Why is the square root of 88 an irrational?

Answer 1

Suppose the square root of 88 is rational. That is, it is of the form p/q where p and q are integers (q ≠ 0) and where p and q have no common factors so that the fraction is in its simplest form.

Then (p/q)2 = 88

p2/q2 = 88

or p2 = 88*q2

Now 11 divides the right hand side so 11 must divide the left hand side. That is 11 divides p2 and since 11 is prime, 11 must divide p.

Therefore p = 11*r where r is some integer.

So now we have (11r)2 = 88*q2

or 121*r2 = 88*q2

which simplifies to 11*r2 = 8*q2

11 divides the left hand side and so must divide the right hand side and since 11 cannot divide 8, it must divide q2 and since 11 is a prime, it must divide q.

But that means that 11 is a common factor of p and q, contrary to p/q being in the simplest form. Therefore no such rational fraction can exist and so the square root of 88 is irrational.