Induction of mathematical knowledge points in Xiaoshengchu

Xiaoshengchu 1 induction of mathematical knowledge points I. Arithmetic

1, additive commutative law: Two numbers are added to exchange the position of addend, and the sum is unchanged.

2. Additive associative law: A+B = B+A.

3. Multiplicative commutative law: a× b = b× a.

4. Multiplicative associative law: a × b × c = a ×(b × c)

5. Multiplicative distribution law: a× b+a× c = a× b+c.

6. The nature of division: a ÷ b ÷ c = a ÷(b × c)

7. Nature of division: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. O is divided by any number that is not O. Simple multiplication: the multiplicand and the end of the multiplier are multiplied by O. You can multiply 1 before o first, and zero does not participate in the operation, and add a few zeros at the end of the product.

8. Division with remainder: dividend = quotient × divisor+remainder

Second, the equation, algebra and equality

Equation: An equation in which the value on the left of the equal sign equals the value on the right of the equal sign is called an equation. Basic properties of the equation: When both sides of the equation are multiplied (or divided) by the same number at the same time, the equation is still valid.

Equation: An equation with an unknown number is called an equation.

One-dimensional linear equation: An equation with an unknown number of degree 1 is called a one-dimensional linear equation. Example method and calculation of learning linear equation of one variable. That is, an example is given to illustrate that the formula is replaced by χ and calculated.

Algebra: Algebra means replacing numbers with letters.

Algebraic expression: Expressions expressed by letters are called algebraic expressions. For example 3x = AB+C.

Three. Volume and surface area

Area of triangle = base × height ÷2. The formula S= a×h÷2.

Area of square = side length × side length formula S= a2

Area of rectangle = length× width Formula S= a×b

Area of parallelogram = base× height Formula S= a×h

Trapezoidal area = (upper bottom+lower bottom) × height ÷2 Formula S=(a+b)h÷2

Sum of internal angles: sum of internal angles of triangle = 180 degrees.

The surface area of a cuboid = (length× width+length× height+width× height )× 2 Formula: S=(a×b+a×c+b×c)×2.

Surface area of cube = side length × side length ×6 Formula: S=6a2.

Cuboid volume = length× width× height formula: V = abh

Volume of cuboid (or cube) = bottom area × height formula: V = abh.

Volume of cube = side length × side length × side length formula: V = a3.

Circumference = diameter × π formula: L=πd=2πr

Area of circle = radius × radius× π formula: S=πr2.

Surface (side) area of cylinder: The surface (side) area of cylinder is equal to the perimeter of bottom multiplied by height. Formula: S=ch=πdh=2πrh.

Surface area of cylinder: the surface area of cylinder is equal to the perimeter of the bottom multiplied by the height plus the area of the circles at both ends. Formula: S=ch+2s=ch+2πr2.

Volume of cylinder: the volume of cylinder is equal to the bottom area multiplied by the height. Formula: V=Sh

Volume of cone = 1/3 bottom× product height. Formula: V= 1/3Sh

Fourth, the score.

Fraction: divide the unit "1" into several parts on average, and the number representing such a part or points is called a fraction.

Comparison of fraction size: Compared with the fraction of denominator, the numerator is large and the numerator is small. Compare the scores of different denominators, divide them first and then compare them; If the numerator is the same, the denominator is big and small.

Addition and subtraction of fractions: add and subtract fractions with the same denominator, only add and subtract numerators, and the denominator remains the same. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.

Fraction multiplied by integer, numerator is the product of fractional and integer multiplication, denominator remains unchanged.

Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.

Law of fractional addition and subtraction: Fractions with the same denominator are added and subtracted, only the numerator is added and subtracted, and the denominator remains unchanged. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.

The concept of reciprocal: 1 If the product of two numbers is 1, we call one of them the reciprocal of the other. These two numbers are reciprocal. The reciprocal of 1 is 1, and 0 has no reciprocal.

A fraction divided by an integer (except 0) is equal to this fraction multiplied by the reciprocal of this integer.

The basic properties of a fraction: the numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction.

The law of division of fractions: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.

True fraction: The fraction with numerator less than denominator is called true fraction.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With a score: write a false score as an integer, and a true score is called with a score.

The basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged.

Xiaoshengchu 2 mathematical knowledge points induction. Integers and decimals

1. The smallest number is 1 and the smallest natural number is 0.

2. Meaning of decimals: Divide the integer "1" into 10, 100, 1000 ... These fractions are one tenth, percentage and one thousandth respectively ... which can be expressed by decimals.

3. The decimal point has an integer part on the left and a decimal part on the right, followed by decimal, percentile and thousandth. ...

4. Classification of decimals: Decimals are limited.

Infinitely cyclic decimal

Endless decimal

Infinite non-repeating decimal

Integers and decimals are numbers written in decimal notation.

6. Properties of decimals: Add 0 or remove 0 at the end of decimals, and the size of decimals remains unchanged.

7. Move the decimal point to the right by one, two and three places ... The original number is enlarged by 10 times, 100 times and 1000 times respectively. ...

The decimal point is shifted to the left by one place, two places and three places ... The original number is reduced by 10 times, 100 times and 1000 times respectively. ...

2. Divisibility of numbers

1. divisible: the integer A is divisible by the integer B (b≠0), and the divisible quotient is exactly an integer with no remainder, so we say that A can be divisible by B, or that B can be divisible by A. ..

2. divisor and multiple: If the number A is divisible by the number B, then A is called a multiple of B and B is called a divisor of A. ..

3. The number of multiples of a number is infinite, the minimum multiple is itself, and there is no maximum multiple.

The divisor of a number is finite, the smallest divisor is 1, and the largest divisor is itself.

4. According to whether it can be divisible by 2, natural numbers that are not 0 are divided into even numbers and odd numbers. Numbers that are divisible by 2 are called even numbers, and numbers that are not divisible by 2 are called odd numbers.

5. According to the divisor of a number, non-zero natural numbers can be divided into three categories: 1, prime number and composite number.

Prime number: If a number has only 1 and two divisors of itself, it is called a prime number. Every prime number has two divisors.

Composite number: a number. If there are other divisors besides 1 and itself, such numbers are called composite numbers. A composite number has at least three divisors.

The smallest prime number is 2 and the smallest composite number is 4.

The prime numbers in 1~20 are: 2,3,5,7,1,13, 17, 19.

The complex number of 1~20 is "4,6,8,9, 10, 12, 14, 15, 16, 18".

6. Features of numbers divisible by 2: Numbers with digits of 0, 2, 4, 6 and 8 can be divisible by 2.

Features of numbers divisible by 5: Numbers with 0 or 5 bits can be divisible by 5.

Xiaoshengchu 3 Mathematical knowledge points induction I. Sum of series

Arithmetic progression: In a column, the difference between any two adjacent numbers is certain. This number of columns is called arithmetic progression.

Basic concepts: the first item: the first number of arithmetic progression, which is generally expressed by a 1;

Number of terms: the number of all arithmetic progression, generally represented by n;

Tolerance: the difference between any two adjacent numbers in a series, generally expressed by d;

General term: a formula representing each number in a series, which is generally represented by an;

Sum of series: the sum of all numbers in this series, usually represented by Sn.

Basic idea: arithmetic progression involves five quantities: a 1, an, d, n and sn, and the general formula involves four quantities. If we know three of them, we can find the fourth; There are four quantities involved in the summation formula. If we know three of them, we can find the fourth one.

Basic formula: general formula: an = a1+(n-1) d;

General term = first term+(number of terms-1) × tolerance;

Sequence and formula: sn, = (a1+an) × n ÷ 2;

Sum of series = (first item+last item) × number of items ÷ 2;

Term number formula: n = (an-a1) ÷ d+1;

Number of items = (last item-first item) ÷ tolerance+1;

Tolerance formula: d = (an-a1) ⊙ (n-1);

Tolerance = (last item-first item) ÷ (item number-1);

Key issues: determine the known quantity and unknown quantity, and determine the formula used.

Second, the principle of addition, subtraction, multiplication, division and geometric counting

Addition principle: If there are n methods to complete a task, the first method has m 1 different methods, the second method has m2 different methods, and the nth method has mn different methods, then there are * * * different methods to complete this task: m1+m2 ...+Mn+Mn.

Key problem: determine the classification method of work.

Basic characteristics: each method can complete the task.

Multiplication principle: If a task needs to be divided into n steps, there are M 1 methods to do the first 1 step. No matter which method is used in the first 1 step, there are always m2 methods in the second step ... No matter which method is used, there are always mn methods in the n step, so there are * *.

Key problem: determine the completion steps of the work

Basic characteristics: each step can only complete some tasks.

Straight line: the trajectory formed by the movement of a point in a straight line or space in one direction or the opposite direction.

Features of straight lines: no end point, no length.

Line segment: the distance between any two points on a straight line. These two points are called endpoints.

Line segment features: there are two endpoints and one length.

Ray: one end of an infinitely extending straight line.

Ray characteristics: only one endpoint; No length

① Line segment counting rule: total number = 1+2+3+…+ (points-1);

(2) Angle law = 1+2+3+…+ (ray number-1);

③ The counting rule of rectangles: number = number of long line segments × number of wide line segments;

④ Rectangular rule of numbers: number = 1× 1+2×2+3×3+…+ number of rows× number of columns.

Knowledge points of mathematics in Xiaoshengchu: principle of addition, subtraction, multiplication and division and geometric counting

Third, prime numbers and composite numbers

Prime number: A number has no divisor except 1 and itself. This number is called prime number, also called prime number.

Complex number: A number has other divisors besides 1 and itself. This number is called a composite number.

Prime factor: If a prime number is a divisor of a number, then this prime number is called the prime factor of this number.

Prime factorization: a number multiplied by a prime number is called prime factorization. Prime factors are usually decomposed by short division. The result of any composite factorization of prime factors is unique.

The standard expression of factorization prime factor: N=, where a 1, a2, a3...an are all prime factors of composite number n, and a 1.

The formula for finding the divisor is: p = (r1+1) × (R2+1) × (R3+1) × …× (rn+1).

Prime number: If the greatest common divisor of two numbers is 1, these two numbers are called prime numbers.

Fourth, divisor and multiple

Factor and multiple: If the integer A is divisible by B, A is called multiple of B, and B is called divisor of A. ..

Common divisor: the common divisor of several numbers is called the common divisor of these numbers; The largest one is called the greatest common divisor of these numbers.

Properties of the greatest common divisor:

1, the quotient obtained by dividing several numbers by their greatest common divisor is a prime number.

2. The greatest common divisor of several numbers is the divisor of these numbers.

The common divisor of several numbers is the divisor of the greatest common divisor of these numbers.

4. Several numbers are multiplied by a natural number m, and the greatest common divisor of the product is equal to the greatest common divisor of these numbers multiplied by m. ..

For example, the divisor of 12 is 1, 2, 3, 4, 6,12;

The divisor of 18 is: 1, 2,3,6,9,18;

Then the common divisors of 12 and 18 are: 1, 2, 3, 6;

Then the greatest common divisor of 12 and 18 is: 6, marked as (12,18) = 6;

The basic method of finding the greatest common divisor;

1. prime factor decomposition: decompose prime factors first, and then multiply the same factors.

2, short division: first find the common divisor, and then multiply.

3, toss division: divide by divisor and remainder every time, and the remainder that can be divisible is the greatest common divisor.

Common multiple: the common multiple of several numbers is called the common multiple of these numbers; The smallest one is called the least common multiple of these numbers.

The multiple of 12 is: 12, 24, 36, 48 ...;

The multiple of 18 is: 18, 36, 54, 72 ...;

Then the common multiples of 12 and 18 are: 36,72,108 ...;

Then the least common multiple of 12 and 18 is 36, and it is recorded as [12,18] = 36;

Properties of the least common multiple:

1, any common multiple of two numbers is a multiple of their least common multiple.

2. The product of the greatest common divisor and the least common multiple of two numbers is equal to the product of these two numbers.

The basic method of finding the least common multiple: 1, short division to find the least common multiple; 2. The method of decomposing prime factors.

20 1720 17 Introduction to Mathematics Review: Approximation and Multiplication

5. Divisibility of numbers

I. Basic concepts and symbols:

1, divisible: If an integer A is divided by a natural number B to get an integer quotient C, and there is no remainder, it is said that A is divisible by B or B is divisible by A, and it is recorded as B | A. ..

2. Common symbols: the symbol | is separable, but the symbol | is inseparable; Because of the symbol "⊙", the symbol "∴";

Second, the separable judgment method:

1. is divisible by 2 and 5: the last digit is divisible by 2 and 5.

2. Divisible by 4 and 25: The number composed of the last two digits is divisible by 4 and 25.

3. It is divisible by 8. 125: The number composed of the last three digits is divisible by 8. 125.

4. Divisible by 3 and 9: The sum of the numbers on each digit is divisible by 3 and 9.

5. divisible by 7:

(1) The difference between the digits in the last three digits and the digits before the last three digits can be divisible by 7.

(2) Remove the last digit one by one and subtract twice the last digit to be divisible by 7.

6. divisible by 1 1:

The difference between the last three digits of (1) and the first three digits can be divisible by 1 1.

② The difference between the sum of odd digits and the sum of even digits can be divisible by 1 1.

③ Remove the last digit one by one, and the last digit can be divisible by 1 1.

7. Divisible by 13:

(1) The difference between the number composed of the last three digits and the number composed of the digits before the last three digits can be divisible by 13.

② Remove the last digit one by one. After subtracting 9 times of the last digit, it can be divisible by 13.

Third, the nature of separability:

1. If A and B are divisible by C, then (a+b) and (a-b) can also be divisible by C. ..

2. If A is divisible by B and C is an integer, then A multiplied by C can also be divisible by B. ..

3. If A can be divisible by B and B and C, then A can also be divisible by C. ..

4. If A is divisible by B and C, then A can also be divisible by the least common multiple of B and C. ..

20 1720 17 Introduction to Mathematics Review: Divisibility of Numbers

Six, the remainder problem

Attribute of remainder:

① The remainder is less than the divisor.

② If the remainder of a and b divided by c is the same, then c|a-b or c | b-a.

③ The sum of A and B divided by C equals A divided by C and B divided by C..

The product of a and b divided by c is equal to the product of a divided by c and b divided by C.

Remainder, congruence and period

I. Definition of congruence:

(1) If the remainder of two integers A and B divided by M is the same, then A and B are said to be congruent with M. ..

② Three integers A, B and M are known. If m|a-b, a and b are said to be congruent to module m, which is marked as a≡b(mod m) and read as congruent to module B.

Second, the nature of congruence:

① Self-nature: A ≡ A (mod m);

② symmetry: if a≡b(mod m), then b ≡ a (mod m);

③ transitivity: if a≡b(mod m) and b≡c(mod m), then a ≡ c (mod m);

④ sum and difference: if a≡b(mod m) and c≡d(mod m), then a+c≡b+d(mod m), a-c ≡ b-d (mod m);

⑤ Multiplicity: If a≡ b(mod m) and c≡d(mod m), then A × C ≡ B× D (mod m);

⑥ Power: If a≡b(mod m), then an ≡ bn (mod m);

⑦ ploidy: if a≡ b(mod m) and integer c, then a× c ≡ b× c (mod m× c);

Third, the preparatory knowledge about multiplier:

① if A=a×b, then ma = ma× b = (ma) b.

② if B=c+d, MB = MC+d = MC× MD

Fourth, the characteristics of the remainder after division by 3, 9, 1 1:

(1) natural number m, where n represents the sum of the digits of m, then M≡n(mod 9) or (mod 3);

(2) a natural number m, where x represents the sum of odd digits of m and y represents the sum of even digits of m, then M≡Y-X or m ≡1-(x-y) (mod11.

5. Fermat's Last Theorem: If P is a prime number, A is a natural number and A is not divisible by P, then ap- 1(mod p).

Mathematics is an important subject in junior high school examination, so we always focus on mathematics when reviewing in junior high school. Because compared with other subjects, mathematics is a subject with relatively large scores. In order to make everyone review better, we have compiled 20 17 common knowledge points of junior high school mathematics for your reference only.

Summary formula of four sum and difference problems of mathematics knowledge points in Xiaoshengchu

(sum+difference) ÷2= large number

(sum and difference) ÷2= decimal

And folding problems.

Sum \ (multiple-1)= decimal

Decimal × multiple = large number

(or sum-decimal = large number)

Difference problem

Difference ÷ (multiple-1)= decimal

Decimal × multiple = large number

(or decimal+difference = large number)

Tree planting problem