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Prove that √2 is an irrational number? |
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Answer» Given √2To prove: √2 is an irrational number.Proof:Let us assume that √2 is a rational number.So it can be expressed in the form p/q where p, q are co-prime integers and q≠0√2 = p/qHere p and q are coprime numbers and q ≠ 0Solving√2 = p/qOn squaring both the side we get,=>2 = (p/q)2=> 2q2 = p2……………………………..(1)p2/2 = q2So 2 divides p and p is a multiple of 2.⇒ p = 2m⇒ p² = 4m² ………………………………..(2)From equations (1) and (2), we get,2q² = 4m²⇒ q² = 2m²⇒ q² is a multiple of 2⇒ q is a multiple of 2Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number√2 is an irrational number. Any root of imperfect square is an \'irrational number\'.As \'2\' is not a perfect square(i.e.4,9,16....) so root of it is surely irrational. |
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