Question

Prove that√6 is an irrational number

Answer 1

Suppose √6 is rational.

Then it can expressed in p/q form.

Let √6 = p/q, where p,q are integers, q≠0 and HCF (p,q) = 1.

Now, √6 = p/q
So, p = √6 q
Squaring
p² = 6 q² -----------(1)

As we see, 6 divides p²
So, 6 divides p ---------(2)

Let p = 6k, k is integer.
Putting in (1)

So, p² = 6q²
So, (6k)² = 6q²
So 36 k² = 6q²
So, 6k² = q²

As we see, 6 divides q²
So, 6 divides q --------(3)


From (2) and (3),

6 divides p and 6 divides q
But we defined that HCF of p and q is 1.

So, this is a contradiction to our supposition.

So, our supposition is wrong.

So, √6 is not rational.

So, √6 is irrational.

Hence proved.