Question

gove that / 2 is irration
یا 2​

Answer 1

Step-by-step explanation:

Let √2 be a rational number, so it can be written in the form of p/q, where p and q are co-primes and q is not equals to zero

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

Squaring both sides,

$({ \sqrt{2} )}^{2} = ( \frac{p}{q}) ^{2}$

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

$2 {q}^{2} = {p}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: (i)$

Now if 2 divides {p}^{2}, then it will also divides p.

Now, let p = 2k, where q is an integer,

On putting p = 2k in equation (i)

$2 {q}^{2} = {(2k)}^{2}$

$2 {q}^{2} = 4 {k}^{2}$

${q}^{2} = 2 {k}^{2}$

Now if 2 divides {q}^{2}, then it also divides q, hence it is a rational number, but it contradicts the fact that √2 is an irrational number, therefore it is as irrational number.

Hence proved