Suppose x is a non-zero rational number and y is irrational. Prove that y/x is irrational.

Step 1: Definition of Rational and Irrational Numbers A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero. An irrational number is a number that cannot be expressed as a ratio of two integers.

Step 2: Assumptions We are given that x is a non-zero rational number, so we can express x as a/b, where a and b are integers and b ≠ 0. We are also given that y is an irrational number, so y cannot be expressed as a ratio of two integers.

Step 3: Proof We need to prove that y/x is irrational. Let's assume, for the sake of contradiction, that y/x is rational. This means that y/x can be expressed as a ratio of two integers. Let's say y/x = c/d, where c and d are integers.

Step 4: Manipulation If y/x = c/d, then y = cx/d. But we know that x = a/b, so y = c(a/b)/d = ca/db.

Step 5: Contradiction But ca/db is a ratio of two integers, which contradicts our assumption that y is irrational. Therefore, our initial assumption that y/x is rational must be false.

Step 6: Conclusion Therefore, y/x is irrational.