Show that √3 is an irrational number

Question

Show that root 3 is an irrational number.

Answer 1

Solution: 
Let us assume, to the contrary, that √3 is rational. 
So, we can find coprime integers a and b(b ≠ 0) 
such that √3 = a/b 
=> √3b = a 
=> 3b² = a² ….(i) [Squaring both the sides] 
=> a² is divisible by 3 
=> a is divisible by 3 
So, we can write a = 3c for some integer c. 
=> a² = 9c² ….[Squaring both the sides] 
=> 3b² = 9c² ….[From (i)] 
=> b² = 3c² 
=> b² is divisible by 3 
=> b is divisible by 3 
=> 3 divides both a and b. 
=> a and b have at least 3 as a common factor. 
But this contradicts the fact that a and b are coprime. 
This contradiction arises because we have assumed that √3 is rational.

Hence, √3 is irrational.