Know the Differences between Rational and Irrational Numbers

There are different kinds of numbers which can be grouped into sets. You may have already come across and even know the sets mentioned below.

1. Natural Numbers: The set of natural numbers starts from 1.

N = {1, 2, 3, 4, 5,…}

2. Whole Numbers: The set of whole numbers starts from 0.

W = {0, 1, 2, 3,…}

3. Integers: The set of integers are nothing but a collection of whole numbers and negative numbers (natural numbers with the minus sign).

I or Z = {…,  –4, –3, –2, –1, 0, 1, 2, 3, 4,…}

Now, answer this question:

In which set does lie?

Think!!!

The answer: In the set of rational numbers.

Any number of the form , where p and q are both integers and q is not equal to zero, is called a rational number.

It can be written as Q =

Now, answer this:

will lie in which set?

Confused?

The answer: In the set of irrational numbers.

Any number which cannot be expressed in the form, where p and q are both integers and q is not equal to zero, is called an irrational number.

A set of irrational numbers is denoted by   .

Now, we will see some important properties of rational and irrational numbers.

1. The sum of two or more rational numbers is always a rational number. This means that the set of rational numbers is closed under addition.
   If x and y are any two rational numbers, then x + y and y + x are also rational numbers.
   
   
2. The difference of two rational numbers is always a rational number. This means that the set of rational numbers is closed under subtraction.
   If x and y are any two rational numbers, then x – y and y – x are also rational numbers.
   
   
3. The product of two or more rational numbers is always a rational number. This means that the set of rational numbers is closed under multiplication.
   If x and y are any two rational numbers, then xy and yx are also rational numbers.
   
   
4. The division of a rational number by a non-zero rational number is always a rational number. This means that the set of rational numbers is closed under division if the divisor ≠ 0.
   If x and y are any two rational numbers, then  when y ≠ 0 and  when x ≠ 0 are also rational numbers.
   
   
5. Addition, subtraction, multiplication and division of any two irrational numbers is not necessarily an irrational number.
   
   

Here are some pointers which will definitely make the concepts of rational and irrational numbers easier for you to understand.

The decimal expansion of

• Rational numbers are either terminating or non-terminating and recurring.

Examples:

 Terminating decimal expansion

 Non-terminating recurring decimal expansion

• Irrational numbers are non-terminating and non-recurring.

Example:  Non-terminating, non-recurring decimal expansion

Terminating: A terminating decimal doesn't keep going. A terminating decimal will have a finite number of digits after the decimal point.  

Non-terminating and recurring (repeating): A non-terminating and repeating decimal is a decimal number which continues endlessly, with the group of digits repeating endlessly. Decimals of this type can be represented as fractions.

Non-terminating and non-recurring (non-repeating): A non-terminating and non-repeating decimal is a decimal number which continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions.

If you want to find out whether the decimal expansion of a given rational number is terminating or non-terminating and repeating without actually performing the division, you need to follow the steps below:

If x = is a rational number.

1. If the prime factorisation of q is of the form 2ⁿ5^m, where m and n are non-negative integers, then x has a decimal expansion which terminates.

Example:

2. If the prime factorisation of q is not of the form 2ⁿ5^m, where m and n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

Example: 

Now, we will solve some problems based on the concepts of rational and irrational numbers.

1. Rationalise the denominator of  .

Solution: 

 2. If x = , find the value of 

Solution:

We know that

(a + b)² = a² + b² + 2ab

3.  Find the two irrational numbers between 3 and 5.

Solution:

If a and b are two rational numbers, then the irrational number between them =

The concepts of rational and irrational numbers are very easy to understand and are a part of both CBSE and ICSE board. Get Selina solutions for Class 9, NCERT solutions for Class 7 and NCERT solutions for Class 8.

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