One possible pair is: 2+sqrt(3) and 5-sqrt(3).
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In between any two rational numbers there is an irrational number. In between any two irrational numbers there is a rational number.
Can be rational or irrational.
You get a product which can be rational or irrational.
The product of 2 rationals must be rational. The product of a rational and an irrational is irrational (unless the rational is 0) The product of two irrationals can be either rational or irrational.
Such a sum is always rational.