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.