NonTerminating Repeating Decimal to Fraction

Introduction

• In non-terminating decimals, the numbers after the decimal point tend to be endless.

• In the non-terminating repeating decimals, a sequence of numbers after the decimal point is recurring in a specific pattern.

• These types of numbers are called Rational numbers. They are in $\mathrm{\frac{p}{q}}$ where q ≠ 0.

• In non-terminating decimals we will be discussing irrational numbers which cannot be represented in fractions.

• Whereas, if the non-terminating decimals have recurring patterns then it is said these decimals can be converted into fractions. In this tutorial, we will learn about non-terminating repeating decimals and how to convert them to fractions.

Decimals

• A number separating an integral part and the fractional part by a point is called a decimal number or decimal.

• The separating point is called the decimal point.

• The numbers after the decimal point are considered single digits since they belong to the fractional part, clearly having a value smaller than one.

For example,

Take a random decimal number 106. 12

1	0	6	.	1	2
Hundreds	Tens	Ones	Decimal Point	Tenths	Hundredths

The Decimal number 106. 12 will be read as Hundred and six-point one-two.

Types of Decimals

There are four types of decimal numbers according to the nature of the numbers in the fractional part. They are as follows −

• Terminating Decimals

• Non- Terminating decimals

• Recurring Decimals

• Non- recurring decimals

Terminating Decimals

If a decimal number has an endpoint, then it is called a Terminating decimal.

$$\mathrm{E.g:\frac{7}{10} = 0.7}$$

If a novel costs $5.46 then it is called a terminating decimal number.

Non- Terminating decimals

If a decimal number has no endpoint, then it is called a Non - Terminating decimal

For example ,Every irrational number is a Non- Terminating decimal.

For example, $\mathrm{ \pi =\frac{22}{7} = 3.141592653589793238\dotso}$

non-terminating decimals can also be non-recurring decimals

Recurring Decimals

If a decimal number has recurring numbers in its fractional part then they are called Recurring Decimals.

The recurring decimals are also called non-terminating repeating decimal numbers.

Say, for example, rational number, $\mathrm{\frac{4}{9}=0.4444\dotso }$

Non-recurring decimals

If a decimal number continues endlessly but doesn't have any repeating pattern then they are called non-repeating decimals.

Non-terminating decimal numbers can be called non-repeating decimals.

Conversion of Non-terminating repeating decimals to Fractions

If a decimal number is non-terminating and has a recurring pattern of a sequence of numbers we can convert it into a fraction by the following technique −

• Take 0.2222… as an example of a Non-terminating repeating decimal.

• To change the number in the ones place value of the decimal 0.2222…

• Multiply the number with the help of 10

• We get 2.222….

• And subtract the number we got from the given decimal 0.2222…

• We get a whole number 2

• Divide the whole number with 9 (if it is a single digit, if the number is a double digit then by 99 and so on)

• The converted fraction is $\mathrm{\frac{2}{9}}$

The above steps in a simple representation −

$$\mathrm{10 × a\: fraction -fraction\: = a\: whole\: number }$$

$$\mathrm{9\: fraction = whole\: number}$$

Solved Examples

1) Convert the decimal number 0.111… to a fraction.

Solution:

• In order to change the number in the ones place value of the decimal 0.1111…

• Multiply the number with the help of 10

• We get 1.111…

• And subtract the number we got from the given decimal 1.111….

• We get a whole number 1

• Divide the whole number with 9

• The converted fraction is $\mathrm{\frac{1}{9}}$

2) Convert the decimal number 0.88888… to a fraction.

Solution:

• In order to change the number in the ones place value of the decimal 0.8888…

• Multiply the number with the help of 10

• We get 8.8888…

• And subtract the number we got from the given decimal 8.888…

• We get a whole number 8

• Divide the whole number with 9

• The converted fraction is $\mathrm{\frac{8}{9}}$

3) Convert the decimal number 0. 171717… to a fraction.

Solution:

• In order to change the number in the ones place and tens place value of the decimal 0. 171717…

• Multiply the number with the help of 100

• We get 17. 171717…

• And subtract the number we got from the given decimal 17.171717….

• We get a whole number 17

• Divide the whole number with 99

• The converted fraction is $\mathrm{\frac{17}{99}}$

4) Convert the decimal number 0. 521521521… to a fraction.

Solution:

• In order to change the number in the ones place, tens and hundreds place value of the decimal 0. 521521521…

• Multiply the number with the help of 1000

• We get 521. 521521521…

• And subtract the number we got from the given decimal 521. 521521521…

• We get a whole number 521

• Divide the whole number with 999

• The converted fraction is $\mathrm{\frac{521}{999}}$

5) Convert the decimal number 0. 656656656…. to a fraction.

Solution:

• In order to change the number in the ones place, tens and hundreds place value of the decimal 0. 656656656….

• Multiply the number with the help of 1000

• We get 656. 656656656….

• And subtract the number we got from the given decimal 656. 656656656…

• We get a whole number 656

• Divide the whole number with 999

• The converted fraction is $\mathrm{\frac{656}{999}}$

6) Convert the decimal number 0.77777… to a fraction.

Solution:

• In order to change the number in the ones place value of the decimal 0.7777…

• Multiply the number with the help of 10

• We get 7.777…

• And subtract the number we got from the given decimal 7.7777…

• We get a whole number 7

• Divide the whole number with 9

• The converted fraction is $\mathrm{\frac{7}{9}}$

7) Convert the decimal number 0. 232323… to a fraction.

Solution:

• In order to change the number in the ones place and tens place value of the decimal 0. 232323…

• Multiply the number with the help of 100

• We get 23.232323…

• And subtract the number we got from the given decimal 23.232323….

• We get a whole number 23

• Divide the whole number with 99

• The converted fraction is $\mathrm{\frac{23}{99}}$

8) Convert the decimal number 0. 454454454… to a fraction.

Solution

• In order to change the number in the ones place, tens and hundreds place value of the decimal 0.454454454…

• Multiply the number with the help of 1000

• We get 454. 454454454…

• And subtract the number we got from the given decimal 454. 454454454…

• We get a whole number 454

• Divide the whole number with 999

• The converted fraction is $\mathrm{\frac{454}{999}}$

FAQs

1. What is a decimal notation?

Denoting numbers in a decimal system is called Decimal notation.

A Decimal system is a numeral system containing integer numbers and non-integer numbers

2. Who invented Decimals?

A Scottish Mathematician John Napier was the first one to introduce the decimal system.

The logarithms were also one of his contributions to the field of Mathematics and Physics.

3. What are irrational numbers?

The numbers which cannot be represented in $\mathrm{\frac{p}{q}}$ where p and q are non-zero integers. These numbers are non-terminating non-recurring decimals. Example: √(2 ) = 1.41421...

4. Who invented fractions?

Mathematician Simon Stevin invented fractions. He is famously known for introducing Decimal Fractions.

5. What are integers?

The number that can be written without the fractional component is called an integer. There are two different types of integers: positive and negative integers.