Question
Give an example of two irrational numbers whose sum as well as product is rational.
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Answer
Let the two irrational numbers be $2+\sqrt{3}$ and $2-\sqrt{3}.$
Sum of these irrational numbers $=\big(2+\sqrt{3}\big)+\big(2-\sqrt{3}\big)=4,$ which is rational
Product of these irrational numbers $=\big(2+\sqrt{3}\big)\big(2-\sqrt{3}\big)=2^2-\big(\sqrt{3}\big)^2=4-3=1,$ which is rational
Sum of these irrational numbers $=\big(2+\sqrt{3}\big)+\big(2-\sqrt{3}\big)=4,$ which is rational
Product of these irrational numbers $=\big(2+\sqrt{3}\big)\big(2-\sqrt{3}\big)=2^2-\big(\sqrt{3}\big)^2=4-3=1,$ which is rational
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