 # Non-Terminating Repeating Decimal to Fraction

Before going into the conversion of non-terminating and repeating decimal to fractions, let us understand what significance do these terms hold. What are terminating, non-terminating, repeating and non-repeating decimals.

Terminating and Non-Terminating decimals-

A terminating decimal that has an ends. It is a decimal, which has a finite number of digits(or terms).

Eg. 0.15, 0.86 etc.

Whereas non-terminating decimals are the one that do not have an end term. It has infinite number of terms.

Eg. 0.5444444….., 0.1111111….., etc.

## Repeating and Non-Repeating decimals:

Repeating decimals are the one, which have a set of terms in a decimal to be repeated in a uniform manner.

Eg. 0.666666…., 0.123123…., etc.

It is to be noted that the repeated term in a decimal are represented by bar on top of the repeated part. Such as $0.333333….. = 0.\bar{3}$.

Whereas non-repeating decimals are the one that do have have repeated terms.

Non-Terminating and non-repeating decimals are said to be an Irrational numbers. Eg. $\sqrt{2} = 1.4142135……$.

The square roots of all the terms (leaving perfect squares) are irrational numbers.

Non- Terminating and repeating decimals are Rational numbers and can be represented in the form of p/q, where q is not equal to 0.

Let us now learn to convert Non-Terminating and repeating decimals in rational form.

(i) Fraction of the type $0.\overline{abcd}$

$\overline{abcd} = \frac{Repeated \; term}{Number \; of \; 9’s \; for \; the ;\ repeated \; terms}$

Example – Convert $0.\overline{7}$ in Rational form

Solution – Here the number of repeated term is only 7, thus number of times 9 to be repeated in the denominator is only one.

$0.\overline{7} = \frac{7}{9}$

Example – Convert 0.125125125….. in Rational form

Solution – The decimal shown above can be written as $0.\overline{125}$.

Here 125 consist three terms to be repeated in a continuous manner. Thus number of time 9 to be repeated in the denominator becomes three.

$0.\overline{125} = \frac{125}{999}$

(ii) Fraction of the type $0.ab..\overline{cd} =\frac{(ab….cd…..) – ab……}{Number \; of \; time \; 9’s \; the \; repeating \; term \; followed \; by \; the \; number \; of \; times \; 0’s \; for \; the \; non-repeated \; terms }$

Example – Convert $0.12\overline{34}$ in a Rational form

Solution – In the given ratio we have 12 to be of the non-repeated form and 34 to be of the repeating form. Thus denominator becomes 9900.

$0.12\overline{34} = \frac{1234 – 12 }{9900} = \frac{1222}{9900}$

Example – Convert $0.00\overline{69}$ in p/q form

Solution- In the given ratio we have 00 to be of the non-repeated form and 69 to be of the repeating form. Thus denominator becomes 9900.

$0.00\overline{69} = \frac{0069}{9900} = \frac{69}{9900}$<

This was all about converting Non-Terminating decimals into Fraction. Practice more questions on conversion such as decimals to percentage etc. at BYJU’S-The learning app.