In this article, we will learn about, How to find the decimal expansion of rational numbers. Earlier we discussed rational numbers and their properties. Every rational number has a unique decimal expansion. Finding it is an important thing one should know.

Inside Story

## Decimal Expansion

To understand the representation of rational numbers in decimal expression, we should know what are rational numbers.

Rational numbers:A number which can be written in the p/q form where q is not equal to zero is called rational number.

**For Example: **-3,4,5/8, 4/5 etc.

To know more about rational numbers read this article, Number System Class 9 NCERT, How To Find Rational Numbers Between Any Two Numbers?

### Types of the decimal expansion of rational numbers:

There are three types of the decimal expansion of rational numbers

**Terminating****Non-terminating and repeating**

#### Terminating Decimal expansion :

Terminating decimal are those decimals which contain a fixed number of digits. They are finite decimals.

** For example:** 2.3, 4.555, 1234.23, 1.132555552 etc.,

All these decimals have a finite number of digits.

**Non-terminating and repeating** decimal:

Non-terminating decimals are those who have infinite number of digits. These decimals have a repeating digits sequence in their decimal expansion.

**Example:** 1.222…,23.454545…, 0.567567567… etc.,

### How to find the decimal expansion of rational numbers

To find the decimal expression of rational numbers we divide the numerator by the denominator.

**Example 1:** Find the decimal expression of 10/3

3. | 3 | 3 | 3 | ||

3 | 1 | 0. | 0 | 0 | 0 |

9 | |||||

1 | 0 | ||||

0 | 9 | ||||

1 | 0 | ||||

0 | 9 | ||||

1 | 0 | ||||

0 | 9 | ||||

1 |

This gives us 10/3=3.3333…

**Example 2:** Find the decimal expression of 7/8

0. | 8 | 7 | 5 | |

8 | 7. | 0 | 0 | |

0 | ||||

7 | 0 | |||

6 | 4 | |||

6 | 0 | |||

5 | 6 | |||

4 | 0 | |||

4 | 0 | |||

0 |

We find that the 7/8 has a terminating decimal expansion of 0.875

**Example 3:** Find the decimal expression of 1/7

0. | 1 | 4 | 2 | |

7 | 1. | 0 | 0 | 0 |

0 | ||||

1 | 0 | |||

0 | 7 | |||

3 | 0 | |||

2 | 8 | |||

2 | 0 | |||

1 | 4 | |||

6 |

This gives us 1/7=0.142857142857… which is a non-terminating and repeating decimal expansion.

**Example 4:** Find the decimal expansion of 15/4.

3. | 7 | 5 | 0 | ||

4 | 1 | 5. | 0 | 0 | 0 |

1 | 2 | ||||

3 | 0 | ||||

2 | 8 | ||||

2 | 0 | ||||

2 | 0 | ||||

0 | |||||

0 | |||||

0 |

We find that the decimal expansion of 15/4 is 3.75, which is terminating.

**Example 5:** Find the decimal expansion of 329/9.

3 | 6. | 5 | 5 | 5 | ||

9 | 3 | 2 | 9. | 0 | 0 | 0 |

2 | 7 | |||||

5 | 9 | |||||

5 | 4 | |||||

5 | 0 | |||||

4 | 5 | |||||

5 | 0 | |||||

4 | 5 | |||||

5 | 0 | |||||

4 | 5 | |||||

5 |

We find that the decimal expansion of 329/9 is 36.555…. which is non-terminating and repeating.

## Conclusion

Hence, we conclude that every rational number has a decimal expansion. depending upon the divisor it could be terminating or non-terminating.

Also, if the rational numbers have a non-terminating decimal expansion then it must be repeated. To find the decimal expansion of a rational number we divide the numerator by the denominator.

##### Tutorial video to watch about how to find the decimal expansion of rational numbers

### Frequently Asked Questions – FAQs

##### Does every rational number has a decimal expansion?

Yes, every decimal number has a unique decimal expansion. It could be terminating or non-terminating.

Interestingly, all the rational numbers have a unique decimal expansion. we can also find the rational number from its decimal expansion.

##### What is terminating decimal expansion?

Terminating decimals are those decimals that contain a fixed number of digits. They are finite decimals

##### What is non-terminating decimal expansion?

Non-terminating decimals are those that have an infinite number of digits. These decimals have a repeating digits sequence in their decimal expansion.

**Read Also**

- Irrational numbers and their properties
- How to find rational numbers between any two numbers?
- How to represent √2 on a number line?
- How To Draw Root 3 And Root 5 On A Number Line?

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