Question

Examine whether √2 is rational or irrational number.

Answer 1

ANSWER:

• √2 is an irrational number.

GIVEN:

• Number = √2

TO FIND:

• whether √2 is a rational or irrational number.

SOLUTION:

Let √2 be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \: \sqrt{2} = \dfrac{p}{q} \\ \implies \: \sqrt{2} q = p \\ \\ \: \: \: \: \: squaring \: both \: sides \: we \: get \\ \implies \: ( \sqrt{2} q) {}^{2} = (p) {}^{2} \\ \implies \: 2q {}^{2} = p {}^{2} ......(i)$

Here:

• 2 divides p²
• Then 2 divides p. ...(ii)

Let p = 2m in eq(i) we get

$\implies \: 2q {}^{2} = (2m) {}^{2} \\ \implies \: 2q {}^{2} = 4m {}^{2} \\ \implies \: q {}^{2} = 2m {}^{2}$

Here:

• 2 divides q²
• Then 2 divides q. ....(iii)

From eq(ii) and (ii)

• 2 is the common factor of p and q.
• Thus our contradiction is wrong.
• So √2 is an irrational number.