No. The product of conjugate pairs is always rational.
So suppose sqrt(y) is the irrational square root of the rational number y. Then
Thus [x + sqrt(y)]*[x - sqrt(y)] = x^2 + x*sqrt(y) - x*sqrt(y) - sqrt(y)*sqrt(y)
= x^2 + y^2 which is rational.
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It is always an irrational number.
In general, no. It is possible though. (2pi)/pi is rational. pi2/pi is irrational. The ratio of two rationals numbers is always rational and the ratio of a rational and an irrational is always irrational.
Whole numbers are always rational
Since the sum of two rational numbers is rational, the answer will be the same as for the sum of an irrational and a single rational number. It is always irrational.
Whole numbers can never be irrational.