Question

रूट 3 इज ए रेशनल नंबर प्रूफ दैट​

Answer 1

Answer:

Step-by-step explanation:

Answer

Let us assume on the contrary that  

3

​  

 is a rational number.  

Then, there exist positive integers a and b such that

3

​  

=  

b

a

​  

 where, a and b, are co-prime i.e. their HCF is 1

Now,

3

​  

=  

b

a

​  

 

⇒3=  

b  

2

 

a  

2

 

​  

 

⇒3b  

2

=a  

2

 

⇒3 divides a  

2

[∵3 divides 3b  

2

]  

⇒3 divides a...(i)  

⇒a=3c for some integer c

⇒a  

2

=9c  

2

 

⇒3b  

2

=9c  

2

[∵a  

2

=3b  

2

]  

⇒b  

2

=3c  

2

 

⇒3 divides b  

2

[∵3 divides 3c  

2

]  

⇒3 divides b...(ii)  

From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.