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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.

358

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