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.