MathLabs

Problem 1

Prove that for every irrational real number a, there are irrational real numbers b and b' such that a+b and ab' are rational, while ab and a+b' are irrational.
Step 5 of 5: Verify all four requirements
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a+b∈Q,ab∉Q,ab′∈Q,a+b′∉Qa+b\in\mathbb Q,\quad ab\notin\mathbb Q,\quad ab'\in\mathbb Q,\quad a+b'\notin\mathbb Q
Detailed analysis

The constructed b and b' are irrational and satisfy exactly the four required rationality and irrationality conditions, completing the proof.