The product of a non-zero rational number and an irrational number is

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Study for the Algebra 1 Honors End-of-Course Test. Study with flashcards and multiple-choice questions, each question has hints and explanations. Get ready for your exam!

Multiple Choice

The product of a non-zero rational number and an irrational number is

Explanation:
Multiplying a nonzero rational number by an irrational number always yields an irrational number. Why that is true: Let r be a nonzero rational number and i an irrational number. Suppose their product ri were rational. Since r is nonzero, you can solve for i as i = (ri)/r. The quotient of two rational numbers is rational, so i would have to be rational. But i is irrational, which is a contradiction. Therefore ri cannot be rational, so it must be irrational. Also, because the rational number is nonzero, the product cannot be zero, and it certainly isn’t undefined. So the product is irrational.

Multiplying a nonzero rational number by an irrational number always yields an irrational number.

Why that is true: Let r be a nonzero rational number and i an irrational number. Suppose their product ri were rational. Since r is nonzero, you can solve for i as i = (ri)/r. The quotient of two rational numbers is rational, so i would have to be rational. But i is irrational, which is a contradiction. Therefore ri cannot be rational, so it must be irrational.

Also, because the rational number is nonzero, the product cannot be zero, and it certainly isn’t undefined. So the product is irrational.

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