solution of the equation can be substituted"!
5. Moving term: after changing the sign, moving the term of the equation from one side to the other is called moving term. The shift term is based on the equality attribute 1.
6. One-dimensional linear equation: An integral equation with only one unknown number, degree 1 and non-zero coefficient is a one-dimensional linear equation.
7. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, a and b are known numbers, a≠0).
8. The simplest form of linear equation with one variable: ax=b(x is unknown, a and b are known numbers, a≠0).
9. General steps for solving a linear equation with one variable: sorting out the equation ... removing the denominator ... dismantling the bracket ... changing the terms ... merging similar terms ... and converting the coefficient into 1 ... (testing the solution of the equation).
3.2, a linear equation application problems
1. Reading analysis method-mostly used for "sum, difference, multiplication and division questions"
Read the stem carefully, find out the key words that express the equal relationship, such as "big, small, many, few, yes, * * *, combination, right, completion, increase, decrease, match-",list the literal equations with these key words, and set the unknown number according to the meaning of the question. Finally, using the relationship between quantity and quantity in the question, fill in the algebraic expression and get the equations.
2. Drawing analysis method-mainly used for "trip problem"
Analyzing mathematical problems with graphics is the embodiment of the combination of numbers and shapes in mathematics. Read the question carefully, and draw the relevant figures according to the meaning of the question, so that each part of the figure has a specific meaning. Finding the equation relationship through graph is the key to solve the problem, so as to obtain the basis of concise equation. Finally, using the relationship between quantity and quantity (unknown quantity can be regarded as known quantity), filling in the relevant algebraic expression is the basis of getting the equation.
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