Question

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)
Responses

You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
You can prove it directly by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.

He’s correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.
He’s correct; it is impossible to prove that the product of a rational number and an irrational number is always irrational.

You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.
You can prove it by choosing random numbers to plug into multiplication problems until he’s convinced.

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

Choose the proof-by-contradiction response. Proof:

Let r be a nonzero rational and x be irrational. Suppose, for contradiction, that r·x is rational. Then
x = (r·x)/r,
which is the quotient of two rationals (r·x and r ≠ 0), hence rational — contradiction. Therefore r·x must be irrational.

(Note: if r = 0 the product is 0, which is rational.)