The decimal part of a number is an infinite decimal that one or several numbers repeat in turn from a certain bit, which is called a cyclic decimal. Cyclic decimals have cyclic nodes (cyclic points), which can be turned into fractions.

If, after the decimal point of an infinite decimal, a number appears circularly from a certain bit to the right, and the ends are connected, it is called a cyclic decimal, and this number is called a cyclic segment. Cyclic decimal can be converted into a fraction by writing it as terms and an infinite geometric series.

Extended data:

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

The infinitesimal decimal of the previous number or section is called cyclic decimal, such as 2. 1666...* (mixed cyclic decimal), 35.232323 ... (cyclic decimal), 20.333333 ... (cyclic decimal), etc. , in which the numbers that appear repeatedly in turn are called cyclic decimals.

The abbreviation of cyclic decimal is to omit all the digits after the first cyclic segment, and add a dot above the first two digits and the last two digits of the first cyclic segment. For example:

2.966666 ... is abbreviated as 2.96(6 plus a dot) (pronounced "2.96, six cycles"), and 6 is a cycle segment;

35.232323 ... is abbreviated as 35.23 (dotted above 23) (pronounced as "35.23,23 cycle"), and 23 is the cycle segment;

36.568568 ... abbreviated as 36.568 (dotted above 568) (pronounced as "36.568,568 cycle"), 568 is the cycle segment;

Cyclic decimals can be converted into fractions by the summation formula of equal ratio series, so cyclic decimals are rational numbers.

Decimal scores are divided into two categories:

1, pure cyclic decimal, and the cyclic node is a molecule;

Write several 9s as denominator, and how many people write several 9s in the loop. Example: 0.3(3 cycles) =3/9 (the cycle part has one digit, so write a 9);

0.347(347 cycles) = 347/999 (write 3 9s in 3 cycles);

2. Mix circulating decimals, and subtract acyclic numbers from the decimal part as molecules;

Write a few 9s, and then write a few 0s as the denominator. If the number of cycles is several, write a few 9s, and if the number of cycles (fractional part) is several, write a few 0s. For example, 0.2 134(34 cycles) =(2 134-2 1)/9900.

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