How do cyclic decimals become fractions?

The decimal that starts the cycle from the first decimal place after the decimal point is called a pure cyclic decimal The method of converting a pure cyclic decimal into a fraction is: the numerator is a number composed of the numbers of a cyclic node; The denominator is 9, and the number of 9 is equal to the number of digits of a cyclic section

If the first few digits after the decimal point do not circulate, and the cycle does not start until a certain digit after it, such a decimal is called a mixed loop decimal The method of turning a mixed loop decimal into a fraction is: the numerator is the difference between the number composed of the number of the non-circular part and the number of the number of the non-circular part, and the denominator is to write a few 9s according to the number of bits of a cyclic section, and then add a few 0s to the number of digits of the non-cyclic part

There are two kinds: pure cycles.

Using 9 as the denominator, there are as many cycles as there are 9s, for example, a cycle of 3 is 3/9, a cycle of 654 is 654/999, a cycle of 9 is 9/9 (1), and so on.

Mixed circulation. Use 9 and 0 as the denominator, first have a few cycle sections on a few 9, and then add a few 0s to a few numbers that have not joined the cycle, and then use the number after the decimal point to subtract the number that has not joined the cycle, for example, the cycle of 3, there is a single digit that has not joined the cycle, just add a 0 behind the 9 as the denominator, and then use 43 minus 4 to make the numerator, to get 39/90, the cycle of 5 is followed by 9 plus 2 0s as the denominator, and then use 145 minus 14 to make the numerator, to get 131 of 900, , 49 cycle, Add 1 0 to 99 as the denominator, subtract 5 from 549 as the numerator, and finally get 545/990, and so on.

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.

Method: The pure cyclic decimal was rewritten as a fraction, and the numerator was a cyclic node.

the number of the number of the composition; Denominator.

Each number is 9, and the number of 9 is the same as the number of numbers in the loop section, and the number that can be divided at the end can be divided again.

2. The fractional of the mixed cycle is a fraction.

Method: The mixed cyclic decimal is rewritten as a fraction, and the numerator is the difference between the number of the fractional part of the circular section minus the number of the non-cyclic part of the decimal part. The first few digits of the denominator are 9, the last digit is 0, the number of 9 is the same as the number of the cyclic section, and the number of 0 is the same as the digit of the non-cyclic part.

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To unify the above conclusions, the characteristics are: if there are a total number of bits in the cyclic section plus the non-cyclic digits, then the denominator is the number of 9+0 digits of the number of digits, the number of 9 is equal to the number of cyclic knots, and the number of 0 is equal to the number of non-cyclic digits; The numerator is equal to = the number that does not cycle after the decimal point plus the number formed by the first cycle section, and then subtracts the number that does not cycle after the decimal point.

The decimal that starts the cycle from the first decimal place after the decimal point is called a pure cyclic decimal The method of converting a pure cyclic decimal into a fraction is: the numerator is a number composed of the numbers of a cyclic node; The denominator is 9, and the number of 9 is equal to the number of digits of a cyclic section

If the first few digits after the decimal point do not circulate, and the cycle does not start until a certain digit after it, such a decimal is called a mixed loop decimal The method of turning a mixed loop decimal into a fraction is: the numerator is the difference between the number composed of the number of the non-circular part and the number of the number of the non-circular part, and the denominator is to write a few 9s according to the number of bits of a cyclic section, and then add a few 0s to the number of digits of the non-cyclic part

Japan's Tetsunori Noguchi in "Oh my God! Mathematics can be learned in this way", which introduces how to convert cyclic decimals into fractions, is introduced as follows:

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 is 1, 2, and 3 three, so the fraction is 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.The cyclic decimal is first multiplied by 100, 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

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1. Pure cyclic decimal into fractions.

Method: The pure cyclic decimal was rewritten as a fraction, and the numerator was a number composed of the numbers of a cyclic section; The denominator is 9, and the number of 9 is the same as the number of numbers in the loop section, and the last can be reduced to another minute.

2. Mixed circulation decimals are converted into fractions.

Method: The mixed cyclic decimal is rewritten as a fraction, and the numerator is the difference between the number of the fractional part of the circular section minus the number of the non-cyclic part of the decimal part. 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.

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.

For example, if the chemical fiber answer is not a fraction, then 100x= so 100x-x=12, that is, 99x=12x=12, 99=4, 33, that is, the score of chemical destruction is 4, and the number is 33