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 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 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 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) 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.