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 2 of 5: Choose b when a squared is rational
Detailed analysis
If a squared is rational, take b=a squared minus a. It is irrational and a+b=a squared is rational. Moreover ab=a(a squared minus 1), which is irrational because a squared minus 1 is a nonzero rational number; it cannot be zero since a is irrational.