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If `x`is a rational number and `y`is an irrational number, thenboth `x + y a n d x y`arenecessarily rationalboth `x + y a n d x y`arenecessarily irrational`x y`isnecessarily irrational, but `x + y`can beeither rational or irrational`x + y`isnecessarily irrational, but `x y`can beeither rational or irrational |
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Answer» No. (xy) is necessarily an irrational only when `x ne O`. Let x be a non-zero rational and y be an irrational. Then , we have to show that xy be an irrational . If possible, let xy be a rational number. Since, quotient of two non- zero rational so , `((xy)/x)` is a rational number ` Rgihtarrow ` Y is a rational number . But, this contradicts the fact that y is an irrational number. thus, our supposition is wrong. Hence , xy is an irrational number. but, when x=0, then xy=0, rational number. |
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