Question

Prove that 7-V5 is irrational.​

Answer 1

Let us assume, to the contrary, that 7-√5 is rational

That is, we can find coprime a  and b (b≠ 0) such that  

7-√5 = a/b

Therefore, 7 - a/b = √5

Rearranging this equation √5 = (7b -a)/b

since a and b are integers,so (7b -a)/b is an rational.

And so √5 is rational

But this contradicts the fact that √5 is irrational.

This  contradiction  has  arisen  because  of  our  incorrect  assumption  that  7-√5 is rational.

So, we conclude that

7-√5 is irrational.

Answer 2

Answer:

Let us assume that 7  

5

​  

 is rational number

Hence 7  

5

​  

 can be written in the form of  

b

a

​  

 where a,b(b  



​  

=0) are co-prime

⟹7  

5

​  

=  

b

a

​  

 

⟹  

5

​  

=  

7b

a

​  

 

But here  

5

​  

 is irrational and  

7b

a

​  

 is rational

as Rational  



​  

=Irrational

This is a contradiction  

so 7  

5

​  

 is a irrational number

Step-by-step explanation: