Question

prove that√2 is an irrational number and hence show that -2√2 is also an irrational number.

Answer 1

Suppose √2 is a rational no.

⇒√2= p/q (where p and q are co-prime integers and q is not equal to 0)

⇒(√2)² = (p/q)²

⇒2 = p²/q²

⇒ q² = p²/2

⇒2q² = p²

⇒q is an integer so p² is divisible 2.

Now,

If p² is divisible by 2,

then p is also divisible by 2.

⇒2 is a factor of p.

Let p=2a (where a is some integer)

⇒q² = (2a)²/2

⇒q² = 2a²

⇒a² = q²/2

⇒2a² =q²

Therefore, a is an integer and q² is divisible by 2.

If q² is divisible by 2

then q is divisible by 2.

⇒2 is a factor of q.

The conclusion is that 2 is a factor of p and q which contradicts our assumption.

therefore, √2 is not a rational no.