Hello everyone. He told us he could of course change the parameters which he will of the proofs we have been working on so I'm testing out some cases but I want to make sure i'm doing it right. Here is an example of a proof the boook had: http://suprfile.com/src/1/3qaagxh/Untitled-1%20copy.jpg [Broken] Now where he has the statement: "If a and b are rational numbers...." I'm changing that to: "If a and b are integers...." and now here is my proof, i think its correct but I have to make sure. If a and b are integers, b != 0, and r is an irrational number, then a+ br is irrational. Proof by Contradiction: Suppose not. Suppose that a and b are integers, b != 0, and r is an irrational number such that a+br is rational. We must obtain a contradiction. Since a, b are integers and a + br are rational, a+br = m/n for some integers m, n with n != 0. Then a + br = m/n br = m/n - a r = (m-an)/bn where (m-an) and (bn) are integers since m, a, n, and b are integers, and bn is nonzero since b is nonzero. Therefore, r is rational, contradicting that r is irrational. Thanks!