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32. Prove that root 5 is irrational​

Answer»

TO PROVE :- Root of 5 is IRRATIONAL

PROOF

We can EASILY prove this by using the cobtradiction METHOD :-

Let root 5 be x/y where x and y are co-primes.

\sqrt{5} = \frac{x}{y} \\ \\ \\ \implies 5 = \frac{ {x}^{2} }{ {y}^{2} } \\ \\ \\ \implies 5 {y}^{2} = {x}^{2} \\ \\ \\ \implies {y}^{2} = \frac{ {x}^{2} }{5} \\ \\ \\ {x}^{2} is \: divisible \: by \: 5 \: so \: x \: is \: also \: divisible \: by \: 5 \\ \\ \\ now \: let \: \frac{x}{5} \: be \: s \\ \\ \\ \frac{x}{5} = s \\ \\ \\ \implies 5s = x \\ \\ \\ \implies 25 {s}^{2} = {x}^{2} \\ \\ \\ \implies 25 {s}^{2} = 5 {y}^{2} \\ \\ \\ \implies {s}^{2} = \frac{ {y}^{2} }{5} \\ \\ \\ hence \: {y}^{2} is \: divisible \: by \: 5 \: so \: y \: is \: also \: divisible \: by \: 5

Hence both x and y are divisible by 5 so x and y aren't co-primes.

Therefore it contradicts our assumption so root 5 is irrational.


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