Prove √7 is not a rational number

Question2:- Prove √7 is not a rational number.

Prove:- Let us suppose that √7 is a rational number if possible and it is in the simplest form of p/q. Then, p and q are integers, having no common factor other than 1, q is not equal to 0.  
p/q ≠ 0  
 

Now, √7 = p/q => p² = (√7.q)² => p² = 7q²                     (i)

=> p² is multiple of 7

=> p is multiple of 7                                                             (ii)

Let p = 7m for some integer m, then

P² =7m => p² =49m²

=>7q² = 49m² =>q² =7m²

=>q² is multiple of 7

=>q is multiple of 7                                                      (iii)

From (ii) and (iii), it follows that 7 is a common factor of p and q.

This contradicts the hypothesis that there is no common factor of p and q other than 1. So our supposition is wrong.

Hence, √7 is not a rational number.
                                                            Proved

Prove √3 is not a rational Number

Question no. 1:- Proved √3 is not a rational Number.

Solution:- Let us suppose if possible √3 is a rational number and it is in simplest form of P/q.

Then, p and q are integers having no common factor other than 1,

Now, √3 = p/q => P = √3.q => p² = (√3.q)² = 3q²                  (i)

=> p² is even and multiple of 3

P is even and multiple of 5                                                 (ii)

Hence, only squares of even integers are even.

Let p = 3k for some integer k.

Let p = 3m for some integer m,

Then,

P= 3m => p² = 9m²   

=> q² = 3m² => q² is a multiple of 3

=> q is a multiple of 3                                                   (iii)

From (ii) and (iii), it follows that 3 is a common factor of p and q. this contradicts the hypothesis that these are no common factor of p and q other than 1. So over supposition is wrong.

Hence, √3 is not a rational number.  Proved