Question

Prove that is irrationnal
$\sqrt{5}$
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Answer 1

Question:-

Prove that is irrationnal

$\sqrt{5}$.

To prove:-

$\sqrt{5}$.is a rational number.

Prove:-

Let √5 be a rational number.

then it must be in form of p/q where,

q is not equal to 0 ( p and q are co-prime).

$\sqrt{5} = \frac{p}{q}$

$\sqrt{5} \times q \: = p$

$squaring \: on \: both \: side.$

$5 {q}^{2} = {p}^{2}$

${p}^{2} is \: divisible \: by \: 5$

$so \: p \: is \: divisible \: by \: 5.$

$p = 5c$

$squring \: on \: both \: side.$

${p}^{2} = 25 {c}^{2}$

$put \: {p}^{2} in \: equ(1)$

$5 {q}^{2} = 25( {c}^{2} )$

${q}^{2} = 5 {c}^{2}$

So, q is divisible by 5.

.

Thus p and q have a common factor of 5.

So, there is a contradiction as per our assumption.

We have assumed p and q are co-prime but here they a common factor of 5.

The above statement contradicts our assumption.

Therefore, √5 is an irrational number.