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Prove that √2 is irrational number? |
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Answer» Prove that 0 is a irrational Let us assume √2 is rational number.a rational number can be written into he form of p/q√2=p/qp=√2qSquaring on both sidesp²=2q²__________(1).·.2 divides p² then 2 also divides p.·.p is an even numberLet p=2a (definition of even number,\'a\' is positive integer)Put p=2a in eq (1)p²=2q²(2a)²=2q²4a²=2q²q²=2a².·.2 divides q² then 2 also divides qBoth p and q have 2 as common factor.But this contradicts the fact that p and q are co primes or integers.Our supposition is false.·.√2 is an irrational number.\xa0 Proof:let us assume √2 is an rational number.Therfore,√2=p/q where p and q are coprime and q is not equal to 0P= √2qSquaring on both sides,(P)square=(√2)squarePsquare=2squareTherefore,qsquare =psquare/2Psquare is divisible by 2P is also divisible by 2. {Equation 1}Let p=2k where k is some constant√2q=2kSquaring on both sides,√2qsquare=2ksquareKsquare=qsquare/2Therefore,qsquare is divisible by 2q is also divisible by 2. {Equation 2}From equation 1 and 2,P and q divides by 2 other than 1This is contradictory to our assumption that p and q are co primesTherefore, √2 is a irrational number see R.S Aggrawal example |
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