Understand Rational Numbers: Concept & Examples

Key Concepts

• Write rational numbers in decimal form: terminating decimals  
• Write rational numbers in decimal form: repeating decimals
• Recognize rational numbers in decimal form

1.2 Understand Rational Numbers 

What is a rational number? 

A rational number is a number that is of the form , where p and q are integers and q ≠ 0. Rational numbers are denoted by Q . 

How to identify rational numbers? 

To identify if a number is rational or not, check the below conditions. 

• It is represented in the form of p/q, where q≠0. 
• The ratio p/q can be further simplified and represented in decimal form. 
• All whole numbers are rational numbers. 

Decimal Representation of Rational Numbers: 

Rational numbers can be expressed in the form of decimal fractions. 

A rational number can have two types of decimal representations: 

• Terminating 
• Non-terminating 

Terminating decimals: 

Terminating decimals are those numbers which come to an end after few repetitions after decimal point. 

Example: 0.5, 2.456, 123.456, etc. are all examples of terminating decimals. 

Non terminating decimals (repeating): 

Non terminating decimals are those which keep on continuing after decimal point. They do not come to an end or if they do it is after a long interval. 

Example: 1/7= 0.1428571…. which is a non-terminating repeating decimal. 

Example 1: 

Convert the fraction, 5/8 to a decimal. 

So,5/8 = 0.625. This is a terminating decimal. 

Example 2: 

Convert the fraction 7/12 to a decimal. 

7/12= 0.583. This is a repeating decimal. 

The bar over the number, in this case 3, indicates the number or block of numbers that repeat unendingly. 

Example 3: 

The length and breadth of a rectangle are 7.1 inches and 2.5 inches respectively. Determine whether the area of the rectangle is a terminating decimal or not. 

Solution: Given, the length of rectangle is 7.1 inches and the breadth of rectangle = 2.5 inches. 

Area of Rectangle = Length × Breadth = 7.1 inches × 2.5 inches =17.75 inches. 

As the number of digits is finite after the decimal point, the area of rectangle is a terminating decimal expansion. 

Example 4: 

Write 5/3 in decimal form. 

Using long division method, we will observe the steps in calculating 5/3

. 

Therefore, 1.666… is a non-terminating repeating decimal and can be expressed as 1.6.

Exercise:

Classify the following decimal fractions as terminating and non-terminating recurring decimals.

1. 0.777…
2. 0.777
3. 4.7182
4. 4.7182
5. 9.1651651…….
6. 9.165
7. 0.52888…….
8. 0.528
9. 72.13
10. 10.605

What have we learnt:

• Understand the meaning of rational numbers
• Identify positive and negative integers
• Write the rational numbers in the form
• Write rational numbers as decimal form
• Differentiate between terminating and non-terminating decimals
• Solve problems on rational numbers

Concept Map