Prove that following numbers are irrationals:
(i) 
(ii) 
(iii) 
(iv) 
(i) Let assume that 
is rational
Therefore it can be expressed in the form of 
, where p and q are integers and q≠0
Therefore we can write 
= 
√7= 
is a rational number as p and q are integers. This contradicts the fact that √7 is irrational, so our assumption is incorrect. Therefore
is irrational
(ii) Let assume that 
is rational
Therefore it can be expressed in the form of 
, where p and q are integers and q≠0
Therefore we can write 
= 
√5= 
is a rational number as p and q are integers. This contradicts the fact that √5 is irrational, so our assumption is incorrect. Therefore
is irrational
(iii) Let assume that 
is rational
Therefore it can be expressed in the form of 
, where p and q are integers and q≠0
Therefore we can write 
= 
√2= 
-4
-4 is a rational number as p and q are integers. This contradicts the fact that √2 is irrational, so our assumption is incorrect. Therefore
is irrational.
(iv) Let assume that 
is rational
Therefore it can be expressed in the form of 
, where p and q are integers and q≠0
Therefore we can write 
= 
√2= 
is a rational number as p and q are integers. This contradicts the fact that √2 is irrational, so our assumption is incorrect. Therefore
is irrational.