Question

Can we first prove tha √2 is irr

Answer 1

Answer:

Yes. We can prove that √2 is irrational by contradiction method.

Step-by-step explanation:

Let us assume that √2 is rational number.

A rational number can be written in the form of p/q.

√2=p/q           (where p and q are co-prime integers and q≠0)

⇒p=√2q

Squaring on both sides

p²=2q²

Here, 2 divides p². Then 2 also divides p. ------------------(1)

Now,

Let p=2k

Put p=2k in eq (1).

p²=2q²

⇒(2k)²=2q²

⇒4k²=2q²

⇒2k²=q²

Here, 2 divides q². Then 2 also divides q.  ------------------(2)

Now, both p and q have 2 as common factor [from (1) and (2)]

But this contradicts the fact that p and q are co primes integers.

Thus, our assumption is wrong.

∴√2 is an irrational number.