# 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.

#### Practise This Question

Median is defined as:
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9398307800292969, "perplexity": 1301.0588521121135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526386.37/warc/CC-MAIN-20190719223744-20190720005744-00220.warc.gz"}