Class 10 - √2 is Irrational

An irrational number is a number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. The number √2 is one of the most important examples used in mathematics to show that not all numbers are rational. In this topic, we will learn the difference between rational and irrational numbers and prove that √2 is an irrational number and cannot be written as a fraction of two integers using proof by contradiction.

Table of Contents

• What are Irrational Numbers
• Difference between Rational and Irrational Numbers
• How to Prove that √2 is Irrational
  
  
• Frequently Asked Questions √2 is an Irrational Number

What are Irrational Numbers

An irrational number is a number that cannot be expressed in the form:

$\frac{p}{q}, q ≠ 0$

where p and q are integers.

Properties of irrational numbers: 

• Their decimal expansion is non-terminating. 

• Their decimal expansion is non-repeating

• They lie on the number line and are part of the real numbers

• There are infinitely many irrational numbers
  
  

Difference between Rational and Irrational Numbers

The following are the differences between rational and irrational number:

How to Prove that √2 is Irrational

To prove that √2 is irrational, we use the method of contradiction.
Let us assume that √2 is a rational number with p and q as co-prime integers and q ≠ 0
⇒ √2 = p/q
On squaring both sides, we get  $2q^{2} = p^{2}$  ---------- (1)
Here,  $2q^{2}$ is a multiple of 2, and hence it is even. Thus,  $p^{2}$ is an even number.
Therefore, p is also even.
So we can assume that p = 2x, where x is an integer.
By substituting this value of p in equation (1) we get
 $2q^{2}= (2x)^{2}$

⇒  $2q^{2} =4x^{2}$

⇒  $q^{2}= 2x^{2}$

⇒  $q^{2}$ is an even number. Therefore, q is also even.

Since both p and q  are even numbers, they have 2 as a common multiple.
But given that p and q are co-prime numbers. This leads to the contradiction that root 2 is a rational number in the form of p/q with "p and q both co-prime numbers" and q ≠ 0.

Thus, √2 is an irrational number by the contradiction method.