# Non terminating decimals

In this chapter we will learn the concept of non terminating decimals with examples and solved problems.

## Non Terminating Decimals Definition

The decimals containing unending number of digits after decimal points are called Non Terminating Decimals.

As the name suggests, in non terminating decimals, the digits after decimals doesn’t terminates.

This decimal contain infinite series of number after decimal point.

### Non Terminating Decimal Examples

17.333333. . . .
It’s a non terminating decimal since the number after decimal point is unending.

0.2857146398 . . .
It’s a non terminating decimal as the decimal point numbers are unending.

0.6666666. . .
It’s a non terminating decimal.
Note that the number 6 after decimal is getting repeated again and again.

Conclusion
Non terminating decimals contain infinite set of numbers after decimal point.

## Types of Non Terminating Decimals

There are two types of non repeating numbers;

(a) Repeating decimals
(b) Non Repeating decimals

### Repeating Decimals

They are non terminating decimals whose digits after decimal points get uniformly repeated again and again.

In repeating decimal you can identify the pattern of number repetition.

Examples of repeating decimals

0.66666. . . . .
It’s a repeating decimal since the digit 6 is repeated infinite times.

14.756756756756. . .
It’s a repeating decimal as the value 756 is repeated infinite times.

### Non Repeating Decimals

It’s a non terminating decimals whose decimal numbers are non-repeating.

\mathtt{\sqrt{2} \ } ⟹ 1.414213. . . .
It’s is a non repeating decimals as the number are not repeated after decimal point.

\mathtt{\sqrt{3} \ } ⟹ 1.732050. . . .
Again it’s a non repeating, non terminating decimal.

Note: All the square root values of numbers (except perfect square) are non repeating, non terminating decimals.

## Fraction with non terminating decimal value

The value of fraction can be found by dividing numerator and denominator.

There are different types of fractions which produce terminating or non-terminating decimal values.

Here we will learn to find the fraction value and identify if it’s terminating or non terminating.

Example 01
Find decimal value of fraction 100/3

Solution
Dividing numerator by denominator.

On dividing the fraction we get ⟹ 33.3333 . . . .

It’s a non terminating decimal as the number 3 after decimal point is repeating again and again.

Example 02
Find the decimal value of fraction 2/7

Solution
Dividing numerator by denominator.

On dividing the fraction we get ⟹ 0.2857 . . .

Note that the above division is unending. The above division steps will go on and on and you will never reach the point of conclusion.

Hence, the above decimal is non terminating.

Example 03
Find value of fraction 500/6

Solution
Divide the numerator with denominator.

On dividing the fraction we get ⟹ 83.3333 . . .

The number is non terminating and repeating decimal.

Example 04
Find the value of fraction 16/45

Solution
Dividing numerator by denominator

On dividing the fraction we get ⟹ 0.35555. . .

The number is a non terminating decimal.

Example 05
Find the value of fraction 7/9

Solution
Dividing numerator by denominator.

After division we get ⟹ 0.7777 . . .

The number is a non terminating decimal.

## Non Terminating decimals – Solved problems

(01) Identify repeating and non repeating decimals.

(i) 11.2222222 . . .
(ii) 105.365365365. . .
(iii) 36.4564523745 . . .
(iv) 11.263636363 . . .
(v) 91.90919297. .

Solution

(i) 11.2222222 . . .
It’s a repeating decimal.
Number 2 is repeated infinite times.

(ii) 105.365365365. . .
Its a repeating decimal.
Number 365 is repeated again and again.

(iii) 36.4564523745 . . .
It’s a non repeating decimal.

(iv) 11.263636363 . . .
It’s a repeating decimal.
Number 63 is repeated after decimal point.

(v) 91.90919297. .
It’s a non repeating decimal.

(02) Find the value of below fraction and check if its terminating or non-terminating decimal.

(i) 7/3
(ii) 6/5
(iii) 11/15
(iv) 24/9

Solution

(i) 7/3
Dividing the numerator with denominator.

After division we get ⟹ 2.3333 . . .

The number is a non terminating decimal since the number 3 after decimal is repeating infinite times.

(ii) 6/5

Dividing numerator by denominator.

Value of fraction ⟹ 1.2

The number is a terminating decimal since it contain finite digits after decimal point.

(iii) 11/15

Dividing numerator by denominator

Value of the fraction ⟹ 0.73333 . . .

It’s a non terminating decimal as the digit 3 is repeating infinite times.

(iv) 24/9

Dividing numerator by denominator.

The value of fraction ⟹ 2.6666 . . .

It’s a non terminating decimal as digit 6 is repeating infinite times.