What is the method and principle of cyclic decimal fractions

1. Pure cyclic decimal fraction The decimal that circulates on the first decimal place after the decimal point is called a pure cyclic decimal. How do you turn it into a fraction? Take a look at the example questions below.

Convert the pure cyclic decimal into a fraction: The fractional part of the pure cyclic decimal can be converted into a fraction, the numerator of this fraction is a number represented by a cyclic section, and the number on the denominator is the same as the number of digits of the cyclic section. The offer to be able to make a contract.

2. Mixed Cycle Decimal Fraction The decimal that is not the first place after the decimal point is called the mixed cycle decimal. How do you turn mixed loop decimals into fractions? Fractionalize the mixed loop.

2) Let's look at the decimal part first, the fractional part of a mixed cyclic decimal can be turned into a fraction, and the numerator of this fraction is the difference between the number of the decimal part before the second cycle and the number of the non-cyclic part of the decimal part. The first digits of the denominator are 9, the last digits are the same number as the number of digits in the loop section, and the number of 0s is the same as the number of digits in the non-cyclic section. 3. After the four rules of cyclic decimal operations are converted into fractions, the four rules of cyclic decimal operations can be carried out according to the four rules of fraction operation.

In this sense, the quadruple operation of cyclic decimal places is the same as the quartile operation of finite decimals, which is also the quadruple operation of fractions. The finite decimal fraction is directly removed from the decimal point, and the denominator corresponds to ten hundred million, etc. Again.

Cycle knots. If there are several people, there is a denominator.

There are a few zeros, a few zeros that do not participate in the cycle, and a few zeros after the denominator are fractional parts.

For example, in the 01 cycle, it is 1 99

In the 1 cycle, it is 1 90

In the 21st cycle, it is 321 990

A multi-digit decimal loop is divided by 0After a few 9s, the dividend is the part of the original decimal that is not zero minus the part of the loop section, for example: the middle 1 cycle is (the middle 81 cycle is (the middle 481 cycle is (the middle 5481 cycle is (the middle 75481 cycle is (

The method of mixing cyclic decimal fractions is to subtract the difference obtained from the non-cyclic fraction by subtracting the fraction of the fraction before the second cyclic section, and use this difference as the numerator of the fraction; The first few digits of the denominator are 9, and the last digit is 0; The number of 9s is the same as the number of digits of a looping section, and the number of digits of 0 is the same as the number of digits of the non-looping section.

The arrows point to the description: a 9 is written in a loop section, and a 0 is written in a non-loop section.

The arrow indicates that there are two digits written as two nines in the circular section, and one bit as a 0 in the non-circular section.

The arrow points to the description: There are two nines written in the loop section, and two zeros are written in the non-loop section.

This method is obviously more complicated than that of pure cyclic decimal fractions, but the arithmetic is still based on the method of pure decimal fractions. That is, the mixed cyclic decimal is first converted into a pure cyclic decimal, and then into a fraction.

The above three example problems can be proved by derivation.

<> derivation results are consistent with the intermediate offset in example (3).

It can be seen that the method of expanding and then shrinking the same multiple is adopted, and the method of pure cyclic decimal fraction is proved to be completely valid.

1. The cyclic decimal cyclic section is 7,2 digits, so the fraction is 72 99 = 1 8That is, if there are a few digits, divide by a few nines. For example, the cyclic section has 1, 2, and 3 three dimensions, so the fraction is 123 999 = 41 333

This megadermal method is only applicable to delayed decimal places where the first family of denier differences after the decimal point is circulated, and if it is not a decimal that starts the cycle from the first digit, the following method must be used.

2. Multiply the cyclic decimal by 100 first, which can be understood as 41+, so the fraction is written as 41 + 6 9 = 41 + 2 3 = 125 3Since we start by multiplying by 100, we divide by 100, i.e. 125 3 100 = 125 300 = 5 12

1. The circulating decimal buried bending infiltration circulation node is 7,2 two places, so the bending ridge is 72 99 = 1 8That is, if there are a few digits, divide by a few nines. For example, the cyclic section is 1, 2, and 3, so it is reduced to a fraction of 123 999 = 41 333

This method only works for decimals that start looping from the first decimal point, and if they don't start looping from the first decimal point, you must use the following method.

2. Multiply the cyclic decimal by 100 first, which can be understood as 41+, so the fraction is written as 41 + 6 9 = 41 + 2 3 = 125 3Since we start by multiplying by 100, we divide by 100, i.e. 125 3 100 = 125 300 = 5 12

The formula for cyclic decimal fractions: ab(ab cyclical) = (ab 99). The law of decimal fraction of pure stove cycle is:

The next loop is used as the numerator, and several 9s are written as the denominator, and the number of 9s is equal to the number of bits of a loop section.

The law of cyclic decimal fractions is that the fraction of this fraction is the difference between the number of the decimal part before the second cycle and the number of the non-cyclic part of the decimal part. The first digits of the denominator are 9, the last digits are the same as the number of digits in the cyclic section, and the number of 0 is the same as the number of digits in the non-cyclic section.

Classification of cyclic decimals:1. Pure cyclic decimals: start the cycle from the decimal place to the tenth place, for example: it is a pure cyclic decimal.

2. Mixed cycle decimal: the cycle does not start from the tenth place after the decimal point, and the cycle starts later, for example: it is a mixed cycle decimal.

Rewrite the pure cyclic decimal as a fraction, and the numerator is a number composed of the numbers of the cyclic section; The denominator is 9, and the number of 9 is the same as the number of numbers in the Gaoxiao cycle.

For example12341234...1234/9999。

Mixed loop decimal fractions.

The mixed cyclic decimal is rewritten as a fraction, and the numerator is the number composed of the number composed of the non-cyclic part and the first cyclic section, and the difference between the number composed of the non-cyclic part of the number is subtracted; The first digits of the denominator are 9, the last digits are 0, the number of 9 is the same as the number of the loop section, and the number of 0 is the same as the number of the non-loop part.

For example