Prove that [tex]{\bf{\red{\sqrt{5}}}}[/tex] is rational.
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Prove that [tex]{\bf{\red{\sqrt{5}}}}[/tex] is rational.
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CORRECT QUESTION PROVE THAT ROOT 5 IS IRRATIONAL NUMBER .
Given:
Proof:
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
So it can be expressed in form p/q where p,q are co-prime integers and q ≠ 0.
On squaring both sides we get,
So 5 divides p.
P is a multiple of 5.
From eq. (i) & (ii), we get,
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.