(1) integer
1, meaning of integer
Natural numbers and 0 are integers.
2. Natural numbers
When we count objects, 1, 2, 3 used to represent the number of objects are called natural numbers.
There is no object, which is represented by 0. 0 is also a natural number.
3. Counting unit
One (1), ten, hundred, thousand, ten thousand, one hundred thousand, one million, ten million and one hundred million are all counting units.
The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method.
4. Numbers
Counting units are arranged in a certain order, and their positions are called numbers.
5. Divisibility of numbers
Integer a divided by integer b(b? 0), the quotient of division is an integer without remainder, so we say that A is divisible by B, or that B is divisible by A. ..
If the number a can be counted by the number b (b? 0) divisible, where a is called a multiple of b and b is called a factor. Multiplication and factor are interdependent.
Because 35 is divisible by 7, 35 is a multiple of 7 and 7 is a factor of 35.
The number of factors of a number is limited, in which the smallest factor is 1 and the largest factor is itself. For example, the factors of 10 are 1, 2,5, 10, where the smallest factor is 1 and the largest factor is 10.
The number of multiples of a number is infinite, and the smallest multiple is itself. The multiple of 3 is: 3, 6, 9, 12, where the minimum multiple is 3 and there is no maximum multiple.
Numbers in units of 0, 2, 4, 6 and 8 can be divisible by 2, for example, 202, 480 and 304 can be divisible by 2.
Numbers in units of 0 or 5 can be divisible by 5, for example, 5,30,405 can be divisible by 5. .
The sum of digits in each digit of a number can be divisible by 3, and this number can be divisible by 3, for example, 12,108,204 can be divisible by 3.
The sum of each digit of a number can be divisible by 9, and so can this number.
A number divisible by 3 may not be divisible by 9, but a number divisible by 9 must be divisible by 3.
The last two digits of a number can be divisible by 4 (or 25), and this number can also be divisible by 4 (or 25). For example,16,404 and 1256 can all be divisible by 4, and 50,325,500 and 1675 can all be divisible by 25.
The last three digits of a number can be divisible by 8 (or 125), and this number can also be divisible by 8 (or 125). For example,1168,4600,5000, 12344 can all be divisible by 8, and 1 125,13375,5000 can all be/kloc-.
A number divisible by 2 is called an even number.
Numbers that are not divisible by 2 are called odd numbers.
0 is also an even number. Natural numbers can be divided into odd and even numbers according to their divisibility by 2.
If a number has only two factors: 1, it is called prime number (or prime number), and the prime numbers within 100 are: 2, 3, 5, 7,1,13, 17.
A number is called a composite number if there are other factors besides 1 and itself. For example, 4, 6, 8, 9 and 10 are all complex numbers.
1 is not a prime number or a composite number, and natural numbers are either prime numbers or composite numbers except 1. If natural numbers are classified according to the number of their factors, they can be divided into prime numbers, composite numbers and 1.
Every composite number can be written as the product of several prime numbers. Every prime number is a factor of this composite number, which is called the prime factor of this composite number. For example 15=3? 5,3,5 is called the prime factor of 15.
Multiplying a composite number by a prime factor is called prime factor decomposition.
For example, decompose 28 into prime factors.
The common factor of several numbers is called the common factor of these numbers. The largest one is called the greatest common factor of these numbers. For example, the factor of 12 is 1, 2, 3, 4, 6,12; The factors of 18 are 1, 2,3,6,9 and 18. Where 1, 2,3,6 are the common factors of 12 and 1 8, and 6 is their greatest common factor.
Two numbers whose common factor is only 1 are called prime numbers, and 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 factor 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 a factor of the larger number, then the smaller number is the greatest common factor of the two numbers.
If both numbers are prime numbers, then their greatest common factor 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 multiples of 3 are 3, 6, 9, 12, 15 and 18, where 6, 12 and 18 are 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 number of common factors of several numbers is limited, while the number of common multiples of several numbers is infinite.
(2) Decimals
1, the meaning of decimal
The integer 1 can be divided into 10, 100 and 1000, 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.
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. What is the highest decimal unit of the decimal part? One tenth? And the smallest unit of the integer part? One? The propulsion rate between them 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. 14 15926
Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals.
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 0.033312.109109.
The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, what is the period of 3.99? 9 ? What is the cycle segment of 0.5454? 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 there is only one number in the circular part, just point a point on it.
(3) scores
1, the meaning of the score
Put the unit? 1? Divide into several parts on average, and the number indicating this part or parts is called a score.
In the score, the middle horizontal line is called the dividing line; The number below the fractional line, called the denominator, represents the unit? 1? How many shares are divided equally; The number below the fractional line is called the numerator, indicating how many copies there are.
Put the unit? 1? Divide 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. Subtraction and total score
Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.
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, a number indicating that one number is a percentage of another number is called percentage, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.
Second, the method
(A) the number of reading and writing
1, integer reading:
Read from high to low, level by level. When reading level 110 million, first read according to the reading method of each level, and then add a at the end? Billion? Or? Wan? Words. The zeros at the end of each stage are not read, and only a few zeros of other digits are read.
2, integer writing:
From high to low, write level by level. If there is no unit on any number, write 0 on that number.
3. Decimal reading:
When reading decimals, the integer part should be read as integers and the decimal point should be read as? Point? The decimal part reads the numbers on each digit from left to right in order.
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 the unit, and the decimal part is written on each digit in sequence.
5. How to read music:
When reading fractions, read the denominator before reading? Share it? Then read the numerator, numerator and denominator as integers.
6. How to write music:
Write the fractional line first, then the denominator, and finally the numerator and the integer.
7. Percentage of reading:
When reading the percentage, read the percentage first, then read the number before the percent sign and read it as an integer.
8, the percentage of writing:
Percentages are usually not written in fractional form, but with a percent sign after the original molecule? %? To show.
(2) The number of rewrites
A large multi-digit number, which is often rewritten for reading and writing convenience? Wan? Or? Billion? Number of units. 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 as numbers in units of 10 thousand or 100 million. The rewritten number is the exact number of the original number.
For example, 1254300000 is rewritten into ten thousand, and the number is125430000; Rewritten into a number of 65.438+025.43 billion in units of hundreds of millions.
2. Approximate quantity:
According to 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 deleted; 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
Compare integer sizes:
Comparing the sizes of integers, the more digits, the greater the number. 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.
Compare the sizes of decimals:
Look at their integer parts first, 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.
Compare the size of the score:
Fractions with the same denominator have larger molecules; 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 places:
There are several decimals, so writing a few zeros after 1 as the denominator and removing the decimal point of the original decimal as the numerator can reduce the number of offer points.
2. Decimal part:
Divide the numerator by the 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. Decimals are converted into percentages:
Just move the decimal point two places to the right and add hundreds of semicolons at the end.
5. Decimal percentage:
To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.
6. Percentage of scores:
Fractions are generally converted into decimals first (three decimal places are generally reserved when they are not used up), and then decimals are converted into percentages.
7. Decimal percentage:
First, rewrite the percentage as the number of components, and put forward a quotation that can be simplified to the simplest score.
(4) Divisibility of numbers
1, decompose a composite number into prime factors, usually 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 method of finding the greatest common factor of several numbers is:
Divide the common factors of these numbers continuously until the quotient is only the common factor of 1, and then multiply all the divisors to get the product, which is the greatest common factor of these numbers.
3. The method of finding the least common multiple of several numbers is:
Divide by the common factor of these numbers (or some of them) until they are coprime (or pairwise coprime), and then multiply 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 is coprime with any natural number; 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 factor of two composite numbers is only 1, these two composite numbers are coprime.
(5) Approximate 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.
Third, nature and law.
(A) the law of quotient invariance
Law of constant quotient:
In division, the divisor and divisor are expanded or reduced by the same multiple (except 0) at the same time, and the quotient remains unchanged.
(B) the nature of decimals
Properties of decimals:
Add or delete zeros at the end of the decimal, and the size of the decimal remains the same.
(3) The movement of decimal position causes the change of decimal size.
1, the decimal point moves one place to the right, and the original number is 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 enough to the left or right, use? 0 "complement bit.
(D) the basic nature of the score
Basic attributes of a score:
The numerator and denominator of a fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains the same.
(5) the relationship between fraction and division
1, dividends? Frequency divider = frequency divider/frequency divider
2. Because zero can't be divisible, the denominator of the fraction can't be zero.
3. Divider is equivalent to numerator and divisor is equivalent to denominator.
Fourth, the significance of surgery
Integer operation
1, integer addition:
The operation of combining two numbers into one number is called addition.
In Djaafari, the added number is called addend, and the added number is called sum. The appendix is a partial figure, and the sum is the total.
Appendix Appendix = and one appendix = and-another appendix.
2, integer subtraction:
Given the sum of two addends and one of them, the operation of finding the other addend 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.
3, integer multiplication:
The simple operation of finding the sum of several identical addends is called multiplication.
In multiplication, the same addend and the same number of addends are called factors. The sum of the same addend is called product.
In multiplication, multiplying 0 by any number will get 0. 1, then multiply any number by any number.
A factor? A factor = product. A factor = product? Another factor
4, integer division:
Given the product of two factors and one of them, the operation of finding the other factor is called division.
In division, the known product is called dividend, the known factor is called divisor, and the calculated factor is called 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.
Dividend? Divider = quotient divisor = dividend? Business dividend = business? 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. The operation of finding the other addend by knowing two addends and one of them.
3. Decimal multiplication:
The meaning of decimal multiplication by integer is the same as that of integer multiplication, which is a simple operation to find the sum of several identical addends; The meaning of multiplying a number by a pure decimal is to find out a few tenths, a few percent and a few thousandths of this number.
4, fractional division:
The significance of fractional division is the same as integer division, that is, knowing the product of two factors and one of them, and finding the other factor.
5. Multiplier:
The operation of finding the product of several identical factors is called power. Like 3? 3 =32
(3) Four Fractions Operation
1, fractional addition:
Fractional addition and integer addition have the same meaning. 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 that of integer multiplication, which is a simple operation to find the sum of several identical addends.
4. 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.