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Four Mathematical Operations

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synonym except (mathematical term) generally referring to division

Division is Four arithmetic operations one of. When the product of two factors is known and one of them is non-zero, the operation of finding the other factor is called division.^[1]

The division of two numbers is also called the ratio of two numbers. if ab = c （ b ≠ 0), use product c Sum factor b To find another factor a The operation of is division, writing c ÷ b , read as c divide b (or b except c ）。 Among them, c be called Dividend ， b be called Divisor , result of operation a It's called quotient.

Chinese name: division
Foreign name: division
Discipline: mathematics
Concept: One of the four operations
Rules: Division rule, constant quotient
Division sign: ÷
Calculation formula: Dividend ÷ divisor=quotient (a ÷ b=c)

Attention: The divisor cannot be 0
than: The division of two numbers is also called the ratio of two numbers.
Quotient invariant property: The divisor and the dividend are multiplied or divided by numbers that are not zero at the same time, and the quotient remains unchanged.
Role: C/b can represent scores
Relation with multiplication: Multiplicative Inverse operation
Four arithmetic operations: +、-、×、÷（ addition 、 subtraction 、 multiplication , division)

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definition

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Given the product of two factors and one of the non-zero factors, the operation of finding the other factor is expressed as a ÷ b=c (b ≠ 0) or a ÷ c=b (c ≠ 0).^[1]

Operational formula

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Dividend ÷ divisor=quotient, for example:

Dividend ÷ quotient=divisor, e.g ：

⇒

merchant

Divider=dividend, for example ：

have remainder Situation of:

Dividend ÷ divisor=quotient remainder (where the remainder is less than the divisor)

↕

Divisor × quotient+remainder=dividend.

Considering that division and multiplication are inverses to each other, and the meaning of multiplication is to find the same number Addend So this situation can also be explained as: the dividend continuously subtracts the divisor until remainder The value is below the divisor. For example: 17 ÷ 5=3... 2, that is, 17 minus 3 5, the remaining 2. If using With fraction In the form of

(Three and two fifths).

Operational property

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1. Dividend expansion (reduction) n Multiple, divisor unchanged, quotient correspondingly expanded (narrowed) n Times.

2. Divider expansion (reduction) n Multiple, dividend unchanged, quotient correspondingly reduced (expanded) n Times.

3. Nature of division: The dividend is divided by two divisors consecutively, which is equal to dividing by the product of these two divisors. Sometimes simple operations can be performed according to the nature of division.

For example: 300 ÷ 25 ÷ 4=300 ÷ (25 × 4)=300 ÷ 100=3.

Application of division

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division

If ab = c , if b ≠ 0, then

； if a ≠ 0, then

。

If the Quotient It must be an integer. If the divisor and the dividend are not factors, there will be a difference between the values d ）Is the remainder.

, i.e

（ a also b Fraction d ）, which also means

。

Usually not defined Divide by zero This form.

In particular, when

As c divide b When the quotient of appears (when the quotient is not an integer), or when it means that division is not performed,

It can represent a number, which is called fraction. At this point,

Reading“ b Fraction c ”。 For example:

It can be understood as the quotient of 10 divided by 5, read as ten fifths.

One sixth plus four sixths equals the sum of one plus four divided by six equals five sixths.

computing method

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Long division

division

Long division Commonly known as "long division", it is applicable to integer division, decimal division Polynomial division (i.e Factorization ）Pay more attention to the calculation process and Quotient Division of, and multiplication and subtraction 。 According to the multiplication table, two integers can be used Long division (Direct division) written calculation. If the dividend has a fractional part (or is decimal point ）, the decimal point can be brought down during calculation; If there is a decimal point in the divisor, shift the decimal point of the divisor and the dividend at the same time until there is no decimal point in the divisor. Abacus You can also do division.

Short division

Short division Commonly known as "short division", it is suitable for fast division and multiple integer synchronous division (so it is often used to calculate the sum of the greatest common factor and Least common multiple ）, binary digital conversion, etc. pay more attention to multiple test and Prime factor The process of division (multiplication) mostly only needs to use the multiplication table of 99 and the multiplication factors of a few integers above 9.

0 cannot be divided

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According to the meaning of division, division is the operation of finding the other factor by knowing the product of two factors and one of them. According to the inverse relationship between division and multiplication, if the divisor is 0, then:

① When the dividend is not 0 (for example, 3 ÷ 0), because "any number multiplied by 0 is equal to 0, and cannot be equal to a number that is not 0 (for example, 3)", then the quotient of the division formula does not exist - that is, 0 times any number cannot be non-zero;

② When the dividend is 0, that is, the division formula 0 ÷ 0. Because "any number multiplied by 0 equals 0", the quotient can be any number - that is, 0 times any number equals 0.

In order to avoid the above two situations, it is stipulated in mathematics that "0 cannot be divided".

Four arithmetic operations

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In mathematics, when Primary operation (plus or minus) and Two-level operation (Multiplication and division) When both are in the same formula, their Operation order It is multiplication and division first, then addition and subtraction. If there are parentheses, they are calculated first in parentheses and then in parentheses. The order of operations at the same level is from left to right. Such operation is called Four arithmetic operations 。

Four are addition subtraction 、 multiplication The calculation rules of division. One four arithmetic Arithmetic It is not necessary to have four kinds of operation symbols. Generally, it refers to the operation of combining the majority into a number by two or more operation symbols and parentheses.

Addition: the operation of combining two numbers into one number/the operation of combining two decimals into one decimal/the operation of combining two fractions into one fraction.

Subtraction: two known Addend The operation of calculating the sum of and one of the addends and the other addend.

Multiplication: A simple operation to find the sum of several identical addends. The meaning of multiplying decimal by integer is the same as that of multiplying integer. A number multiplication pure decimal It is to find the number of tenths and percentages of this number... Fraction times integer has the same meaning as integer multiplication.

Division: the operation of finding the other factor by knowing the product of two factors and one factor. AND INTEGER Meaning of division Same.

Example ：

1 ° multiplication: ① how many is how many; ② Find how many times a number is; ③ Calculate the area and volume of the object; ④ Find the fraction or percentage of a number.

2 ° division: ① divide a number into several equal parts and find one of them; ② Find out how many other numbers are in a number; ③ Find a number when you know its fraction or percentage; ④ Find how many times one number is another.

3 ° addition: ① sum; ② subtraction Inverse operation 。

4 ° Subtraction: ① Find the remainder; ② Comparison; ③ addition Inverse operation 。

Addition and subtraction are mutually inverse operations; Multiplication and division are mutually inverse operations; Multiplication is a simple operation of addition.

factor

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definition

integer A Can be an integer B to be divisible by ， A be known as B Multiple of, B It's called A Factor of or Divisor ，

(on Natural number For example: 6 ÷ 2=3, 1, 2, 3 and 6 are the factors of 6.

The factors of 6 are: 1 and 6, 2 and 3.

The factors of 10 are: 1 and 10, 2 and 5.

The factors of 15 are: 1 and 15, 3 and 5.

calculation Maximum common factor or Least common multiple When, the factor needs to be Prime factor 。 The former is the product of all prime factors on the left, excluding the final factor at the bottom; The latter needs to be multiplied together with the final factor.

classification

A In division, if the divisor is divided by the divisor, the quotients obtained are all natural numbers without remainder, the divisor is a multiple of the divisor, and the divisor and quotient are factors of the divisor.

B : divide a composite number into several Prime number In the form of multiplication, such prime numbers are called the composite numbers Prime factor 。

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