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Prove that underroot 5 is irrational |
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Answer» Given: √5 We need to prove that √5 is IRRATIONAL Proof: Let us assume that √5 is a rational number. Sp it t 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 |
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