The first part: the meaning of numbers.

Natural number:

Fraction: divide the unit "1" into several parts on average, and the number representing such one or several parts is called a fraction. The quotient of the division of two integers can also be expressed as a fraction, that is, a ÷ b = (b ≠ 0).

3. Decimal number:

The method of judging whether a fraction can be converted into a finite decimal;

The denominator of the simplest fraction is decomposed into prime factors, of which only two factors 2 and 5 can be converted into finite decimals. (Denominator 8) The prime factor of the decomposition is 2×2×2, only 2, so it can be reduced to a finite decimal. For example, the denominator of 20 is 2×2×5, which can only be decomposed into a finite number of decimals by 2 and 5. For example, the denominator in 15 is 3×5, not 2 and 5, but 3 and 5, so it cannot be reduced to a finite decimal. )

4. Percent: The number indicating that one number is the percentage of another number is called percentage, also called percentage or percentage. Percentages are usually expressed as "%".

Percentage: "A few percent" means "a few tenths". Such as: 60% = 60%, 35% = 35%

Discount: "How much discount" means 10% of the original price, such as 50% discount and 78% discount.

Note: Percent is a special kind of score, which can only represent the score, not the quantity. So you can't use the calculation unit after the percentage.

5, integer and decimal places table:

Integer fraction decimal point

fractional part

... billion, million, million.

Numbers … billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions, billions

The unit of counting is 1000 billion billion billion, one thousandth of a billion.

6. Basic properties of division, fraction, decimal and ratio.

Application of basic attributes

Division divisor and divisor are multiplied or divided by the same number (except 0), and the quotient remains unchanged. Calculate fractional division and some simple calculations

The numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction remains the same. Reduced fraction and general fraction of fractions

Decimal Add 0 or remove 0 at the end of the decimal, and the size of the decimal remains the same. Simplify the decimal to 0.3400.

The first and last items of the ratio are multiplied or divided by the same number (except 0), and the ratio remains unchanged. Become the simplest integer ratio.

7. Decimals, fractions and percentages.

Part II: Divisibility of Numbers

1, factor and multiple:

The number of factors of a number is limited, in which the smallest factor is 1 and the largest factor is itself.

(For example: 15 The minimum factor is 1 and the maximum factor is 15. )

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

(For example: 3 1 The minimum multiple is 3 1 and there is no maximum multiple. )

2 is a multiple of 2, 3 and 5:

The characteristic of multiples of 2 is that the numbers of 0, 2, 4, 6 and 8 can be divisible by 2. (e.g. 302)

A multiple of 3 is characterized by being divisible by 3 after adding the numbers on each bit. (For example: 324 3+2+4 = 9 is divisible by 3)

A multiple of 5 is represented by a number of 0 or 5 digits. (For example, 15,105,230)

The application of divisor: Observe the unit of numerator denominator and you will soon know that it can be divisible by 2.

By observing the denominator of the numerator, we know that these numbers can be divisible by 2 and 3 at the same time.

By observing the denominator, we can know that it can be divisible by 3 and 5 at the same time.

3. Prime numbers and composite numbers, prime factors and factorization prime factors

Prime number: A number greater than 1 has only 1 and its own two factors. Such numbers are called prime numbers. (e.g. 3 1)

The prime numbers within 20 are: 2, 3, 5, 7, 1 1, 13, 17, 19, and the smallest prime number is 2.

Composite number: A number has other factors besides 1 and itself. Such numbers are called composite numbers. (e.g. 25, 30) The smallest composite number is 4.

1 is neither prime nor composite.

Prime factor: Every composite number can be written as the product of several prime numbers, where each prime number is a factor of this composite number.

Prime factor decomposition: a composite number is expressed by multiplying several prime factors, which is called prime factor decomposition. (For example: 18 = 2× 3× 3)

4. Maximum common factor and minimum common multiple, prime number:

Maximum common factor: the common factor of several numbers is called the common factor of these numbers, and the largest is called the greatest common factor of these numbers.

Least common multiple: the common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the least common multiple of these numbers.

Prime number: Two numbers whose common factor is only 1 are called prime numbers. (e.g. 5 and 7)

Two simple methods for judging prime numbers;

(1) If both numbers are prime numbers, they must be prime numbers. (For example, 3 and 1 1 are prime numbers)

② The natural numbers adjacent to two numbers must be prime numbers. (Articles 8 and 9)

(3) The number with the larger prime number among the two numbers must be a prime number.

5. Two special cases of finding the greatest common factor and the least common multiple.

If two numbers are prime numbers, the greatest common factor of these two numbers is 1, and the least common multiple is their product.

If the larger number of two numbers is a multiple of decimal, then the smaller number is the greatest common factor of these two numbers; The larger number is the least common multiple of these two numbers.

(For example, 7 and 1 1, 2 and 17, 5 and 7, 8 and 9 are prime numbers, so the greatest common factor is 1 and the least common multiple is their product.

7 and 14, 15 and 45, 25 and 75 are multiples, so the greatest common factor is a smaller number and the smallest common multiple is a larger number. )

The third part, the operation of numbers.

Examples of law or property

Addition additive commutative law: a+b = b+a.

Additive associative law: (a+b)+c = a+(b+c) 42+56 = 56+12.

42+79+58=79+(42+58)

The essence of subtraction: A-B-C = A-(B+C)

Or: a-(b+c) = a-b-c8.29-3.6-6.7 = 8.29-(3.6+6.7)

13.42—(3.42+5.98)= 13.42—3.42—5.98

Multiplication multiplication commutative law: ab = ba

Law of multiplicative association: (ab) c = a (bc)

Multiplication and distribution law: (a+b) c = AC+AC 4325 = 2543.

865 125=65( 1258)

(+)× 16= 16×+ 16×

The nature of division: ABC = a (BC) 326254 = 326 (254)

Part four: A preliminary understanding of algebra.

1, simple equation:

Equation (1): An equation with an unknown number is called an equation. (For example, it is an equation, but 3+25 is not an equation, 5+36 >; 100 is not an equation either. )

(2) Methods of solving equations: There are six forms.

A, one addend = and-another addend b, minuend = difference +meic, meiosis = minuend-difference.

D, one factor = product ÷ another factor e, dividend = quotient × divisor f, divisor = dividend ÷ quotient.

2. Ratio and proportion.

The basic property of (1) ratio: both the first term and the last term of the ratio are multiplied or divided by the same number (except 0), and the ratio remains unchanged.

The basic property of proportion: in proportion, the product of two internal terms is equal to the product of two external terms.

(2) The difference between finding proportion and simplifying proportion:

Conventional method results

Find the proportion according to the meaning of the ratio, and divide the former by the latter. This is business.

Simplified ratio According to the basic properties of ratio, the ratio is simplified to the simplest integer ratio. (The method is to divide by the greatest common factor at the same time. In the fractional ratio, the former item and the latter item are multiplied by the least common multiple at the same time. In the fractional ratio, the former item and the latter item are multiplied by the same multiple at the same time to become integers, and then converted. ) is a ratio.

3. Scale: The ratio of the distance on the map to the actual distance is called the scale. Rulers are divided into digital rulers and line rulers.

1) 2) distance on the map = actual distance × scale 3) actual distance = distance on the map ÷ scale

4. Proportional distribution: the general steps to solve the application problem of proportional distribution:

(1) Find the total number of copies first. (sum of all ratios)

(2) Write out how many parts account for the total. (The denominator is the total number of shares, and the proportion of each part is the numerator)

(3) Find the number of each part. (Multiply the total amount by a score)

The fifth part is the measurement of quantity.

1, commonly used units of measurement and their rates.

(1) length, area and unit of volume:

Length unit: kilometers, meters, decimeters, centimeters, millimeters. ...

Area unit: square kilometers, hectares, square meters, square decimeters, square centimeters. ...

Unit of volume: cubic meter, cubic decimeter (liter) and cubic centimeter (milliliter). ...

(2) Weight unit: tons, kilograms and grams.

(3) Time unit: year, month, day, hour, minute and second;

2. The judgment method of flat year and leap year:

Generally, a year that can be divisible by "Nian Nian" in a normal year is a leap year, and a year that cannot be divisible is a normal year.

The year of one hundred years should be "four hundred years", the year that can be divisible is a leap year, and the year that cannot be divisible is a normal year.

3, the transformation unit name:

X forward speed

Name of superior unit, name of subordinate unit.

progressive tax rate

The sixth part, the preliminary understanding of geometry.

1, line: straight line, ray and line segment;

2. Angle: acute angle, right angle, obtuse angle, right angle and rounded corner;

3. Triangle: acute triangle, obtuse triangle, right triangle, isosceles triangle and equilateral triangle.

Quadrilateral: rectangle, square, parallelogram and trapezoid. ...

5. Circle: (1) A circle has countless radii and countless diameters.

In the same or equal circle, all radii are equal and all diameters are equal. The diameter is twice the radius.

(2) The ratio of the circumference to the diameter of a circle is called pi.

Expressed in letters, pi is a fixed infinite acyclic decimal, usually with a value of 3. 14.

6, the perimeter and area of the plan

(1) The sum of all the edges enclosing a graph is called the perimeter of this graph.

(2) The size of the surface or closed plane figure of an object is called their area.

(3) The perimeter and area of various plane figures.

Graphic perimeter area

The circumference of a rectangle = (length× width) ÷2

C = (a+b) × 2 Area of rectangle = length× width.

s=ab

Circumference of a square = side length ×4

C = area of c =4a rectangle = side length × side length.

s=a2

Area of parallelogram = base × height

S = ah

Area of triangle = base × height ÷2

s=ah÷2

Trapezoidal area = (upper bottom+lower bottom) × height ÷2

s=(a+b) h÷2

Circumference = π× diameter

C = d or c = 2Rs =

7, three-dimensional graphics

(1) Common three-dimensional figures are: cuboid, cube, cylinder, cone and sphere.

(2) Surface area and volume: Surface area: The sum of all the areas of a three-dimensional figure is called its surface area. Volume: The size of the space occupied by a three-dimensional figure is called its volume. Volume: The volume of the object that a container can hold is called the volume of the container.

(3) Formulas for calculating the surface area and volume of various three-dimensional figures.

Named surface area volume

Surface area of cuboid = (length× width+length× height+width× height) ×2.

S = (AB+AH+BH) × 2 Volume = Length× Width× Height

v=abh

The volume of a straight cylinder

= bottom area × height

Surface area of cube = side length × side length ×6

S = 6a2 Volume = side length × side length × side length

v=a3

Surface area of cylinder = side area+two bottom areas

Cylinder volume = bottom area × height

Cone cone volume = x bottom area x height

The seventh part, simple statistical knowledge.

(1) Statistical charts are divided into bar charts, line charts and fan charts.

(2) The characteristics of each statistical chart:

Bar chart: it is easy to see the figures of various quantities.

Broken line statistical chart: it is not only easy to see the quantity of various quantities, but also reflect the change of quantity.

Department statistics chart: it can clearly show the relationship between partial quantity and total quantity.

The eighth part, the common basic quantitative relations.

1, part number+part number = total-part number = part number.

2. Smaller number+difference = larger number-smaller number = difference-difference = smaller number.

"More" can sometimes be called "expensive", "overproduction" and "exceeding" according to the specific situation. "Less" means "cheap", "less production" and "saving".

3. Number of copies (average) × number of copies = total number of copies/number of copies (average) = total number of copies/number of copies = number of copies (average)

The quantitative relationship of "average number of copies, number of copies and total number of copies" has a specific statement according to the specific situation of the topic. For example:

(1) Travel problem:

Speed x time = distance (certain) is inversely proportional,

Distance/speed = time (certain) is proportional. Distance/time = speed (certain) is proportional.

(2) Meeting questions:

Speed and X Meeting Time = Distance (Certain) Inverse Ratio

Distance ÷ meeting time = speed is proportional (certain) Distance ÷ speed is proportional = meeting time (certain).

Total round-trip distance/total round-trip time = average round-trip speed.

(3) the price problem:

Unit price × quantity = total price (certain) is inversely proportional.

Total price/unit price = quantity (certain) is directly proportional to the total price/quantity = unit price (certain) is directly proportional.

(4) Agricultural production problems:

Single output × quantity = total output (certain) is inversely proportional.

Total output ÷ quantity = single output (certain) is proportional to the total output ÷ single output = quantity (certain) is proportional.

(5) Workload problem:

Work efficiency × working hours = total workload (certain) "inversely proportional"

Total amount of work ÷ working time = working efficiency (certain) "proportional"

Total amount of work ÷ work efficiency = working time (certain) "proportional"

4. A multiple × multiple = several multiples/multiple = a multiple/multiple = multiple

5. The general method to solve the problem of score (percentage) application:

(1) Find its score = the score of its quantity/unit "1".

(2) Find its quantity = the quantity of unit "1" × the quantity of its fraction.

(3) Find the unit "1" (key), and the quantity of the unit "1" = whose quantity ÷ whose score.

6. How to get scores for application questions and copywriting questions (the question is: what percentage, what percentage):

(1)A is a fraction of B? How many times is A bigger than B? What is the percentage of B in A?

Method: First, change the word "yes" to "now", and then both parties.

(2) How much is A more than B? How much is A less than B?

Methods: (large-small) compare the numbers after the words.

The ninth part, supplementary knowledge.

1, common decimals, fractions and percentages.

mark

Decimal 0.5 0.25 0.75 0.20.4 0.60.80.125 0.375 0.625 0.875 0.1.050.04

Percentage 50% 25% 75% 20% 40% 60% 80%12.5% 37.5% 62.5% 87.5%10% 5% 4%

2. Square value of1~ 20

12= 1 22=4 32=9 42= 16 52=25 62=36 72=49 82=64 92=8 1 242=576

1 12= 12 1 122= 144 132= 169 142= 196 152=225 162=256 / kloc-0/72=289 182=324 192=36 1 252=625

3. Cubic value of1~10

13= 1 23=8 33=27 43=64 53= 125 63=2 16 73=343 83=5 12 93=729 103= 1000

4. Common values.

5. Reciprocal: Two numbers whose product is 1 are reciprocal. To ask for the reciprocal of a number (except 0), just switch the numerator and denominator.

6. Some special relationships between positive and negative proportions.

(1) The diameter of a circle is proportional to its radius ()

The circumference of a circle is proportional to its diameter (or radius).

The area of a circle is not proportional to its radius (or diameter or circumference).

(2) The surface area of a cube is proportional to the bottom area. ()

The sum of the sides of a cube is proportional to the length of the sides. (Sum of sides ÷ Length of sides = 12)

The volume of a cube is out of proportion to the bottom area. ()

(3) The side length of a square is proportional to its circumference. ()

The area of a square is out of proportion to its side length. ()

The circumference of a rectangle is constant, and the length (width) is out of proportion to the circumference.

(4) The floor area is fixed, and the square brick area is inversely proportional to the number of blocks. (number of copies × number of copies = total (certain))

The floor space is fixed, and the side length of the square brick is out of proportion to the number of blocks.

(5) The number of copies subscribed by Young Pioneers is directly proportional to the amount of money. (Total price ÷ quantity = unit price (certain))

(6) The working hours are fixed, and the time for making each part is directly proportional to the number of parts made.

(Total amount of work ÷ work efficiency = working hours (certain))

(7) If two numbers are reciprocal, then the two numbers are inversely proportional.

7. Some main algorithms

(1) Integer addition and subtraction law: number alignment (2) Decimal addition and subtraction law: decimal point alignment.

(3) Integer-decimal multiplication rule: the last bit is aligned. (4) the law of addition and subtraction of fractions with the same denominator: add and subtract molecules, and the denominator remains the same.

(5) Addition and subtraction of fractions with different denominators: divide the fractions first, and then add and subtract according to the same denominator.

(6) Law of fractional multiplication: numerator is multiplied by numerator, denominator is multiplied by denominator.

(7) Law of fractional division: A divided by B (except 0) equals the reciprocal of A multiplied by B..

(8) Law of Fractional Multiplication: First, turn a fraction into a false fraction, and then use fractional multiplication to calculate.

8. Several key formulas.

1, rectangle perimeter = (length+width) ×2 rectangle area = length × width.

2. Square perimeter = side length ×4 square area = side length × side length

3. Triangle area = base × height ÷2

4. parallelogram area = base x height

5. Trapezoidal area = (upper bottom+lower bottom) × height ÷2

6. The surface area of a cuboid = length× width× 2+length× height× 2+width× height× 2.

7. cuboid volume = length× width× height (or: bottom area× height)

8. Surface area of cube = side length × side length ×6

9. Volume of cube = side length × side length × side length (or: bottom area × height)

10, area of circle = π× radius× radius ()

1 1, circumference = π× diameter or 2×π× radius ()

12, given the diameter of the circle (d), find the radius. Radius = Diameter ÷2 ()

13. Given the circumference (c) of a circle, find its radius. Radius = perimeter ÷2÷3. 14 ()

14. Surface area of cylinder: (calculated in three steps)

(1) cylindrical side area = bottom circumference × height ()

Known cylinder bottom diameter (d): ()

The radius of the cylinder bottom is known (r): ()

② Bottom area: ()

③ Surface area = side area+two bottom areas () How many bottom surfaces are there in practical application?

15, volume of cylinder = bottom area (circular area) × height () ()

16, volume of cone = x bottom area (circular area) x height () ()

17, annular area = outer circle area (large circle)-inner circle area (small circle)