Pure Recurring Decimal

Decimal fractions are often repeated after the division process. Some rational numbers produce the decimal fractions into recurring terms. However, irrational numbers mostly produce recurring decimals by converting them into decimals. Pure recurring decimals also fall under the periodicity of the recurring decimals, which are represented in a certain way.

Recurring Decimal- Definition and Types

Recurring decimals are the type of decimals fractions where one or more digits in decimals are repeated often. Suppose the fraction ⅔ is divided then the decimal form comes as 0.6666…. . Therefore, the number 6 is repeatedly used. Another example of repeating decimal is that if 22 is divided by 7 the result comes as 3.142857142857…. The repeated part is 142857, which can be termed as a recurring decimal. There are various methods in which the recurring decimal can be converted into rational and vice versa. 

The recurring decimal is of two types: the pure recurring decimal and the mixed recurring decimal. Pure recurring decimals are decimals in which all the digits are repeated. An example of a pure recurring decimal can be 0.4̅8̅. In the mixed recurring decimal fractions, some of the digits are repeated and some are not repeated after the decimals. Examples of the mixed repeated decimal are 0.42̅6̅ 7. In order to convert the mixed decimal fractions the difference of numerator and the numbers after the decimal points are required. The repeated digits are taken only once. The number of zeroes is placed after the denominators depending on the periodicity. However, the pure recurring decimals are converted into simpler steps.

Pure Recurring Decimal 

Pure recurring decimals have certain properties. The cyclic property of the pure recurring decimals is in which a cyclic permutation of the digits are found. Pure recurring decimals are generally reduced to a proper fraction in the form of p/q. In this decimal, then q is equal to prime to 10. It can be noted that in the decimal fraction if the q is not prime to 10 and if the decimal begins with a non-recurring portion then it is not considered as a pure recurring decimal. The pure recurring decimal can be considered when p is prime to q and is less than nine and q is prime to 10.

Conversion of Pure Recurring Decimal

Pure recurring decimals fractions can be converted into simpler fractions. The repeated digits are usually written in the numerator that to once. The number 9 is placed in the denominator. However, the number of nines to be written depends on the decimal digits. 

Suppose in case of the 0. ̅3, it can be written as 3/9 since there is only one repeated digit. Therefore, only a single nine is placed in the denominator. The pure recurring decimal can be converted into such a way and then made into a simpler form of ⅓.

Another example of conversion can be of a pure recurring decimal fraction can be taken by using 0.̅5̅7̅. It can be written as 57/99. There are only two digits in repetition. Therefore, two nines are written in the denominators. The pure recurring decimal fraction can be further written into 19/33.

The steps are required to convert the pure recurring decimal into the vulgar fractions.

• At first, the bar needs to be removed from the top of the decimal fraction by putting it equal “n” variable. 
• The repeating digits need to be written at least twice. 
• Finding the number of digits having bars on its heads. 
• Therefore, it must be noted that the conversion of pure recurring decimals can be determined by multiplying 10, 100, 1000 and so on. 
• 10 need to be multiplied on the pure recurring decimal if the repeating decimal has 1 place of repetition. 
• 100 need to be multiplied if the pure recurring decimal has 2 places of repetition.
• 1000 or more need to be multiplied if the pure recurring decimal has 3 or more places of repetition.
• The number needs to be subtracted which is obtained from the first step from the second step
• Both sides of the equation are divided with the coefficient of “n”.

The number of zeroes increases depending on the places of repetition in the pure recurring fractions. At last, the vulgar fraction is obtained from the pure recurring decimal, which is in its lowest form.

Conclusion

From the above discussion, it can be concluded that there are various types of decimal fractions. The recurring fractions are also categorised into pure recurring fractions and mixed recurring fractions. Pure recurring fractions are such types of fractions in which all the digits after the decimal are repeated. There are simple conversion techniques in which the pure recurring fractions are converted into simpler or vulgar fractions.