Prove that is irrational.


Let is a rational number.

we can find two co-prime numbers p, q (q ≠ 0) such that


Let ‘p’ and ‘q’ have a common factor other than 1.




Therefore, p2 is divisible by 5 and it can be said that ‘p’ is divisible by 5.


Let p = 5k, where k is an integer


(5k)2 = 5q2 this mean that q2 is divisible by 5 and hence, q is divisible by 5.


q2 = 5k2 this implies that p and q have 5 as a common factor.


And this is a contradiction to the fact that p and q are co-prime.


Hence, cannot be expressed as or it can be said that is irrational.


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