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Question

# Question 1 Let x and y be rational and irrational numbers respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

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Solution

## Yes. ( x + y) is necessarily an irrational number. Example: Let us assume x = 2 , y = √3 Then, x+y=2+√3 If possible, let's consider x+y=2+√3 be a rational number. Consider. a=2+√3 On squaring both sides, we get a2=(2+√3)2 [Using identity (a+b)2=a2+b2+2ab] ⇒a2=22+(√3)2+2(2)(√3)⇒a2=4+3+4√3⇒a2−74=√3 According to our assumption, if a is rational, then a2−74 is rational. And, if a2−74 is rational, then √3 is rational. But, this contradicts the fact that √3 is an irrational number. Thus, our assumption x + y is a rational number, is wrong. Hence, x + y is an irrational number. This example clearly explains that the addition of a rational and an irrational number leads to an irrational number.

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