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Question

# Give an example of each, of two irrational numbers whose: (i) difference is a rational number. (ii) difference is an irrational number. (iii) sum is a rational number. (iv) sum is an irrational number. (v) product is a rational number. (vi) product is an irrational number. (vii) quotient is a rational number. (viii) quotient is an irrational number.

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Solution

## (i) Two numbers whose difference is also a rational number. e.g. √2,√2 which are irrational numbers. ∴ Difference = √2−√2 = 0 which is also a rational number. (ii) Two numbers whose difference is an irrational number. e.g. √3 and √2 which are numbers. Now difference = √3−√2 which is also an irrational number. (iii) Let two irrational numbers be √3 and −√3 which are irrational numbers. Now sum = √3 + −√3 = √3−√3 = 0 Which is a rational number. (iv) Let two numbers be √5,√3 which are irrational numbers. Now sum = √5+√3 which is an irrational number (v) Let number be √3+√2 and √3−√2 which are irrational numbers. Now product = (√3+√2) (√3−√2) = 3 - 2 = 1 which is a rational number. (vi) Let numbers be √3 and √5, which are irrational number. Now product = √3 × √5 √3×5 = √15 which is an irrational number. (vii) Let numbers be 6 √2 and 2√2 which are irrational numbers. Quotient = 6√22√2 = 3 which is a rational number. (viii) Let numbers be √3 and √5 which are irrational numbers. Now quotient = √3√5 = √35 which is an irrational number.

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