Question

prove that root 5 is an irrational number. solve it in detail

Answer 1

Hey there !

Thanks for the question !

Here's the answer !

Assumption: Let √5 be a rational number.

Proof:

$Rational \ Numbers \ are \ of \ the \ form \ \frac{p}{q}. \\\\ Here \ p \ and \ q \ are \ coprime \ and \ q \neq{0}. \\\\ => \sqrt{5} = \frac{p}{q} \\\\ Squaring \ on \ both \ sides \ we \ get, \\\\ => ( \sqrt{5} )^2 = \frac{p^2}{q^2}. \\\\ => 5 = \frac{p^2}{q^2}. \\\\ Transposing \ q^2 \ we \ get, \\\\ => 5q^2 = p^2 \hspace{1cm} = Equation 1 \\\\ => q^2 = \frac{p^2}{5} \\\\ => p^2 \ divides \ 5 \ as \ well \ as \ p \ divides \ 5.$$\\\\ So \ let \ the \ quotient \ be \ k. \\\\ => \frac{p}{5} = k \\\\ => p = 5k \\\\ Substituting \ this \ in \ Equation \ 1 \ we \ get, \\\\ => q^2 = \frac{ (5k)^2}{5} \\\\ => 5q^2 = 25k^2 \\\\ => q^2 = 5k^2 \\\\ => \frac{q^2}{5} = k^2 \\\\ => q \ divides \ 5 . \\\\ This \ implies \ that \ p \ and \ q \ have \ a \ common \ HCF. \ This \ contradicts \ the \ fact \ that \ p \ and \ q \ are \ co-prime$

$Hence \ our \ assumption \ was \ wrong. \ Hence \sqrt{5} \ is\ irrational.$

Hope my answer helped !

Answer 2

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