Question

Prove that
$\sqrt{5}$
is irrational ​

Answer 1

❗Gm❗

Que :- Prove that √5 is a irrational number.

Ans :- We know that every rational number can be expressed in the form of p/q where p & q are integers and q is not = to 0.

Let √5 be a rational number.

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

(p & q are co-primes)

${( \sqrt{5}) }^{2} = {( \frac{p}{q}) }^{2}$

(Squaring both side)

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

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

➜ 5 divides p²

i.e. 5 divides p.

➜Let p = 5m

p² = 25m²

5p² = 25m²

q = 5m²

➜ 5 divides q

i.e. 5 divides q²

➜ P & q have atleast two common factors. But this contradicts the fact that P & q are co-primes.

➜Thus, √5 is not a rational number.

➜√5 is a irrational number.

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