Proof: let us assume that √7 be rational.
then it must in the form of p / q [q ≠ 0] [p and q are co-prime]
√7 = p / q
√7 x q = p
squaring on both sides
7q2= p2 (i)
p2 is divisible by 7
p is divisible by 7
p = 7c [c is a positive integer] [squaring on both sides ]
p2 = 49 c2 (ii)
Substitute p2 in eq (i), we get,
7q2 = 49 c2
q2 = 7c2
q is divisible by 7
Thus q and p have a common factor 7.
There is a contradiction
As our assumption p & q are co - prime but it has a common factor.
So that √7 is an irrational.
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