Attempted solution for Section 3.6 Question 3 (by direct proof)

3. We are asked to prove the statement: "rQ with r non-zero, if (s is irrational), then (r·s is irrational).

To prove this by a direct proof, we assume that r is a non-zero rational number and s is an irrational number and try to show that r·s is an irrational number.

Proof Suppose that r is a non-zero rational number and s is an irrational number.

Since r is a non-zero rational number, there exist integers a and b such that r = a / b where a, b0.

Now, since we know nothing about the product r·s, and we don't know how to multiply the fraction a/b by an irrational number, we are stuck. In a case such as this, it is better to use a proof by contradiction or a proof by contraposition.

Back to Section 3.6