. Prove that root 5 is a irrational

Best Answer


Let root is a rational number.

we can find two co-prime numbers  p, q (q ≠ 0) such that root 5 = p/q

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

p = root 5q

p2 = 5q2

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, root 5 cannot be expressed as p/q or it can be said that root 5 is irrational.


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