The first volume of grade seven summarizes the knowledge points of mathematics.
Monomial and polynomial
1, algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)
2. The sum of several monomials is called polynomial. Each monomial is called a polynomial term, and the term without letters is called a constant term.
Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation.
monomial
1, an algebraic expression of the product of numbers and letters, is called a monomial.
2. The single numerical factor is called the single coefficient.
3. The index of all the letters in the monomial and the number of times called the monomial.
4. A single number or letter is also a monomial.
5. The coefficient of monomial with only letter factor is 1 or-1.
6. A number is a monomial, and its coefficient is itself.
7. The degree of a single nonzero constant is 0.
8. A single item can only contain multiplication or power operation, and cannot contain other operations such as addition and subtraction.
9. The coefficient of the monomial includes the symbol before it.
10, when the coefficient of the monomial is a fraction, it should be turned into a false fraction.
When 1 1 and the single coefficient is 1 or-1, the number "1" is usually omitted.
12, the number of monomials is only related to letters, and has nothing to do with the coefficient of monomials.
multinomial
The sum of 1 and several monomials is called a polynomial.
2. Each monomial in a polynomial is called a polynomial term.
3. The term without letters in polynomial is called constant term.
4. A polynomial has several terms, which are called polynomials.
5. Every term of polynomial includes the symbol before the term.
6. Polynomials have no concept of coefficient, but have the concept of degree.
7. The degree of the degree term in a polynomial is called the degree of the polynomial.
Integral expression
1, monomials and polynomials are collectively called algebraic expressions.
2. Both monomials and polynomials are algebraic expressions.
3. Algebraic expressions are not necessarily monomials.
4. Algebraic expressions are not necessarily polynomials.
5. Algebraic expressions with letters in denominator are not algebraic expressions; It is a fraction to learn in the future.
The seventh grade, the first volume, the summary of mathematical knowledge points, the second part
Unit 1 Rational Numbers
1. 1 positive and negative numbers
Books that have been studied before, except 0, are called negative numbers with the minus sign "-"in front of them.
Numbers other than 0 that I learned before are called positive numbers.
The number 0 is neither positive nor negative, it is the dividing line between positive and negative numbers.
In the same question, positive numbers and negative numbers have opposite meanings.
1.2 rational number
1.2. 1 rational number
Positive integers, 0 and negative integers are collectively called integers, and positive and negative fractions are collectively called fractions.
Integers and fractions are collectively called rational numbers.
1.2.2 axis
The straight line that defines the origin, positive direction and unit length is called the number axis.
Function of number axis: All rational numbers can be represented by points on the number axis.
Note: The origin, positive direction and unit length of (1) axis are indispensable.
⑵ The unit length of the same shaft cannot be changed.
Generally speaking, if it is a positive number, the point representing a on the number axis is on the right side of the origin, and the distance from the origin is a unit length; The point representing the number -a is on the left of the origin, and the distance from the origin is one unit length.
1.2.3 reciprocal
Numbers with only two different symbols are called reciprocal.
Two points representing the opposite number on the number axis are symmetrical about the origin.
Add a "-"sign before any number, and the new number represents the antonym of the original number.
1.2.4 absolute value
Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A.
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.
Rational numbers are represented on the number axis, and the order from left to right is from small to large, that is, the number on the left is smaller than the number on the right.
Compare the sizes of rational numbers: (1) Positive numbers are greater than 0, 0 is greater than negative numbers, and positive numbers are greater than negative numbers.
(2) Two negative numbers, the larger one has the smaller absolute value.
Addition and subtraction of rational number 1.3
1.3. 1 addition of rational numbers
Law of rational number addition:
(1) Add two numbers with the same sign, take the same sign, and add the absolute values.
⑵ Add two numbers with different absolute values, take the sign of the addend with larger absolute value, and subtract the number with smaller absolute value from the number with larger absolute value. Two opposite numbers add up to 0.
(3) When a number is added to 0, the number is still obtained.
When two numbers are added, the positions of the addend are exchanged and the sum is unchanged.
Additive commutative law: A+B = B+A.
Add three numbers, first add the first two numbers, or add the last two numbers first, and the sum remains the same.
Additive associative law: (a+b)+c=a+(b+c)
1.3.2 subtraction of rational numbers
The subtraction of rational numbers can be converted into addition.
Rational number subtraction rule:
Subtracting a number is equal to adding the reciprocal of this number.
a-b=a+(-b)
Multiplication and division of rational number 1.4
The rational number multiplication of 1.4. 1
Rational number multiplication rule:
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.
Any number multiplied by 0 is 0.
Two numbers whose product is 1 are reciprocal.
Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative.
When two numbers are multiplied, the exchange factor and the product are in the same position.
ab=ba
Multiply three numbers, first multiply the first two numbers, or multiply the last two numbers first, and the products are equal. C=a (BC)
Multiplying a number by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products. a(b+c)=ab+ac
Writing specification for multiplication of numbers and letters;
(1) Multiplies a number with a letter, omitting the multiplication sign or using "".
(2) Numbers multiplied by letters. When the coefficient is 1 or-1, 1 should be omitted.
(3) The band score is multiplied by letters, and the band score becomes a false score.
If any rational number is represented by the letter X, the product of 2 and X is 2x, and the product of 3 and X is 3x, then the formula 2x+3x is the sum of 2x and 3x, 2x and 3x are the terms of this formula, and 2 and 3 are the coefficients of these two terms respectively.
Generally speaking, when combining formulas with the same letter factor, it is only necessary to combine their coefficients, and the obtained results are used as coefficients, and then multiplied by the letter factor, that is,
ax+bx=(a+b)x
In the above formula, X is the letter factor, and A and B are the coefficients of ax and bx respectively.
Support removal rules:
There is a "+"before the brackets. Remove brackets and the "+"in front of brackets, and nothing in brackets will change its sign. There is a "-"before the brackets. Remove brackets and the "-"sign in front of brackets, and change all the symbols in brackets. The factors outside brackets are positive numbers, and the symbols of the items in the formula after removing brackets are the same as those of the corresponding items in the original brackets; The factor outside the bracket is negative, and the sign of each item in the formula after the bracket is opposite to that of the corresponding item in the original bracket.
1.4.2 division of rational numbers
Rational number division rule:
Dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.
Answer? b = a÷ 1
b(b？ 0)
Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. 0 divided by any one is not equal.
All the numbers of 0 get 0.
Because the division of rational numbers can be converted into multiplication, the operation can be simplified by using the operational nature of multiplication. The mixed operation of multiplication and division often turns division into multiplication first, then determines the sign of the product, and finally calculates the result.
1.5 power of rational number
1.5. 1 power?
The operation of finding the product of n identical factors is called power, and the result of power is called power. In, a is called the base and n is called the exponent. When an is regarded as the result of the n power of a, it can also be read as the n power of a. ..
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 a positive integer is 0.
Operation sequence of rational number mixed operation:
(1) first power, then multiply and divide, and finally add and subtract;
(2) unipolar operation, from left to right;
(3) If there are brackets, do the operation in brackets first, and then press brackets, brackets and braces in turn.
1.5.2 scientific counting method
Represents a number greater than 10 as. 10n (where a is an integer with only one bit and n is a positive integer).
Use scientific notation to represent n-bit integers, where the exponent of 10 is n- 1.
1.5.3 divisors and significands
A number that is close to the actual number but still different from the actual number is called a divisor.
Accuracy: an approximate value is rounded to the nearest place, so it is accurate to the nearest place.
From the first non-zero digit to the last digit on the left of a number, all digits are valid digits of this number.
For the number a expressed in scientific notation? 10n, which specifies that its valid number is ..
The seventh grade, the first volume, the summary of mathematical knowledge points, the third part
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.
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 as 1? ; Divide the angle of 1 degree into 60 equal parts, each called an angle of 1 minute, and write it as 1? ; Divide the angle of 1 into 60 equal parts, each part is called the angle of 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.
The four angles formed by two vertical lines are all 90.
(3) Verticality is a special case of intersection.
(4) vertical notation: 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.
Related articles:
1. Three math reviews in grade one.
2. The first book summary of mathematical knowledge points.
3. Summarize the knowledge points of the first volume of mathematics in grade one.
4. Summary of knowledge points in the first grade mathematics textbook