A concept
(1) integer
The meaning of 1 integer
Natural numbers and 0 are integers.
2 natural number
When we count objects, 1, 2, 3 ... the numbers 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, ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, 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 digits
Counting units are arranged in a certain order, and their positions are called numbers.
Divisibility of 5 numbers
When the integer A is divided by the integer b(b ≠ 0), the quotient is an integer with no remainder, so we say that A is divisible by B, or that B is divisible by A. ..
If the number A is divisible by the number B (b ≠ 0), then A is called a multiple of B, and B is called a divisor of A (or a factor of A). Multiplication and divisor are interdependent.
Because 35 is divisible by 7, 35 is a multiple of 7, and 7 is a divisor of 35.
The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. For example, the divisor of 10 is 1, 2,5, 10, where the smallest divisor is 1 0 and the largest divisor 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 ... The minimum multiple is 3, but 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 the numbers in each bit of a number can be divisible by 3, so this number can be divisible by 3. For example, 12,108,204 can all 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.
A number with only two divisors of 1 is called a prime number (or prime number), and the prime numbers within 100 are: 2, 3, 5, 7,1,13, 17.
If a number has other divisors besides 1 and itself, then it is called a composite number. For example, 4, 6, 8, 9 and 12 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 divisors, 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, and 3 and 5 are called prime factors of 15.
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 there is only one number in the circle, just click 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
Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.
The denominator of a numerator 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 that one number is the percentage of another number, which is called percentage, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.
Two methods
(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; Rewritten into a number of 65.438+025.43 billion in units of hundreds of millions.
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 number with the largest tenth digit is the largest; 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) 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.
Three properties and laws
(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. Divider is equivalent to numerator and divisor is equivalent to denominator.
The significance of four operations
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 addend = and-another addend.
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 = a product = a 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.
Frequency divider/frequency divider = frequency divider = frequency divider/frequency divider = quotient × frequency divider
(2) Four decimal places operation
1. Fractional 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:
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 a few tenths, a few percent and a few thousandths of this number.
4. Decimal 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. Power:
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 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.
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 their sum is unchanged, that is, A+B = B+A.
2. Additive associative law:
Add three numbers, first add the first two numbers, then add the third number; Or add the last two numbers first, and then add the first number, and their sum remains the same, that is, (a+b)+c=a+(b+c).
3. Multiplicative commutative law;
When two numbers are multiplied, the position of the exchange factor remains unchanged, that is, a× b = b× a.
4. Multiplicative associative 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. Multiplicative 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 remains unchanged, that is, a-b-c=a-(b+c).
(5) Algorithm
1. integer addition calculation rules:
The same numbers are aligned from the low order. When the numbers add up to ten, they will advance to the previous number.
2. The integer subtraction calculation rules:
The same numbers are aligned from the lower number. If the number of digits is not reduced enough, the number of digits from the previous digit is reduced by ten, and then it is combined with the number of digits in the radix and then reduced.
3. The integer multiplication calculation rules:
First, 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. The integer division calculation rules:
Divide from the high order of the dividend, which 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 rules:
First, calculate the product according to the calculation rules of integer multiplication, and then look at the factor * * *, how many decimal places there are, just 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. The divisor is an integer fractional division calculation rule:
First of all, according to the law of integer division, 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. The divisor is a decimal division calculation rule:
First, move the decimal point of the divisor to make it an integer, then move the decimal point of the divisor to the right by several digits (the number of digits is not enough, make up "0"), and then calculate according to the division rule that the divisor is an integer.
8. The same denominator fraction addition and subtraction calculation method:
Add and subtract fractions with denominator, only add and subtract numerators, and the denominator remains the same.
9. Different denominator fractions addition and subtraction calculation method:
Divide first, and then calculate according to the addition and subtraction law of fractions with the same denominator.
10. Calculation method of fractional addition and subtraction: