Question

Prove that 2 is a rational number

Answer 1

2 can be written in the form of p/q
that is 2/1 
∴2 is a rational no.
∴hence proved

Answer 2

Solution :—

We shall prove this by the method of contradiction. If possible, let us assume that √2 is a rational number. Then ,

$= > \sqrt{2} = \frac{p}{q}$

where, where p and q are integers having no common factor and q ≠ 0.

$= > 2 = \frac{p^{2} }{q ^{2} }$

$= > {p}^{2} = 2 {q}^{2}$

$= > {p}^{2} \: is \: an \: \: even \: integer$

$= > p \: is \: an \: even \: integer \:$

Note :- if P is not even , then p = 2m + 1, M € Z.

if P is not even , then p = 2m + 1, M € Z. p^2 = (2m+1)^2 = 4m^2 + 4m + 1, which is odd. this is a contradiction to the fact that p^2 is even.

$= > p = 2m \: where \: m \: is \: an \: integer \:$

$= > {p}^{2} = {4m}^{2}$

$= > {2q}^{2} = {4m}^{2}$

$= > {q}^{2} = {2m}^{2}$

$= > {q}^{2} \: is \: an \: even \: integer \:$

$= > q \: is \: an \: even \: integer \:$

So , both p and q are even integers and therefore have a common factor 2. But, this contradicts that P and Q have no common factor.

Hence , √2 is not a rational number.