How to find the decimal expansion of rational numbers?

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.

How to find the decimal expansion of rational numbers

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

  1. Terminating
  2. Non-terminating and repeating
image 32 How to find the decimal expansion of rational numbers

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.,

image 30 How to find the decimal expansion of rational numbers

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

3.333
310.000
9
10
09
10
09
10
09
1

This gives us 10/3=3.3333…

Example 2: Find the decimal expression of 7/8

0.875
87.00
0
70
64
60
56
40
40
0

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

Example 3: Find the decimal expression of 1/7

0.142
71.000
0
10
07
30
28
20
14
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.

3.750
415.000
12
30
28
20
20
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.

36.555
9329.000
27
59
54
50
45
50
45
50
45
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

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