Question

prove that
$\sqrt{2}$
is an irrational number​

Answer 1

Answer:

It is irrational because it cannot be come in the form of p/q

Hope it helps..

Answer 2

Hey!!

Answer:

Let √2 be a rational number,

Thus, √2 can be represented in the form of p/q. where p and q are co prime integers. and q≠0.

=>√2=p/q

=>q√2=p

on sq. both sides;

=>2q²=p².....(1)

This means that, 2 divides p².

,i.e., 2 divides p.

So, we can write that,

=>p=2m; for any integer 'm'.

on sq. both sides.

=>p²=4m².....(2)

Now, on substituting the value of p² in eq.(1);

=>4m²=2q²

=>2m²=q²

This means that 2 divides q².

,i.e., 2 divides q.

But, 2 also divides p(proved above).

But, we know that p and q are co prime numbers with no common factor other than 1.

This, contradiction arises because of our wrong assumption that √2 is rational.

Therefore, √2 is an irrational number.

Hence, proved.

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