Question

Prove that 3-√5 is an irrational number, given that √5 is irrational.​

Answer 1

Answer:

Let us assume that 3−  

5

​  

 is a rational number

Then. there exist coprime integers p, q,q



=0 such that

     3−  

5

​  

=  

q

p

​  

 

=>  

5

​  

=3−  

q

p

​  

 

Here, 3−  

q

p

​  

 is a rational number, but  

5

​  

 is a irrational number.

But, a irrational cannot be equal to a rational number.

This is a contradiction.

Thus, our assumption is wrong.

Therefore 3−  

5

​  

 is an irrational number.

Step-by-step explanation:

Answer 2

Step-by-step explanation:Let us assume that 3− 5 is a rational number

Then.

there exist coprime integers p, q,q is not equal to 0 such that

3− root 5 = p/q

=> root 5 =3− p/q

Here, 3−p/q is a rational number, but root5 is a irrational number.

But, a irrational cannot be equal to a rational number.

This is a contradiction.

Thus, our assumption is wrong.

Therefore 3− root 5 is an irrational number.

Please thank and rate