Question

1. Prove that √5 is irrational​

Answer 1

..Given:

√5We need to prove that √5 is irrational

Proof:Let us assume that √5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0⇒ √5 = p/qOn squaring both the sides we get,⇒5 = p²/q²⇒5q² = p² —————–(i)p²/5 = q²So 5 divides pp is a multiple of 5⇒ p = 5m⇒ p² = 25m² ————-(ii)From equations (i) and (ii), we get,5q² = 25m²⇒ q² = 5m²⇒ q² is a multiple of 5⇒ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number√5 is an irrational number.

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Answer 2

Step-by-step explanation:

Let √5 be a rational number.

$then \: it \: must \: be \: in \: form \: of   \frac{q}{p}  where, \\  q \neq 0     ( p and q  are \: co-prime)$

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

√5 × q = p

Suaring on both sides,

(√5q)² = p²

5q² = p² _ _ _ _ _ _ (i)

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

So 5 divides p

p is a multiple of 5

p = 5m

p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

q² = 5m²

q² is a multiple of 5

q is a multiple of 5

Hence, p,q have a common factor 5.

This contradicts our assumption that they are co-primes.

Therefore, p/q is not a rational number √5 is an irrational number.

Hence proved

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