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InterviewSolution
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Prove that 3+√5 is irratonal |
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Answer» Let us assume 3+root 5 to be rational.It means that it can be written in the form of p/q where p and q are coprime and q is not equal to zeroL.H.S*p/q=3+root5*p/q-3=root5L.H.S i.e.p/q-3 is rational but R.H.S i.e. root 5 is irrational So our assumption is wrong Therefore 3+root5 is irrational. Hence proved Let us assume that\xa03+5\u200b\xa0is a rational number.Now,3+5\u200b=ba\u200b\xa0[Here a and b are co-prime numbers]5\u200b=[(ba\u200b)−3]5\u200b=[(ba−3b\u200b)]Here,\xa0[(ba−3b\u200b)]\xa0is a rational number.But we know that\xa05\u200b\xa0is an irrational number.So,\xa0[(ba−3b\u200b)]\xa0is also a irrational number.So, our assumption is wrong.3+5\u200b\xa0is an irrational number.Hence, proved. |
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