# 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.

\(0.00\overline{69} = \frac{0069}{9900} = \frac{69}{9900}\)