Question

sums & products of rational & irrational numbers quick check
ryiaz says it is impossible to prove that multiplying a non - zero rational number by an irrational number always results in an irrational product. how can you show that a proof is possible? (1 point)
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.
you can prove it by choosing random numbers to plug into multiplication problems until hes convinced.
hes correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
you can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.

Explanation:

Proof by contradiction is a valid logical method for this claim. Assume the product is rational, then use the properties of rational numbers (closure under division by non-zero rationals) to show this contradicts the irrational factor being non-rational. Random examples do not constitute a proof, direct proof is not feasible here, and the claim is provable so Rylaz is incorrect.

Answer:

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.