Question

Ryiaz says it is impossible to prove that multiplying a non-zero rational number by an irrational always results in an irrational product. how can you show that a proof is possible?
A. he's correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
B. you can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
C. you can prove it by choosing random numbers to plug into multiplication problems until he's convinced.
D. use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. this is a contradiction. according to the closure property, if the product is rational, both factors must be rational. so if one factor is irrational, the product must be irrational.

Answer 1

The best choice to show that a proof is possible is:

**D. use proof by contradiction: assume the opposite is true and the product of a non-zero rational number and an irrational number is always rational. this is a contradiction. according to the closure property, if the product is rational, both factors must be rational. so if one factor is irrational, the product must be irrational.**

This method effectively demonstrates the claim by starting with the assumption that a non-zero rational number multiplied by an irrational number results in a rational product, and then logically arriving at a contradiction based on the definitions and properties of rational and irrational numbers.