When two numbers are divided and cannot be divided, the quotient must be a circular decimal, right?

This statement is true.

The decimal system of the previous digit or section of the number is repeated sequentially from a decimal point after the decimal point.

Infinite, called cyclic decimals, such as mixed cyclic decimals), cyclic decimals), cyclic decimals), etc.

If it is said that two integers are divided, if they are not divided, the quotient is indeed a circular decimal.

This involves the definition of real numbers. At first, people had a very vague understanding of irrational numbers, and they didn't know how to express them. By the middle of the nineteenth century, this prompted mathematicians to pay attention to and deal with the problem of irrational numbers.

Through more than half a century of efforts, a rigorous theory of real numbers in many different forms but essentially equivalent has been established. All forms of constructive real number theory first define irrational numbers from the rational number, that is, all the gaps between the advantageous points on the number of weeks can be determined by the rational number in a certain way, such as approximation, etc., and it is proved that all irrational numbers can be represented by infinite acyclic decimals. (Obviously, cyclic decimal numbers are naturally not irrational numbers, but rational numbers).

Far from it. The above is to add some background. A rational number is an extension of an integer number.

Integers, fractions are collectively referred to as rational numbers; Or the fraction m n is called a rational number, where m,n is the integer n≠0; Or integers, finite decimals, and infinitely cyclic decimals are collectively referred to as rational numbers. The above is the definition.

To put it another way in layman's terms, all fractions, multiplied by a certain number by a certain number, can be transformed into the following form: 99 ......900……0, the number of digits after the zero is the number of digits of the non-cyclic section after the decimal point, and the number of digits corresponding to the nine corresponds to the number of digits of the cyclic section (this is pushed by yourself......In this way, all fractions of nature can be expressed in the form of "cyclic decimals" (for the time being, the cyclic section after the non-cyclic decimal is 0).

No, because the quotient may be an infinite non-cyclic decimal. Counter-examples can be given as follows:

2, it is inexhaustible, but the quotient is infinite and does not circulate decimals, so it is not right.

The correct way to say it is:

When two rational numbers are divided by each other, the quotient must be a cyclic decimal when it is not divided.

Wrong. It is also possible that there are infinite non-cyclic decimals.

Two numbers are divided, and if they are not divided, the quotient must be a cyclic decimal. is wrong. The correct way to say this is: two rational numbers are divided, and if they are not divided, the quotient must be a cyclic decimal.

Division is one of the four operations. Knowing the product of two factors and one of the non-zero factors, the operation of finding the other factor is called division.

The division of two numbers is also called the ratio of two numbers. If ab=c(b≠0), the operation of finding another factor a using the product c and the factor b is division, written as c b, which is read as c divided by b (or b divided by c). where c is called the dividend, b is called the divisor, and the result a is called the quotient.

Operational properties of division:

1. The dividend expands (shrinks) n times, and the divisor remains unchanged, and the quotient also expands (shrinks) n times accordingly.

2. The divisor expands (shrinks) n times, the dividend remains unchanged, and the quotient shrinks (expands) n times accordingly.

3. The nature of division: the dividend-given is divided by two divisors consecutively, which is equal to the product of the two divisors. Sometimes simple operations can be performed depending on the nature of division.

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

Error analysis: There are two cases of quotient when the division is not exhausted in division: one is a circular decimal, that is, the decimal part of a number, from a certain digit, a number or multiple numbers are repeated in turn, such a number is called a circular decimal; The second is an infinite non-cyclic decimal, that is, an infinite non-cyclic decimal refers to an infinite number of digits after the decimal point, but there is no periodic repetition or no regular decimals, such as the circumferential law Answer:

There are two cases of the quotient when dividing inexhaustibly in division: one is a cyclic decimal, and the other is an infinite non-cyclic decimal

When two integers are divided, the quotient cannot be an infinite non-cyclic decimal, and the specific analysis is as follows:

1. Infinite non-cyclic decimal numbers are also called irrational numbers;

2. Irrational numbers are all real numbers that are not rational numbers;

3. Irrational numbers refer to numbers that cannot be expressed as the ratio of two integers within the range of real numbers;

4. Rational numbers are composed of all fractions and integers, and can always be written as integers, finite decimals or infinite cyclic decimals, and can always be written as the ratio of two integers, such as 21 7 and so on.

To sum up, it can be seen that an infinite non-cyclic decimal is not a real number of rational numbers, and an integer is a rational number, so the quotient obtained by dividing two integers is also a rational number, and the definition of an infinite non-cyclic decimal is contrary to this, so an infinite non-cyclic decimal cannot be written as a ratio of two integers, and the quotient cannot be an infinite non-cyclic decimal when two integers are divided.

If the numerator and denominator are both rational numbers, the conclusion is correct.

An uncyclic decimal is an irrational number, and the operation of a rational number cannot arrive at it.

We know that fractions are rational numbers. When a fraction is divided into decimals, the result cannot be infinite and non-cyclical, otherwise it becomes an irrational number, which contradicts "fractions are rational numbers". Therefore, when the fractional decimal is not divided, the result must be a cyclic decimal.

Conversely, cyclic decimals can be converted into fractions.

Some primary school teachers believe that when two numbers are divided inexhaustibly, the quotient may be a cyclic decimal or an infinite non-cyclic decimal. This perception is wrong.

If we assume that the natural number a is divided by the natural number b, which is inexhaustible, then the quotient must be an infinitesimal decimal. In the process of division, the remainder of each division is smaller than the divisor, and the remainder can only be 、...b 1, so that at most (b 1) remainders in a row are under the guise of each other The remainder must be the same as one of the previous (b 1) remainders, and the remainder is repeated, and the quotient is repeated continuously, so that a circular decimal is obtained. If the divisor is 17, the quotient will repeat from the 18th position at most; If the divisor is 43, the quotient is repeated from the 44th position at most.

As long as you have the patience to divide all the time, the quotient will be repeated from the (divisor +1) digit at most.

What if it's decimal division? Depending on the invariant nature of the quotient in division, decimal division can be converted to integer division.

To sum up, if the division of two numbers cannot be exhausted, the quotient must be a cyclic decimal. In the same way, if a simplest fraction cannot be reduced to a finite decimal, it must be reduced to a cyclic decimal.

Two numbers are divided, and if they are not divided, the quotient is not necessarily a cyclic decimal, because it is also possible .

If two integers are divided, if the integer quotient is not obtained, there will be two cases: one is to get a finite decimal; The other, get an infinite decimal.

The decimal infinitesimal decimal of the previous digit or section of the number is repeated sequentially and continuously, starting with a decimal place after the decimal point. The abbreviation for cyclic decimal is to omit all the numbers after the first cyclic stanza and add a small dot above the first and last two digits of the first cyclic stanza.

Analysis: There are two situations when the quotient is not divided in division: one is a cyclic decimal, that is, the decimal part of a number, from a certain digit, a number or multiple numbers are repeated in turn, such a number is called a cyclic decimal;

The second is an infinite non-cyclic decimal, that is, an infinite non-cyclic decimal refers to an infinite number of decimal places after the decimal point, but there is no periodic repetition or no regular decimals, such as the circumferential law

Answer: There are two cases when the quotient is not divided in division:

Not (not necessarily), because it may be an irrational number (or an infinite non-cyclic decimals) for example:

2=2/2, the quotient is an infinite non-cyclic decimal, not a cyclic decimal.

It should be said:

Not necessarily, it can also be an unreasonable non-cyclic decimal.

The quotient must be a cyclic decimal when the division is not exhausted, and this sentence is false.

There are two possibilities for the quotient when dividing is not exhaustive in division:

The first possibility is that the quotient is an infinite non-cyclic decimal.

The second possibility: the quotient is a cyclic decimal.

Pi is an infinite non-cyclic decimal number.

There are two cases of quotient when dividing is not exhaustive in division:

One is a cyclic decimal, and the other is an infinite non-cyclic decimal, such as the circumferential law

So the answer is:

When the division is inexhaustible, the quotient must be a cyclic decimal. （×

There are two possibilities for the quotient when dividing is not exhaustive in division:

The first possibility is that the quotient is an infinite non-cyclic decimal: the second possibility: the quotient is a cyclic decimal. Pi is an infinite non-cyclic decimal number.

Pure decimal: A decimal part of an integer that is 0 is called a pure decimal, and a pure decimal is smaller than 1.

For example, they are all pure decimals. A pure decimal is less than 1, which is 0form.

A pure decimal is a number between 0 and 1, (greater than 0 and less than 1), and in layman's terms, it is a few tenths of a (.

With decimals: Decimals with integer parts that are natural numbers (except 0) are called decimals, and decimal numbers are greater than 1.

Such as: etc. Loop Section:

The decimal part of a decimal place, from a certain digit, there is one or several numbers that repeat in succession and again is called a circular section.

loop), which has a loop knot of 35.

Pure cyclic decimals:

The first place in the decimal part of the cycle section is called a pure circular decimal. If it is a pure cyclic decimal, it is also a pure decimal.

Mixed loop decimals:

Cyclic sections that do not start with the first decimal part are called mixed cyclic decimals.

Such as finite decimals:

The number of digits in the decimal part is a finite decimal and is called a finite decimal place.

Infinitesimal Decimal: The number of digits in the decimal part is an infinite decimal number, which is called an infinite decimal.

The quotient is not necessarily a cyclic decimal, but also an infinite decimal.

The two numbers are divided by each other, and it is not stated what the two numbers are.

Two rational numbers are divided, and if they are not divided, the quotient must be a cyclic decimal.

One is irrational numbers.

One is an irrational number, and the result is an irrational number.

When two irrational numbers are divided, it may be a rational number or an irrational number.

1. The common types of rational numbers are as follows.

1.Integer: All integers are rational numbers.

2.Decimals: Finite decimals and infinitely cyclic decimals in decimal classification are rational numbers.

3.Fractions: Because all fractions are either equivalent to a finite decimal or an infinite loop decimal.

That is, the result of fractions into decimals is either a finite decimal or an infinite loop decimal. And both types of decimals are rational numbers, so, all fractions are rational numbers.

2. The common types of irrational numbers are as follows.

1.Infinite does not loop decimals.

Such as pi. Napierian logarithm.

The base number e, etc.

2.There are endless numbers of squares in the radical formula: such as the square root of 2 and the cube root of 5.

The fourth power of 7 is the root of the punch and so on.

Note: The sum, difference, product, and quotient of two rational numbers (divisor.

not 0) is still a rational number. The sum, difference, product, and Shang of two irrational numbers can be rational numbers or irrational numbers.

1) The sum, difference, product, and quotient of irrational numbers are rational numbers: such as e+(1-e), e-e, the square of "root number 2", e e, etc.

2) The sum and difference product quotient of irrational numbers is irrational numbers: +e, -e, xe, e.

The wrong quotient is an infinite decimal, which can be an infinite non-cyclic decimal or an infinite cyclic decimal.