Repeating Decimal to Fraction Conversion

What is Repeating Decimal?

A repeating decimal, also known as a recurring decimal, is a decimal representation of a number with periodic digits (values that occur at frequent intervals) and an infinitely repeated part that is not empty. A number can be seen to be rational if and only if its decimal representation repeats or terminates (i.e. all except finitely many digits are zero).


The repetend or reptend is a digit series that can be replicated indefinitely. Since the zeros can be omitted and the decimal terminates before these zeros, this decimal representation is considered a terminating decimal rather than a repeated decimal where the repetend is a zero.

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What is a Fraction?

A fraction (from the Latin fractus, which means "broken") denotes a portion of a whole or, more broadly, any number of equivalent parts. In daily English, a fraction denotes the number of pieces of a certain scale, such as one-half, eight-fifths, or three-quarters. The numerator and denominator of positive general fractions are all natural numbers. The denominator means how many of those components make up a unit or a whole, while the numerator represents a number of equivalent parts. Since zero pieces will never make up a whole, the denominator cannot be zero. The numerator 3 of the fraction 3/8, for example, means that the fraction represents three equal parts, and the denominator 8 indicates that the fraction represents three equal parts.

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Types of Decimals

  • Terminating Decimal

  • Non Terminating Decimal 

  • Recurring Decimal

  • Non-recurring Decimal 

What is Terminating and Non Terminating Decimal?

The two main classifications of decimals are terminating and non terminating numbers, let us understand them in detail.


What is a Terminating Decimal?

Terminating Decimal: A decimal number with a finite number of digits after the decimal point is known as a terminating decimal.


What is Non Terminating Decimal?

Non Terminating Decimal: A non-terminating decimal is one that continues indefinitely. To put it another way, a non-terminating decimal is formed when a fraction is represented in decimal form but still has a remainder, regardless of how much the long division procedure is carried out.


What is a Recurring Decimal?

Recurring Decimal: A repeating decimal, also known as a recurring decimal, is a decimal representation of a number with periodic digits (values that occur at frequent intervals) and an infinitely repeated part that is not empty. A number can be seen to be rational if and only if its decimal representation repeats or terminates (i.e. all except finitely many digits are zero).


What is a Non-Recurring Decimal?

Non-recurring Decimal: A non-recurring decimal is one in which the digits do not repeat. The number 0.101001000100001..., for example, is apparently non-recurring. Most numbers that are not specified as fractions of integers, to begin with, have non-recurring decimal expansions; for example, the square root of 2 is 1.4142135623730950488016887242..., and its digits do not repeat. These are referred to as irrational numbers.

Conversion Repeating Decimal to Fraction

Following are the steps to convert repeating decimal to fraction:

  • Allow x to be the repeating decimal to convert to a fraction.

  • Look for the repeating digit of the repeating decimal(s).

  • To the left of the decimal point, place the repeated digit(s).

  • To the right of the decimal point, place the repeated digit(s).

  • Subtract the left sides of the two equations using the two equations you discovered in steps 3 and 4. Then, take the right sides of the two equations and subtract them.

  • Only make sure that the difference is positive on both sides when you deduct.

Non Terminating Repeating Decimals are Rational

The repetitive terms are used in non-repeating decimals. Irrational numbers are described as non-terminating and non-repeating decimals. Non-terminating and repeated decimals are Rational numbers and can be expressed as p/q, where q is greater than zero.

Let us take the example of 0.004 to understand the concept,

Answer: Start by writing the following simple equation to transform 0.004 repeating into a fraction:

n = 0.004 ……..( 1)

Step 2: Since the repeated block (4) has 1 digit, multiply all sides by 1 followed by 1 zero, i.e., by 10.

10 × n = 0.044 …….(2)

Step 3: Deduct equation 1 from equation 2 to eliminate the repeated block (or repetend).

10 × n = 0.044

 1 × n = 0.004

 9 × n = 0.04

The fraction above has a decimal numerator. We would convert it to an integer by multiplying it by 100. We can multiply the denominator by the same sum as we multiply the numerator. As a result,

\[\frac{0.04}{9}\] = \[\frac{4}{900}\]

On simplification, we get,

n = \[\frac{1}{25}\]

Therefore, we can say that 0.004can be represented as \[\frac{1}{25}\] which is in \[\frac{p}{q}\] form, hence repeated non terminating decimals are rational.

FAQs (Frequently Asked Questions)

1. What Is the Method for Converting a Repeating Decimal to a Fraction?

Answer: Make an equation in which x equals the decimal sum, y is the number of decimal places. Make a second equation by multiplying both sides of the first by 10y. Subtract the result of the second equation from the result of the first. Calculate x. Cut the fraction in half.

2. Can You Write Repeated Decimals as Fractions?

Answer: Any terminating decimal representation can be written as a decimal fraction, that is, a fraction with a denominator that is a power of ten. It is also possible to write it as a ratio of the form k/2n5m.

3. Can We Convert Fraction to Decimal?

Answer: Yes, fractions can be converted to decimals. Divide the numerator by the denominator to transform a fraction to a decimal. 

4. What are the Two Main Types of Decimals?

Answer: The two types of decimals are terminating and non terminating decimals. An example of Non terminating decimal is the value of and an example of a terminating decimal is 1.42.