Multiplying a composite number by a prime factor is called prime factor decomposition.
For example, decompose 28 into prime factors.
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. For example, the divisor of 12 is 1, 2, 3, 4, 6,12; The divisors of 18 are 1, 2,3,6,9 and 18. Where 1, 2,3,6 are the common divisors of 12 and 1 8, and 6 is their greatest common divisor.
The common divisor is only 1, which is called prime number. There are the following situations:
1 is coprime with any natural number.
Two adjacent natural numbers are coprime.
Two different prime numbers are coprime.
When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime.
When the common divisor of two composite numbers is only 1, these two composite numbers are coprime. If any two numbers are coprime, they are said to be coprime.
If the smaller number is the divisor of the larger number, then the smaller number is the greatest common divisor of these two numbers.
If two numbers are prime numbers, their greatest common divisor is 1.
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. For example, the multiple of 2 is 2,4,6,8, 10, 12, 14, 16, 18. ...
The multiple of 3 is 3,6,9, 12, 15, 18 ... where 6, 12, 18 ... are the common multiples of 2 and 3, and 6 is their least common multiple. .
If the larger number is a multiple of the smaller number, the larger number is the least common multiple of the two numbers.
If two numbers are prime numbers, then the product of these two numbers is their least common multiple.
The common divisor of several numbers is finite, while the common multiple of several numbers is infinite.
(2) Decimals
The meaning of 1 decimal
Divide the integer 1 into 10, 100, 1000 ... a tenth, a percentage, a thousandth ... 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. ...
Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.
In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.
2 Classification of decimals
Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals.
With decimals: decimals whose integer part is not zero are called with decimals.
For example, 3.25 and 5.26 are all decimals.
Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals.
For example, 4 1.7, 25.3 and 0.23 are all finite decimals.
Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal.
For example: 4.33...3. 145438+05926 ...
Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals.
For example: ∈
Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal.
For example: 3.555 … 0.0333 …12.15438+009 …
The decimal part of cyclic decimal is called the cyclic part of cyclic decimal.
For example, the period of 3.99 ... is "9", and the period of 0.5454 ... is "54".
Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal.
For example: 3.111.5656 ...
Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222 …… 0.03333 ……
When writing a cyclic decimal, for simplicity, the cyclic part of the decimal only needs one cyclic segment, and a dot is added to the first and last digits of this cyclic segment. If the loop
Festivals only
A number, just click on a point on it. For example: 3.777 ... Jane writing 0.5302302 ... Jane writing.
.
(3) scores
1 significance of the score
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.
In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.
Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.
2 Classification of scores
True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.
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 fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.
3 Reduction and comprehensive score
Turn a fraction into a fraction equal to it, but the numerator and denominator are smaller.
, called approximate score.
The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.
Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.
4) Percentage
1 indicates what percentage of one number is another.
This is called percentage, also called percentage.
Or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.
two
way
(A) the number of reading and writing
1. integer reading method: from high to low, read step by step. When reading the 110 million level, first read according to the reading method of the 100 million level, and then add a word "100 million" or "10 thousand" at the end. The zeros at the end of each stage are not read, and only a few zeros of other digits are read.
2. Writing of integers: from high to low, writing step by step. If there is no unit on any number, write 0 on that number.
3. Decimal reading method: When reading decimals, the integer part is read by integer reading method, the decimal point is read as "dot", and the decimal part reads the numbers on each digit from left to right in sequence.
4. Decimal writing: When writing decimals, the integer part is written as an integer, the decimal point is written in the lower right corner of each digit, and the decimal part is written on each digit in turn.
5. How to read fractions: When reading fractions, read the denominator first, then the "fraction", and then the numerator. Both numerator and denominator read integers.
6. How to write the fraction: write the fraction first, then the denominator, and finally the numerator and the integer.
7. Reading method of percentage: When reading percentage, read the percentage first, and then read the number before the percentage symbol. When reading, read it as an integer.
8. Writing of percentage: percentage is usually expressed by adding a percent sign "%"after the original molecule instead of a fraction.
(2) The number of rewrites
In order to facilitate reading and writing, a large multi-digit number is often rewritten as a number in units of "10,000" or "100 million". Sometimes, if necessary, you can omit the number after a certain number and write it as an approximation.
1. exact number: in real life, for the convenience of counting, larger numbers can be rewritten into numbers in units of ten thousand or hundreds of millions. The rewritten number is the exact number of the original number.
For example, 1254300000 is rewritten into ten thousand, and the number is125430000; Rewrite into
In billions.
The number is 65.438+25.4 million.
2. Approximation: According to the actual needs, we can also use a similar number to represent a larger number and omit the mantissa after a certain number.
For example: 13024900 15 The mantissa after omitting 100 million is1300 million.
3. Rounding method: If the highest digit of the mantissa to be omitted is 4 or less, the mantissa is removed; If the digit with the highest mantissa is 5 or more, the mantissa is truncated and 1 is added to its previous digit. For example, the mantissa after omitting 3.459 billion is about 350,000. After omitting 472509742 billion, the mantissa is about 4.7 billion.
4. Size comparison
1. Compare the sizes of integers: compare the sizes of integers, and the number with more digits will be larger. If the numbers are the same, view the highest number. If the number in the highest place is larger, the number is larger. The number in the highest bit is the same. Just look at the next bit, and the bigger the number, the bigger it is.
2. Compare the sizes of decimals: first look at their integer parts, and the larger the integer part, the larger the number; If the integer parts are the same, the tenth largest number is larger; One tenth of the numbers are the same, and the number with the largest number in the percentile is the largest. ...
3. Compare the scores: the scores with the same denominator and the scores with large numerator are larger; For numbers with the same numerator, the score with smaller denominator is larger. If the denominator and numerator of a fraction are different, divide the fraction first, and then compare the sizes of the two numbers.
(3) the number of mutual
1. Decimal component number: There are several decimals, so writing a few zeros after 1 as denominator and removing the decimal point after the original decimal point as numerator can reduce the number of quotation points.
2. Fractions become decimals: numerator divided by denominator. Those that are divisible are converted into finite decimals, and some that are not divisible are converted into finite decimals. Generally three decimal places are reserved.
3. A simplest fraction, if the denominator does not contain other prime factors except 2 and 5, this fraction can be reduced to a finite decimal; If the denominator contains prime factors other than 2 and 5, this fraction cannot be reduced to a finite decimal.
4. Decimal percentage: Just move the decimal point to the right by two places, followed by hundreds of semicolons.
5. Decimal percentage: Decimal percentage, just remove the percent sign and move the decimal point two places to the left.
6. Convert fractions into percentages: usually, first convert fractions into decimals (three decimal places are usually reserved when they are not used up), and then convert decimals into percentages.
7. Decimalization of percentage: First, rewrite percentage into component quantity and put forward a quotation that can be simplified to the simplest score.
(4) Divisibility of numbers
1. Usually a composite number is decomposed into prime factors by short division. Divide this complex number by a prime number until the quotient is a prime number, and then write the divisor and quotient in the form of multiplication.
2. The way to find the greatest common divisor of several numbers is to divide the common divisors of these numbers continuously until the quotient obtained is only the common divisor of 1, and then multiply all the common divisors to get the product, which is the greatest common divisor of these numbers.
.
3. The method of finding the least common multiple of several numbers is: divide by the common divisor of these numbers (or part of them) until it is coprime (or pairwise coprime), and then multiply by all the divisors and quotients to get the product, which is the least common multiple of these numbers.
4. Two numbers that become coprime relations: 1 and any natural number coprime.
;
Two adjacent natural numbers are coprime;
When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime;
When the common divisor of two composite numbers is only 1, these two composite numbers are coprime.
(5)
Simplified points and general points
Reduction method: divide the denominator by the common divisor of the denominator (except 1); Usually, we have to separate it until we get the simplest score.
General division method: first find the least common multiple of the denominator of the original fraction, and then turn each fraction into a fraction with this least common multiple as the denominator.
three
Nature and law
(A) the law of quotient invariance
Law of quotient invariance: in division, the dividend and divisor expand or shrink by the same multiple at the same time, and the quotient remains unchanged.
(B) the nature of decimals
The nature of decimals: add or remove zero at the end of decimals, and the size of decimals remains unchanged.
(3) The movement of decimal position causes the change of decimal size.
1. If the decimal point moves one place to the right, the original number will be expanded by 10 times; If the decimal point is moved two places to the right, the original number will be expanded by 100 times; If the decimal point is moved three places to the right, the original number will be enlarged by 1000 times. ...
2. If the decimal point moves one place to the left, the original number will be reduced by 10 times; If the decimal point is moved two places to the left, the original number will be reduced by 100 times; If the decimal point is moved three places to the left, the original number will be reduced by 1000 times. ...
3. When the decimal point is not moved to the left or right, use "0" to make up the digits.
(D) the basic nature of the score
The basic nature of a fraction: both the numerator and denominator of the fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains unchanged.
(5) the relationship between fraction and division
1. divider/divider = divider/divider
2. Because zero can't be divisible, the denominator of the fraction can't be zero.
3. Dividends
Equivalent to numerator, divisor equivalent to denominator.
four
Importance of operation
Integer operation
1 integer addition: the operation of combining two numbers into one number is called addition.
-In Djaafari, the added numbers are called addends, and the added numbers are called sums. The appendix is a partial figure, and the sum is the total.
-Appendix+Appendix = Sum
One addend = and-the other addend.
Integer subtraction: the operation of finding the sum of two addends and one of them is called subtraction.
In subtraction, the known sum is called the minuend, the known addend is called subtraction, and the unknown addend is called difference. The minuend is the total number, and the subtraction and difference are the partial numbers respectively.
-Addition and subtraction are reciprocal operations.
Integer multiplication: The simple operation of finding the sum of several identical addends is called multiplication.
-In multiplication, the same addend and the number of the same addend are called factors. The sum of the same addend is called product.
-In multiplication, if 0 is multiplied by any number, 0. 1 times any number.
-coefficient x coefficient = product
One factor = product ÷ another factor
Integer division: the operation of finding the product of two factors and one of them is called division.
-In division, the known product is called the dividend, the known factor is called the divisor, and the calculated factor is called the quotient.
-Multiplication and division are reciprocal operations.
-In division, 0 cannot be divided. Because 0 is multiplied by any number to get 0, any number divided by 0 can't get a definite quotient.
-Divider-Divider = quotient
Divider = Divider
Dividend = quotient × divisor
(2) Four decimal places operation
1. decimal addition: the meaning of decimal addition is the same as that of integer addition. It is an operation that combines two numbers into one number.
2. Decimal subtraction: Decimal subtraction and integer subtraction have the same meaning. Know the sum of two addends and one of them, and find the other addend.
3. Decimal multiplication: Decimal multiplication of integers has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends; The significance of multiplying a number by a pure decimal is to find a few tenths, a few percent and a few thousandths of this number.
4. Decimal division: Decimal division has the same meaning as integer division, that is, by knowing the product of two factors and one of them, the operation of finding the other factor.
multiplication
The operation of finding the product of several identical factors is called power. For example, 3 × 3 =32
(3) Four Fractions Operation
1. Fractional addition: Fractional addition has the same meaning as integer addition.
It is an operation that combines two numbers into one number.
2. Fractional subtraction: The significance of fractional subtraction is the same as that of integer subtraction. The operation of finding the other addend by knowing two addends and one of them.
3. Fractional multiplication: The significance of fractional multiplication is the same as integer multiplication, and it is a simple operation to find the sum of several identical addends.
Two numbers whose product is 1 are called reciprocal.
5. Fractional division: Fractional division has the same meaning as integer division. It is an operation to find the other factor by knowing the product of two factors and one of them.
(4) Operation law
1. additive commutative law: When two numbers are added, the positions of addends are exchanged, and the sum is unchanged, that is, A+B = B+A.
2. The law of addition and association: when three numbers are added, the first two numbers are added first, and then the third number is added; Or add the last two numbers first, and then add the first number, and their sum is unchanged, that is, (a+b)+c=a+(b+c).
3. Multiplicative commutative law: When two numbers are multiplied, the position of the commutative factor remains unchanged, that is, a× b = b× a..
4. Multiplication and association law: multiply three numbers, first multiply the first two numbers and then multiply the third number; Or multiply the last two numbers first, and then multiply them with the first number, and their products are unchanged, that is, (a×b)×c=a×(b×c).
5. Multiplication and distribution law: When the sum of two numbers is multiplied by a number, you can multiply the two addends by this number, and then add the two products, that is, (a+b) × c = a× c+b× c.
6. The essence of subtraction: If you subtract several numbers from a number continuously, you can subtract the sum of all subtractions from this number, and the difference is unchanged, that is, a-b-c=a-(b+c).
(5) Algorithm
1. Integer addition calculation rule: the same digits are aligned, starting from the low order. When the numbers add up to ten, they will advance to the previous number.
2. Integer subtraction calculation rules: align with the same number, starting from the low order. If the number of digits is not reduced enough, subtract it from the last digit, merge it with the standard number, and then subtract it.
3. Calculation rules of integer multiplication: multiply the number on each bit of one factor by the number on each bit of another factor, then multiply the number on which bit of the factor, align the end of the multiplied number with which bit, and then add the multiplied numbers.
4. Calculation rules of integer division: divide from the high order of the dividend, and the divisor is a few digits, depending on the first few digits of the dividend;
If the division is not enough, look at another place and the quotient is written on the dividend. If any number is not quotient 1, a "0" placeholder should be added. The remainder of each division should be less than the divisor.
5. Decimal multiplication rule: first, calculate the product according to the calculation rule of integer multiplication, and then look at the factor * * *, how many decimals there are, count a few from the right side of the product and point to the decimal point; If the number of digits is not enough, make up with "0".
6. Calculation rules of fractional division with divisor as integer: First, divide according to the rules of integer division, and the decimal point of quotient should be aligned with the decimal point of dividend; If there is a remainder at the end of the dividend, add "0" after the remainder to continue the division.
7. Division calculation rules with divisor as decimal: first move the decimal point of divisor to make it an integer, then move the decimal point of divisor to the right by several digits (add "0" if there are not enough digits), and then calculate according to the division rules with divisor as integer.
8. Calculation method of addition and subtraction of fractions with the same denominator: addition and subtraction of fractions with the same denominator, only addition and subtraction of numerators, and the denominator remains unchanged.
9. Calculation method of addition and subtraction of scores with different denominators: divide the scores first, and then calculate according to the addition and subtraction law of scores with the same denominator.
10. Calculation method of fractional addition and subtraction: add and subtract the integer part and the decimal part respectively, and then combine the obtained numbers.
1 1. Calculation rules of fractional multiplication: Fractions are multiplied by integers, and the product of fractional numerator and integer multiplication is taken as numerator, with the denominator unchanged; Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.
12. Calculation rule of fractional division: A number divided by B number (except 0) equals the reciprocal of A number multiplied by B number.
(6)
operation sequence
1. The operation order of four decimal operations is the same as that of four integer operations.
2. The operation sequence of fractional four operations is the same as that of integer four operations.
3. Mixed operation without brackets: the operations at the same level are operated from left to right in turn; Two-stage operation
Multiply first, then divide, then add and subtract.
4. Mixed operation with parentheses: first calculate the parentheses, then calculate the parentheses, and finally calculate the parentheses.
5. First-level operation: Addition and subtraction are called first-level operation.
6. Secondary operation: Multiplication and division are called secondary operations.
five
App application
(A) the application of integers and decimals
1 Simple application problem
( 1)
Simple application problem: an application problem that only contains a basic quantitative relationship or is solved by one-step operation, usually called a simple application problem.
(2)
Steps to solve the problem:
A understand the meaning of the question: understand the content of the application question and know the conditions and problems of the application question. When reading a question, read, think and understand the meaning of every sentence in the question without losing words or adding words. You can also repeat the conditions and questions to help you understand the meaning of the questions.
B selection algorithm and column calculation: this is the central work to solve application problems. Starting with what to say and ask, according to the given conditions and questions, and connecting with the significance of four operations, this paper analyzes the quantitative relationship, determines the algorithm, answers and marks the correct unit name.
C test: according to the conditions and problems of the application questions, check whether the listed formulas and calculation processes are correct and whether they meet the meaning of the questions. If mistakes are found, correct them immediately.
D answer: according to the calculation results, answer orally first, and then gradually transition to written answer.
(3) Solve the problem of addition application:
An application problem of finding the total number: what is the known number A, what is the number B, and what is the sum of the two numbers A and B.
Find a number greater than the number. Application problem: Know what A number is, how much more B number is than A number, and find what B number is.
(4) Solve the application problem of subtraction:
A Finding the residual application problem: removing a part from the known number and finding the residual part.
The application problem of finding the difference between two numbers by -b: Given the numbers of A and B, find how much A is more than B, or how much B is less than A. ..
The application of c to find the number less than the number: what is the known number a, how much is the number b less than the number a, and how much is the number B.
(5) Solve the problem of multiplication application:
An application problem of seeking the sum of common addends: knowing the same addend and the number of the same addend, find the sum.
The application problem of finding the multiple of a number is: how many times is one number, how many times is another number, and how much is another number?
(6) Solve the problem of division application:
A divide a number into several parts on average, and find out how much each part is: know a number, divide it into several parts on average, and find out how much each part is.
B. Find an application problem, in which one number contains several other numbers: given a number, how many copies are there in each number, and how many copies can you find?
C the application problem of finding a number that is several times that of another number: given the number A and the number B, finding a larger number is several times that of a smaller number.
D know how many times a number is, and find the application problem of this number.
(7) Common quantitative relations:
-Total price = unit price × quantity
-Distance = speed × time
-Total amount of work = working hours × working efficiency
-Total output = single output × quantity
(9)
Reduction problem: We call it reduction problem, and look for the application problem of an unknown after four operations.
-The key to solving the problem is to clarify the relationship between each step of change and the unknown.
-law of solving problems: starting from the final result
Based on this, the original number is deduced step by step by using the inverse operation method of the original problem.
-List the quantitative relations according to the operation order of the original question, and then calculate and deduce the original number by the inverse operation method.
-Pay attention to the operation sequence when solving the restore problem. If you need to add and subtract first, don't forget to write parentheses when calculating multiplication and division later.
example
There are four classes in grade three in a primary school 168 students. If four classes are transferred from three to three, from three to two, from two to one, and from two to four, then the number of students in four classes is equal. How many students are there in four classes?
Analysis: When the number of four classes is equal, it should be 168 ÷ 4. Take Class Four as an example. It transfers three people to Class Three and two people from Class One, so the number of people in the original four classes minus three plus two equals the average. The original number of class four is 168 ÷ 4-2+3=43 (people).
The original number of a class is 168 ÷ 4-6+2=38 (people); The original number of class two is 168 ÷ 4-6+6=42 (people).
The original number of class three is 168 ÷ 4-3+6=45 (people).
(10) Tree planting problem: This kind of application problem takes "tree planting" as its content. Any application problem of studying the four quantitative relations of total distance, plant distance, number of segments and number of plants is called tree planting problem.
-The key to solving the problem: To solve the problem of planting trees, we must first judge the terrain and distinguish whether the graph is closed, so as to determine whether to plant trees along the line or along the perimeter, and then calculate according to the basic formula.
-law of problem solving: plant trees along the line.
-tree = number of segments+1 tree = total distance/plant spacing+1
-Plant spacing = total distance ÷ (tree-1)
Total distance = plant spacing × (tree-1)
-Planting trees along the periphery
-Tree = total distance ÷ plant distance
-Distance between plants = total distance.
-Total distance = plant spacing × trees
example
30 1 pole is buried along the road, and the distance between every two poles is 50m.
. Later, it was completely revised and only 20 1 was buried. Find the distance between two adjacent ones after modification.
Analysis: this question is to bury telephone poles along the line, and the number of telephone poles is reduced by one. The formula is 50× (301-1) ÷ (201-1) = 75 (m).
(1 1) profit and loss problem: It was developed on the basis of equal share.
His characteristic is to distribute a certain number of goods to a certain number of people equally. In the two distributions, one is surplus, the other is insufficient (or both are surplus), or both are insufficient). The problem of finding the right quantity of goods and the number of people participating in the distribution is called profit and loss problem.
-The key to solving the problem: The key point of profit and loss problem's solution is to find the difference of the quantity of goods that the distributor did not get in the two distributions, and then find the difference of goods in each distribution (also called total difference). The final difference is divided by the previous difference to get the number of distributors, and then get the quantity of goods.
-Law of problem solving: total difference ÷ per capita difference = number of people
-The solution of total difference can be divided into the following four situations:
-First redundancy, second insufficiency, total difference = redundancy+insufficiency.
-the first time is just right, and the second time is redundant or insufficient.
, total variance = redundant or insufficient
-First redundancy, second redundancy, total difference = large redundancy-small redundancy
-Not enough the first time, not enough the second time,
Total Difference = Large Shortage-Small Shortage
Chapter II Weights and Measures
A length
What is length?
Length is a measure of one-dimensional space.
(2) Common length units
* kilometers (km) * meters (m) * decimeters (dm) * centimeters (cm) * millimeters (mm) * microns (um)
(3) Conversion between units
* 1mm = 1000mm * 1cm = 10mm * 1cm = 1cm * 1m = 1000mm * 1km。
Two areas
(1) What is the area?
Area is the size of the plane occupied by an object. The measurement of the surface of three-dimensional objects is generally called surface area.
(2) Public area unit
* square millimeter * square centimeter * square decimeter * square meter * square kilometer
(3) conversion of area units
* 1 cm2 =100mm2 * 1 cm2 =1cm2 * 1 m2 =100mm2.
* 1 ha = 10000 m2 * 1 km2 = 100 ha.
Three volumes and volumes
(1) What are volume and volume?
Volume is the size of the space occupied by an object.
Volume, the volume of objects that can be accommodated in boxes, oil drums, warehouses, etc. , usually called their volume.
(2) Common units
1 unit of volume
* cubic meter * cubic decimeter * cubic centimeter
2 unit of volume * L * mL
(3) Unit conversion
1 unit of volume
* 1 m3 = 1000 cubic decimeter; * 1 cubic decimeter = 1000 cubic centimeter
2 unit of volume
* 1 l = 1000 ml; * 1 l = 1 m3; * 1 ml = 1 cm3
Four qualities