No. A rational plus an irrational is always an irrational.
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Yes. In fact, a rational plus or minus an irrational will always be irrational.
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.
It is irrational.
Yes. Any irrational number can be divided by itself to produce 1, which is a rational number.
No, never.