Question

Prove that √5 is a irrational.​

Answer 1

$\huge† \huge \bold{\: \pmb {\purple{ Answer } }}$

$\sf\small\underline\red{Given:-}$

√5

$\sf\small\underline\red{to \: prove:-}$

We need to prove that √5 is irrational

$\sf\small\underline\red{Proof:-}$

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/q

On squaring both the sides we get,

⇒5 = p²/q²

⇒5q² = p² —————–(i)

p²/5 = q²

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

Answer 2

Answer:

Prove That Root 5 is Irrational by Contradiction Method

Assuming if p was a prime number and p divides a2, then p divides a, where a is any positive integer. Hence, 5 is a factor of p2. ... This contradicts our assumption that √5 = p/q. Therefore, the square root of 5 is irrational.

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