A repeating or recurring decimal is often a way of which represents rational numbers throughout base 10 in math. The decimal representation of an number is reportedly repeating if the idea becomes periodic.

A repeating decimal is often a decimal that has a digit, or any block of numbers, that repeat over and over and over again without ever finishing. The part that repeats is usually shown by placing dots over the first and last digits of the repeating pattern, or sometimes a line over the pattern.The digit sequence repeated at infinitely is called the repetend or reptend.
Examples: $\frac{1}{7}$ = 0.142857142857.......

$\frac{1}{3}$ = 0.33333.

In mathematics, we deal with different types of numbers and arithmetic operations on them. When we divide two numbers, usually we get result as terminating decimal, repeating decimal, irrational number or natural number.

Terminating decimal have a limited number of digits and no remainder in the quotient. Repeating decimals repeat one or more digits in the same pattern infinitely. Whereas irrational numbers can not be expressed as a fraction and do not have a repeated pattern of digits in the quotient.

## Definition

Non terminating decimals could be the continuous decimal variety. It has been taking place till the worth infinitive. It could be the very big worth sometimes these no terminating decimals tend to be rounded to its nearest values. However the terminating decimal number could be the fixed value of the decimal value. A decimal number which has a decimal part having repeating digits is called repeating decimals.

For Example: 0.7 = 0.777777.....

Here the digit is repeated after one digit.

0.05 = 0.05050505.....

Here the digit is repeated after two digits.

0.34216 = 0.342163421634216....

Here the digit is repeated after five digits.

## Convert Repeating Decimals to Fractions

Fractions describes the number of parts a certain size is usually where as some sort of decimal has ten as its base. Fractions and decimals are usually equivalent representations in the part of an entire. Decimals are your representation of number in mix of whole and decimal part.
Steps to Convert Repeating Decimals to Fractions:

Step 1:  Name the given fraction as x

Step 2: Multiply this fraction by a power of 10, such that the power is equal to the number of repeating digits in the decimal.

Step 3:  Subtract the equation in step 1 from the equation in step 2.

Step 4: Represent x as a fraction.

## Convert Fraction to Repeating Decimal

A repeating decimal may also be expressed as the infinite series. That's, a repeating decimal might be regarded as the sum of an infinite amount of rational numbers. More common practice to symbolize repeating decimals would be to put a dot or a bar above the digit that's repeated.

Best way to identify repeating decimal: If a simple  frcation is in lowest terms and its denominator has a prime factor other than 2 and 5, then the frcation equals an infinite repeating decimal.

Let us see with the help of an example how to find repeating decimal:

Example: Write $\frac{6}{11}$ as a decimal.

Solution: Divide 6 by 11 using long division. Add a decimal point and atleast  one zero.

Step 2:  Divide. When we subtract 55 from 60, we are left with remainder 5. Add another zero, now we have another 50. Divide again. When we subtract 44 from 50, the remainder is 6 again. Add another zero, again we are left with 60 (starting point).

Step 3: If we proceed division process, will left with 0.54545454....(repeating decimal).

Usually, repeating decimals are written using a bar over the digit that repeat.

This implies, $\frac{6}{11}$ = $0.\bar{54}$

## Examples

Given below are some examples on repeating decimals.

Example 1: Convert 0.4646... to fractions.

Solution:

Let x = 0.4646

Step 1: Multiply  x by 100

Then 100x = 46.4646

Step 2: Subtract x from 100x

100x - x = 46.4646 - 0.4646

99x = 46

Step 3: Divide both sides by 99

x = $\frac{46}{99}$

The fraction cannot be simplified any further.

Hence 0.4646.... = $\frac{46}{99}$

Example 2: Convert 0.55555......... to fraction.

Solution:

Step 1: Consider x to be 0.55555.....  ...1

In the given problem we see that only one digit is repeated so multiply the above equation by 10.

Now we get, 10x = 5.5555  ---2

Step 2: Subtract equation 1 from equation 2
9x = 5

Step 3: $\rightarrow$ x = $\frac{5}{9}$

The fraction cannot be simplified, therefore $\frac{5}{9}$ is in its simplest form.

Hence, 0.55555 = $\frac{5}{9}$

Example 3: Write $\frac{2}{3}$ as a repeating decimal.

Solution: Use long division method to divide 3 by 2.

atleast one zero.

Subtract 18 from 20, we are left with remainder 2. Add another zero, now we have another 20. Divide again. When we subtract 18 from 20, the remainder is 2 again. Add another zero, again we are left with 20. If we repeat this process infinity times also will left with same number.

Write the repeating decimals using a bar over the repeated digit.

This implies, $\frac{2}{3}$ = $0.\bar{6}$