Which statement describes the product of a nonzero rational number and an irrational number?

Which statement describes the product of a nonzero rational number and an irrational number?

• A. Rational
• B. Irrational
• C. Zero
• D. Not defined

Answer: (B)

When you multiply a nonzero rational number by an irrational number, the result must be irrational. Here’s why: let r be a nonzero rational and i be irrational, and suppose their product ri is rational. Then you could solve for i as i = (ri)/r. Since ri is assumed rational and r is a nonzero rational, their quotient would be rational, which would make i rational—contradicting that i is irrational. Therefore the product cannot be rational and must be irrational. It can’t be zero either, because r is nonzero and i is not zero, so their product can’t vanish. So the description that fits is an irrational product.

When you multiply a nonzero rational number by an irrational number, the result must be irrational. Here’s why: let r be a nonzero rational and i be irrational, and suppose their product ri is rational. Then you could solve for i as i = (ri)/r. Since ri is assumed rational and r is a nonzero rational, their quotient would be rational, which would make i rational—contradicting that i is irrational. Therefore the product cannot be rational and must be irrational. It can’t be zero either, because r is nonzero and i is not zero, so their product can’t vanish. So the description that fits is an irrational product.