1, integer range

Integers include natural numbers and negative integers, or integers consist of positive integers, zero integers and negative integers.

(1) natural number

Meaning of natural numbers: When we count objects, the numbers 0, 1, 2, 3, 4, 5, ... are called natural numbers. The number of natural numbers is infinite and there is no maximum natural number.

Basic unit of natural number: Any natural number other than "0" is composed of several "1", so "1" is the basic unit of natural number. 1 is also the smallest number.

Meaning of "0": "0" means that there is no object, and it occupies a place in counting, which means that there is no counting unit on this number. "0" can also indicate the starting point, demarcation point, etc. "0" is the smallest natural number.

Two meanings of natural number: if a natural number is used to represent the number of objects, it is called cardinal number; If natural numbers are used to represent the order in which objects are arranged, they are called ordinal numbers.

(2) Positive numbers

Definition of positive numbers. Numbers like 8,16,200 ...................................................................................................................................................

The writing and reading of positive numbers can also be preceded by a+sign, for example, +8 is pronounced as: plus eight. You can usually omit the "+"sign.

(2) Negative numbers

Definition of negative numbers Numbers like-1, -5,-132 are called negative numbers. "One" is called negative sign.

The writing and reading of negative numbers begin with "one", for example,-15 is pronounced as: negative fifteen. The larger the number, the smaller the negative number.

"0" is neither positive nor negative.

(4) The connection and difference between integer and natural number.

Natural numbers are all integers, and integers are not all natural numbers, including negative integers.

2. How to read and write integers

According to our country's counting habit, integers start with one digit, and every four digits are one level. Unit, ten, hundred and thousand are levels, indicating how many ones; Ten thousand, one hundred thousand, one million and ten million are all ten thousand, which means how many ten thousand; Billions, billions, billions, billions is billions, which means billions.

Counting unit Integers and decimals are numbers written in decimal, where one (one), ten, hundred ... are counting units of integers. Counting units are arranged in a certain order.

The position occupied by each counting unit of a digit is called a digit. For example, the "5" in 9357 is the second digit from the right, that is, the digit of "5" is ten digits.

The number of digits means that a number consists of several digits, including digits. For example, 1234 occupies four digits, which means four digits.

Decimal counting method Decimal counting method refers to all decimal one, decimal one ten, decimal hundred, decimal one thousand ... The ratio between every two adjacent counting units is "ten". This counting method is called decimal counting method.

(2) reading and writing integers

Integer reading method When reading integers, read them step by step from high to low, and read them according to the reading method of each level when reading one billion and ten thousand. Just add the words "one billion" and "ten thousand" after them, and the zeros at the end of each level will not be read. Other numbers will have zeros or only one zero will be read after several zeros.

When writing integers, write them step by step from high to low. If there is no unit on any number, write 0 on that number.

3. Comparison of integer sizes

Comparing the sizes of two integers, the more integer digits, the greater the number; If the integer digits are the same, we should look at the digits on the same digit in turn from the high digit, and the digits with larger digits on the same digit are larger.

Decimal of knowledge point

1, the meaning of decimal

Divide the integer "1" into 10, 100, 1000 ... such 1 is a few tenths, a few percent, and a few thousandths ... which can be expressed in decimals. One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths.

1, decimal reading and writing

The propulsion rate between the highest counting unit "one tenth" of the decimal part and the lowest counting unit "one" of the integer part is also ten.

(2) Decimal reading and writing

When reading decimals, the integer part is read as an integer, the integer part is read as a "zero" and the decimal point is read as a "dot". The decimal part can read the numbers on each digit in sequence.

When writing a decimal, the integer part should be written as an integer. If the integer part is zero, you should write "0" in the lower right corner of each bit, and then write the number on each bit of the decimal part in turn.

3. Comparison of decimal size

Comparing the sizes of two decimals, first look at their integer parts, and the number with the larger integer part will be larger; The same is true for the integer part, where is the largest number in the tenth digit; One tenth of the numbers are the same, and the number with the largest number in the percentile is the largest. ...

4. Rewrite numbers and find approximate values

Rewrite and omit the number (1) to write the mantissa after one digit of the number as an approximation.

For the convenience of reading and writing, large numbers are often abbreviated to "10,000" or "100 million". For example: 2365500 = 2365500 (rewrite the number as "ten thousand"). Sometimes you can omit the mantissa of this number and write it as an approximation as needed. For example, 2,365,500 ≈ 2.37 million (the mantissa after ten thousand digits is omitted), sometimes it is required to keep a decimal approximation. Such as: 7.62983≈7.6 (keep one decimal place).

When taking divisors, methods such as rounding, one-step method and one-step method are often used to omit the mantissa after a certain digit of a number.

(2) Similarities and differences between "rewriting" large numbers and "seeking divisor"

The same points are all counting units that change the original number. Use "100 million" or "10,000" as the unit as needed.

The "rewriting" of different points only changes the unit of the number, but does not change the size of the number, which is represented by "=". "Finding divisor" is a rounding method, or "one-in method" or "one-out method", which not only changes the unit of numbers, but also changes the size of numbers, and is represented by "≈".

5. Classification and nature of decimals

Classification of (1) decimals

Decimals are divided into pure decimals and decimals according to whether the integer part of decimals is 0 or not.

Decimals with an integer part of 0 are called pure decimals.

A decimal whose integer part is not 0 is called a decimal. (All pure decimals are less than 1, and all with decimals are greater than or equal to 1. )

Decimals can be divided into finite decimals and infinite decimals according to whether the multiples of decimal parts are limited or not.

A decimal with a limited number of digits in the decimal part is called a finite decimal.

Infinite decimal The fractional part with infinite digits is called infinite decimal.

Infinite decimal can be divided into infinite acyclic decimal and infinite cyclic decimal.

Cyclic decimal An infinite decimal, in which one or several numbers are repeated in turn from a certain position in the decimal part, is called an infinite cyclic decimal.

The number in which the decimal part of a cyclic decimal repeats in turn is called the cyclic segment of the cyclic decimal.

A simple way to write cyclic decimals When writing cyclic decimals, for the sake of simplicity, generally only the first cyclic segment is written, and a dot is added to the first and last digits of the cyclic segment.

(2) the nature of decimals

Add "0" or remove "0" after the decimal point, and the size of the decimal point remains the same (note: it is after the decimal point, not after the decimal point). )

(3) The movement of decimal position leads to the change of decimal size.

Move the decimal point to the right by one, two or three places ... The decimal point will be expanded to the original 10 times, 100 times, 1000 times ... Move the decimal point to the left by one, two or three places ... The decimal point will be restored to its original position. ...

(4) Common quality units, RMB units, time units and frankness between units.

(5) The judgment method of normal year and leap year.

The Gregorian calendar year is a multiple of 4, usually a leap year. The Gregorian calendar year is an integer, and it must be a multiple of 400 to be considered as a leap year.

Scores of three knowledge points

1, the meaning unit of a fraction "1" is divided into several parts on average, and the number representing such one or several parts is called a fraction.

2. Decimal unit "1" unit is divided into several parts on average, indicating a part of the score, which is called decimal unit.

3. Classification of scores

(1) Fractions with numerator less than denominator are called true fractions.

(2) Fractions with numerator greater than or equal to denominator are called false fractions.

4, the basic nature of the score The numerator and denominator of the score are multiplied or divided by the same number (except 0) at the same time, and the size of the score remains unchanged. This is called the basic nature of music score.

5. Relationship between Fraction and Division (1) The numerator of Fraction is equivalent to the dividend of Division, the denominator of Fraction is equivalent to the divisor of Division, and the fractional line is equivalent to the divisor of Division. (2) In division, the divisor in the fraction cannot be 0 and the denominator cannot be 0. Division is meaningless, and the denominator is 0.

6. The process of simplifying a fraction into a fraction that is equal to it and has smaller numerator and denominator is called simplification.

7, simplest fraction numerator, denominator is a prime number fraction called simplest fraction.

8. Comprehensive score refers to the conversion of scores of different denominators into scores of the same denominator equal to the original score, which is called comprehensive score.

9. Comparison of the size of fractions Two fractions with the same denominator, the larger the numerator, the greater the score; For two fractions with the same numerator, the fraction with smaller denominator is larger.

10, Decimal Decimal Decimal According to the relationship between fraction and division, the fraction is converted into a division formula, and then the decimal can be obtained by calculation.

There are two kinds of fractional decimals: the general numerator can be divided by the denominator to get a finite decimal, such as = 0.4; One is to divide the numerator by the denominator to get an infinite decimal, such as = 0. 142857. ...

1 1, decimal places have several decimal places, so write a few zeros after 1.

Mom, after removing a molecule from the original decimal point and converting it into a component number, the offer score can be reduced.

12, the relationship between the basic properties of fractions and the basic properties of decimals

The basic properties of fractions are consistent with those of decimals. Add "0" after the decimal point.

Or removing "0" is equivalent to expanding (or reducing) the numerator and denominator of the corresponding score to the original 10 times (or), 100 times (or), 1000 times (or). ...

"Space and Graphics" section

1, a preliminary understanding of graphics

(1) Three-dimensional graphics in life

Reading material: Euler formula

(2) Drawing 3D graphics: ① From 3D graphics to views; ② From View to 3D Graphics

(3) Surface development diagram of three-dimensional graphics

(4) Plane graphics

Reading materials: Tangram.

(5) The most basic graphics: points and lines ① points and lines; ② Comparison of line segment lengths

(6) Angle: ① Comparison and calculation of angles; ② Special relationship of angles.

(7) intersection line: ① vertical line; (2) The angle in the intersection line.

(8) Parallel lines: ① Identification of parallel lines; ② Characteristics of parallel lines

2. Polygons

(1) triangle

(2) The sum of the inner and outer angles of a triangle.

(3) Tile laying

(4) Use regular polygons to lay the floor.

Reading materials: color patterns

Theme learning: mosaic of graphics

3, the transformation of graphics

(1) translation: ① graphic translation; (2) the characteristics of graphics

(2) Rotation: ① Rotation of graphics; (2) the characteristics of rotation; ③ rotationally symmetric graphics; ④ Centrally symmetric figure

(3) Axisymmetry: ① Axisymmetry in life; ② Understanding of axial symmetry; ③ isosceles triangle

Reading materials: (1) cut the five-pointed star; (2) symmetrical puzzles; (3) Time and date

(4) Potential transformation: ① enlargement and reduction of graphics; ② Draw similar figures.

4. Proposition and proof

(1) Definition, Proposition and Theorem

(2) Proof and recognition.

5. Graphic congruence

Congruence of (1) graph

(2) congruent triangles's identification and its nature.

(3) Ruler drawing: ① Draw line segments; 2 draw corners; ③ Draw a line segment; ④ Draw an angular bisector.

6. Similarity of graphics

(1) similar graphs and their characteristics

(2) similar triangles: ① Identification of similar triangles; (2) the characteristics of similar triangles.

(3) Graphics and coordinates

7. Solve the triangle

(1) measurement

(2) Pythagorean theorem

(3) acute angle trigonometric function

(4) Solving the right triangle

8. Parallelogram

(1) parallelogram: ① the concept of parallelogram; ② Recognition of parallelogram; ③ Characteristics of parallelogram

(2) rectangle: ① the concept of rectangle; ② Rectangular recognition; ③ The characteristics of rectangle

(3) Diamonds: ① the concept of diamonds; ② Diamond identification; ③ rhombic features

(4) Square: ① the concept of square; ② Identification of squares; ③ the characteristics of the square

Reading material: quadrilateral transformation

Topic learning: midpoint quadrangle

9. circle

The basic element of (1) circle

(2) Symmetry of the circle

(3) Angle of the circle

(4) the positional relationship with the circle: ① the positional relationship between the point and the circle; (2) the positional relationship between a straight line and a circle; ③ The positional relationship between circles.

(5) Some calculation problems in the circle: ① Arc length and sector area; ② The lateral area and total area of the cone.

1, statistics

Scientific notation: numbers greater than 10 can be expressed as A* 10N, where 1 is less than or equal to a and less than 10, and n is a positive integer.

Sector statistical chart: ① A circle is used to represent the population, and each sector in the circle represents a different part of the population. The size of the sector reflects the percentage of this part in the population. This kind of statistical chart is called departmental statistical chart. (2) In the sector statistical chart, the percentage of each part in the whole is equal to the ratio of the central angle of the sector corresponding to this part to 360 degrees.

Advantages and disadvantages of various statistical charts: bar chart: the specific figures of each item can be clearly displayed; Broken line statistical chart: can clearly reflect the changes of things; Department statistical chart: it can clearly show the percentage of each part in the total.

Approximate value and significant figures: ① The measurement result is approximate value. (2) When taking the divisor of a number by rounding method, it means that the divisor is rounded to the nearest place. (3) for a divisor, from the first number on the left that is not 0 to the most accurate number, all numbers are called the significant digits of this number.

Average number: For the number n, X 1, X2…XN, we call (X 1+X2+…+XN)/N the arithmetic average number n, and write it as x (the upper one is horizontal).

Weighted average: the importance of each data in a set of data may be different, so when calculating the average value of this set of data, each data is often given a weight, which is the weighted average.

Median and mode: ①N data are arranged in order of size, and the data in the middle position (or the average of the two data in the middle) is called the median of this group of data. ② The data with the highest frequency in a group of data is called the pattern of this group of data. Advantages and disadvantages: average: all data participate in the operation, which can make full use of the information provided by the data, so it is commonly used in real life, but it is easily affected by extreme values; Median: the calculation is simple, and it is less affected by extreme value, but it can't make full use of all data information; Pattern: when the number of repetitions of each data is roughly equal, the pattern often has no special meaning.

Survey: ① A comprehensive survey of the respondents for a certain purpose is called a general survey, in which all the respondents are called the whole and each object that constitutes the whole is called an individual. (2) Select some individuals from the population for investigation, which is called sampling investigation, and select some individuals from the population as a sample of the population. Sampling survey only investigates a small number of individuals in the population, so it has the advantages of small survey scope and saving time, manpower, material resources and financial resources, but its survey results are often not as accurate as those obtained by census. In order to obtain more accurate survey results, the main samples should be representative and extensive when sampling.

Frequency and frequency: ① frequency is the frequency of each object, and the ratio of the frequency of each object to the total frequency is the frequency. (2) When the collected data take values continuously, we usually group the data properly first, and then draw the histogram of frequency distribution.

2. Possibility

Possibility: ① Some things that we can be sure will happen are called inevitable events; Some things we can be sure will not happen. These things are called impossible events. Inevitable events and impossible events are certain. There are many things that we are not sure will happen. These things are called uncertain events. (3) Generally speaking, the possibility of uncertain events is different.

Probability: ① People usually use 1 (or 100%) to indicate the possibility of inevitable events and 0 to indicate the possibility of impossible events. The fairness of the game means that both sides have the same possibility of winning. (3) The probability of inevitable events is 1, and it is marked as p (inevitable events) =1; The probability of an impossible event is 0, and it is recorded as p (impossible event) = 0; If a is an uncertain event, then 0 < p (a) < 1.