Prove that √3 is irrational

Solution:

We will prove that √3 is irrational using the contradiction method.

Let’s assume √3 is a rational number in the form of p/ q where p and q are coprime integers and q ≠ 0.

⇒ √3 = p/ q

⇒ √3 q = p………….. be equation (1).

Take squares on both sides of equation (1).

⇒ 3q2= p2

∵ 3 is a prime number that divides p2, so 3 divides p.

⇒ 3 is a factor of p. 

Therefore, p is a number that divides q.

Let p = 3a where a is a whole number.

Substitute the value of p in equation (1)

⇒ 3q2= (3a)2

⇒ 3q2= 9a2

⇒ q2= 3a2

⇒ q2 / 3 = a2 …………….. be equation (2)

⇒ Since 3 is a factor of q. 

From equation 1 and 2, we can conclude that

3 is a factor of p 

3 is a factor of q. 

3 is a factor of both p and q.

This leads to the contradiction to our assumption that p and q are co-primes

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

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Prove that √3 is irrational

Summary:

Hence proved that √3 is an irrational number with p and q having common factors other than 1

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