Class 9 - Decimal Expansion of Real Numbers: Types, Solved Examples & Practice Questions

Real numbers and their decimal expansion is an important topic in the number system chapter of class 9 maths. This topic covers the terminating decimal expansion, non-terminating repeating and non-repeating decimal expansions. Let’s understand each of these decimal expansions one by one along with examples.

Table of Contents

• Decimal Expansion of Real Numbers
• Solved Examples on Decimal Expansion of Real Numbers
• Practice Questions on Decimal Expansion of Real Numbers

Decimal Expansion of Real Numbers

Decimal expansion of real numbers is one of the ways to represent real numbers. Expressing real numbers in decimal form is known as Decimal Expansion. For example, ½ in decimal form is equal to 0.5.

Based on how far the number appears after a decimal point real numbers can be classified into three major categories:

• Terminating Decimal

• Non-terminating and Repeating Decimal

• Non-terminating and Non-repeating Decimals

Terminating Decimal numbers:

Terminating decimal numbers are numbers in which the numbers after a decimal terminate after a certain digit. These numbers are rational numbers as all terminating decimal numbers can be converted into fractions.

Here are some examples of terminating decimal numbers:

12=0.5

34=0.75

15=0.2

78=0.875

Non-terminating and Repeating Decimal

Some decimals number after decimal never end, but they follow a pattern. These are called non-terminating repeating decimals. In such numbers, one digit or a group of digits repeats again and again. Even though these decimals do not end, they are still rational numbers because they can be expressed as fractions.

For example:

13=0.3333…

27=0.285714285714…

16=0.1666…

In all these numbers, the pattern of digits after decimal keeps on repeating endlessly.

Non-terminating and Non-repeating Decimals

The third type of decimal expansion is called non-terminating and non-repeating decimals. In this one the digits after decimal keep going on forever but they usually do not follow any fixed pattern like non-terminating repeating decimals. These numbers cannot be written as simple fractions and are irrational numbers.

For example:

2=1.4142135…

π=3.1415926…

Read more:

• Real numbers
• Class 9 - Operations on Real Numbers
• Class 9 - Exponents Rules 

Solved Examples on Decimal Expansion of Real Numbers

Example: Show that 3.142678 is a rational number. In other words, express 3.142678 in the form p/q, where p and q are integers and q ≠ 0.

Solution: We have 3.142678 which can be represented in fraction form as: 31426781000000

Hence, 3.142678 is a rational number.

Practice Questions on Decimal Expansion of Real Numbers

Check which of the following decimal expansion is terminating or non-terminating (recurring/non-recurring):

• 38
• 715
• 920
• 116
• 5
• 1325